the value of information sharing in a multi-product supply chain with product substitution
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The value of information sharing in a multi-productsupply chain with product substitutionMuthusamy Ganesh a , Srinivasan Raghunathan b & Chandrasekharan Rajendran aa Department of Management Studies , Indian Institute of Technology Madras , Chennai,600036, Indiab School of Management , The University of Texas at Dallas , Richardson, TX, 75083, USAPublished online: 22 Oct 2008.
To cite this article: Muthusamy Ganesh , Srinivasan Raghunathan & Chandrasekharan Rajendran (2008) The value ofinformation sharing in a multi-product supply chain with product substitution, IIE Transactions, 40:12, 1124-1140, DOI:10.1080/07408170701745360
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IIE Transactions (2008) 40, 1124–1140Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/07408170701745360
The value of information sharing in a multi-product supplychain with product substitution
MUTHUSAMY GANESH1, SRINIVASAN RAGHUNATHAN2,∗and CHANDRASEKHARAN RAJENDRAN1
1Department of Management Studies, Indian Institute of Technology Madras, Chennai, 600036, India2School of Management, The University of Texas at Dallas, Richardson, TX 75083, USAE-mail: [email protected]
Received August 2006 and accepted September 2007
The extant literature on the value of information sharing within a supply chain has investigated only the case in which the supplychain manufactures and distributes a single product to customers. In this paper, the case in which a supply chain distributes multipleproducts is considered. These products may also be substitutable in the sense that a consumer may be willing to buy an alternateproduct when the customer’s preferred product is out of stock. It is shown that substitutability among products generally reducesthe value of information sharing and that the reduction is increasing in the degree of substitution. This result occurs because of tworeasons. First, the demand-pooling effect of substitution reduces demand variance, and the reduction in demand variance because ofsubstitution is higher when the degree of substitution is higher. Second, substitution increases the base level profit when informationis not shared. The reduction in the value of information sharing because of substitutability is higher when the number of substitutableproducts is lower, the demands of products are less correlated or when the number of products whose information is shared is higher.When information about the demands of only a subset of products is shared, the value of information sharing under substitution ishigher than that under no substitution under certain conditions. However, in these conditions the inventory holding and shortagecosts are significantly low, relative to the profit margin, causing the value of information sharing to be low. The key implication ofthese findings is that if substitution effects are ignored, then there is a risk of overestimating the value of information sharing. Theoverestimate can be very significant when the degree of substitution is higher, the number of substitutable products is smaller, demandsof these products are less correlated and more independent or when the number of products whose information is shared is higher.
Keywords: Value of information sharing, supply chain, multiple products, substitution
1. Introduction
Information sharing among firms within a supply chain hasbeen a cornerstone of recent innovations in supply chainmanagement. It is well known that Wal-Mart and Proctor& Gamble have been sharing point-of-sale and real-timeinventory information for a long time. Other companiessuch as Campbell Soup, Dell and Cisco have also investedin similar information sharing strategies (Dong and Xu,2002). The primary benefit of sharing demand and inven-tory information is a reduction in the bullwhip effect and,consequently, a reduction in inventory holding and short-age costs within the supply chain (Forrester, 1958; Sterman,1989; Lee et al., 1997a, 1997b).
The value of information sharing within a supply chainhas been extensively analyzed by researchers. The extant lit-erature has investigated the case in which the supply chain
∗Corresponding author
manufactures and distributes a single product to customers.However, modern supply chains, even when there is a sin-gle manufacturer and a single retailer, often manufactureand distribute multiple products (or multiple varieties of aproduct) to satisfy diverse customer preferences. The abilityto satisfy heterogeneous customer preferences by providingmore product variety has been noted as a critical successfactor in retailing (Kima et al., 2005). A study by the USFederal Reserve Bank documented the dramatic increase inproduct variety in almost every industry during the 1980sand 1990s (Federal Reserve Bank of Dallas, 1998). Table 1reveals the magnitude of the increase in product variety insome industries. Similar increases are also observed in au-tomobile (new vehicle models rose from 140 to 260), enter-tainment (the number of TV channels increased from five to185) and pharmaceutical (the number of over-the-counterpain relievers rose from 17 to 141) industries. One reasonwhy managing multiple varieties of a product is challeng-ing is that the varieties are often substitutable (Mahajanand Ryzin, 2001). That is, when a variety that a customer
0740-817X C© 2008 “IIE”
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Table 1. Product variety statistics
Food products Household items Beverages
Item 1980 1998 Item 1980 1998 Item 1980 1998
Meals 159 671 Detergents 12 48 Milk, yogurt 26 255Meat 42 234 Paper towels 11 126 Health drinks 4 70Soup 119 291 Deodorizers 53 372 Soft drinks 26 252
is looking for is unavailable, the customer may buy anothervariety of the same product, which suggests that decisionsabout a variety have ramifications on decisions about oth-ers. Furthermore, the demands of different varieties of thesame product are likely to be correlated. The primary ob-jective of this paper is to offer insights into how the valueof information sharing within a two-level supply chain isaffected when the supply chain distributes several substi-tutable products as opposed to a single product or a set ofnon-substitutable products.
We perform our analysis using a two-level supply chainthat consists of a manufacturer and a retailer. The sup-ply chain distributes a set of substitutable products. Wemodel substitutability by allowing a fraction of customersto choose alternate products when their preferred productis out of stock; the fraction measures the degree of sub-stitution. Our analysis has led to the following significantfindings. First, substitutability among products reduces thevalue of information sharing in most situations, and ahigher degree of substitution results in larger decreases ofthe value of information sharing. The primary reason forthis result is the demand-pooling effect of product substi-tution. Demand-pooling reduces the demand variance perproduct, which leads to a lower value of information shar-ing. The secondary reason is that product substitution in-creases the base level profit, i.e., the profit when informationis not shared. Second, the reduction in the value of infor-mation sharing because of substitutability, for any degree ofsubstitution, is higher when demand is less correlated, whenthe number of substitutable products is smaller or when thenumber of products whose demand information is sharedis higher. Independent or weakly correlated demands en-hance the demand-pooling effect of substitution. On theother hand, a larger number of substitutable products di-minish the pooling effect. Third, when information aboutdemands of only a sub-set of products is shared, the value ofinformation sharing can be higher under substitution thanunder no substitution. We observe this generally when thecorrelation between demands is sufficiently high and thenumber of products for which demand is shared is suffi-ciently low. The seemingly counter-intuitive finding is theoutcome of the interplay of the pooling effect of substitu-tion and the impact of demand correlation on the varianceof the pooled demand. Demand correlation increases thevariance of the pooled demand, but substitution decreasesthe variance. When the number of products whose informa-
tion is shared is small and the demand correlation is suffi-ciently high, the former effect dominates the latter, resultingin a higher value of information sharing under substitution.As the number of products whose demand information isshared increases, the impact of the first effect reduces rel-ative to that of the second, resulting in a smaller value ofinformation sharing under product substitution. The keyimplication of our findings to supply chain managementpractice is that if substitution effects are ignored, then thereis a risk of overestimating the value of information sharing.The overestimate can be very significant when the degree ofsubstitution is large, demands are independent, the numberof substitutable products is small, or when the number ofproducts whose information is shared is large.
The rest of the paper is organized as follows. We reviewthe relevant literature in Section 2. The modeling frame-work is discussed in Section 3. Sections 4 and 5 present thetheoretical results of our analysis. Results from a numericalsimulation are discussed in Section 6. Finally, we concludethe paper with a summary in Section 7.
2. Literature review
There is a substantial literature on savings in inventoryholding and shortage costs to the manufacturer when theretailer shares its information. Most papers in this streamof research use a two-level supply chain with a single manu-facturer and a single retailer. Bourland et al. (1996) derivedthe benefits of information sharing when the review periodof the manufacturer is not synchronized with that of theretailer. Metters (1997) showed that sharing informationcan reduce the bullwhip effect and increase profitability.Gavirneni et al. (1999) studied the value of informationsharing for a finite capacity supplier facing demand froma single retailer. Lee et al. (2000) studied the benefit of de-mand information sharing when the underlying demandprocess faced by the retailer follows a first-order auto-regressive, AR(1), process. Raghunathan (2001) showedthat the results derived by Lee et al. (2000) overestimatethe benefit of demand information sharing if the manu-facturer uses the entire order history to do its forecast.Chen, Drezner, Ryan, and Simchi Levi (2000) quantifiedthe bullwhip effect in a supply chain when players use anAR(1) model to forecast their demands and showed that thebullwhip effect will exist even when demand information is
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1126 Ganesh et al.
shared. Chen, Ryan, and Simchi Levi (2000) investigatedthe impact of forecast methods and demand patterns on thebullwhip effect. Reddy and Rajendran (2005) developed asimulation model to compare the value of partial and fullinformation sharing. Gavirneni (2006) showed that whenthe wholesale price alternates between a high and a lowlevel, information sharing significantly benefits the manu-facturer. Chu and Lee (2006) studied how the incentive toshare information is related to the cost of sharing in thespecific context of a supply chain that manufactures andsells newsvendor-type products.
A few papers have investigated models with a singlemanufacturer and multiple retailers. Raghunathan and Yeh(2001) analyzed the optimal number of information shar-ing retailers when a manufacturer distributes a productthrough several retailers. Cachon and Fisher (2000) ana-lyzed a model with one supplier and several identical re-tailers and concluded that accelerating the physical flow ofgoods through a supply chain is significantly more valuablethan exchange of information. Aviv (2001, 2002) investi-gated the value of collaborative forecasting and of integrat-ing retailer forecasts into the manufacturer’s replenishmentprocess. Zhao et al. (2002) considered a supply chain modelwith capacity constraints. Raghunathan (2003) analyzedthe value of demand information sharing in the context ofan N retailer version of Lee et al. (2000). Huang and Iravani(2005) analyzed how selective information sharing by oneof two retailers affects the value of information sharing.They identified conditions under which information fromonly one retailer captures most of the savings that can beobtained when information is received from both retailers.
Different from the work mentioned above, Gavirneni andTayur (1998) compared the value of information sharingto the value derived from delayed differentiation, anothermechanism to reduce demand uncertainty. All of the abovepapers report that there are benefits to sharing demandinformation. They all consider supply chains with a singleproduct. Our work differs from the previously cited workin that we consider a supply chain setting that deals withmultiple substitutable products.
Another stream of research has investigated how infor-mation sharing affects pricing decisions within a supplychain. These papers do not investigate inventory-related is-sues. Li (2002) analyzed a model that included a manufac-turer and several competing retailers and showed that re-tailers will not voluntarily share information. Zhang (2002)considered a model in which each retailer sells a differentproduct developed from the same base product suppliedby the manufacturer and allows these products to be eithersubstitutes or complements. Li and Zhang (2005) analyzedthe impact of three information sharing scenarios betweenretailers and the manufacturer, with varying degrees of con-fidentiality. Mishra et al. (2007) showed that both the man-ufacturer and the retailer have incentives to share distortedinformation. We do not consider pricing decisions in thispaper.
There is a significant body of literature on the effectsof substitution among products. For example, researchershave studied the problem of economic order quantity andstocking level when a firm sells substitutable products(Veinott, 1965; Ignall and Veinott, 1969; McGillivray andSilver, 1978; Parlar and Goyal, 1984). Because the exactsolution for the optimal inventory level is unavailable evenfor a simple two-product model with substitution, Rajaramand Tang (2001) developed heuristics to determine the ap-proximate inventory level. The literature on product sub-stitution can be classified into two main categories. In thefirst category, firms may choose to fill demand for oneproduct using the inventory of another, perhaps, that of ahigher quality product, to avoid losing the sale. Research onsuch “one-way substitutability” includes Bitran and Dasu(1992), Bassok et al. (1999), Rao et al. (2004), and Honhonet al. (2006). In the second, substitution decisions are notdirected by firms; rather, they are made by customers. Ourpaper models substitution of the second type. Smith andAggrawal (2000) and Mahajan and Ryzin (2001) developedmodels that capture dynamic customer arrivals within asubstitutable products context. Dynamic customer arrivalscapture the more realistic scenario of different productsgoing out of stock at different time periods. Parlar (1988),Pasternack and Drezner (1991) and Drezner et al. (1995)investigated the impact of substitution in a competitive set-ting in which consumers may go to another retailer whentheir preferred retailer is out of stock. The difference be-tween our work and the extant literature on product substi-tution is that whereas we focus on the value of informationsharing when products are substitutable, the prior literaturefocused on the impact of substitution on optimal inventorylevels.
3. Modeling framework
We consider a two-level supply chain, consisting of a sin-gle manufacturer and a single retailer, which distributesN products. Consumer demands for these products occurat the retailer. The demand for a product follows a sim-ple, order one, auto-regressive, AR(1), process. Inventorymodels that assumed AR(1) demand process include Miller(1986), Kahn (1987), Lee et al. (2000), Raghunathan (2001)and Raghunathan (2003). While the adoption of an AR(1)model simplifies the analysis, the qualitative nature of ourresults is not dictated by the specific demand model. Thevalue of information sharing comes primarily from a reduc-tion in the demand variance, and the impact of substitutionis primarily on the extent of this reduction. Thus, while themagnitude of the value of information sharing under sub-stitution and no substitution will depend critically on thedemand model used, the qualitative impact of substitutionand other model parameters will not. Table 2 summarizesthe notation used in our model.
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Value of information sharing 1127
Table 2. Summary of model notation
Variable Definition
i, j Product identifiert Time period identifierDit Customer demand for product i during time period td Base level for the customer demand for the retailerdm Base level for the customer demand for the manufacturerH Holding cost rate for the manufacturerh Holding cost rate for the retailerK Service level for manufacturerk Service level for retailerN Total number of productsn Number of products whose information is sharedP Shortage cost rate for the manufacturerp Shortage cost rate for the retailerQ Expected demand for a single productr n/NSit Order-up-to level for the retailerTnis
it Manufacturer’s order-up-to level for no substitution with no information sharingT is
it Manufacturer’s order-up-to level for no substitution with information sharingT
′nisit Manufacturer’s order-up-to level for perfect substitution with no information sharing
T′isit Manufacturer’s order-up-to level for perfect substitution with information sharing
Vno−subs Value of information sharing for no substitution caseVperf−subs Value of information sharing for perfect substitution caseW Profit contribution per unit for manufacturerw Profit contribution per unit for retailerYit Retailer order quantity for the product i during time period tρ Auto-correlation coefficient in the AR(1) modelρm Correlation coefficient in the AR(1) model for the manufacturerρr Correlation coefficient across product demands during a time periodξit Random component of the customer demand for product i during time period tδit Random component of the retailer’s demand for product i during time period tσ Standard deviation for the customer demandα Degree of substitution
We have the demand of product i during period t as
Dit= d + ρDi(t−1)+ξit , (1)
where d > 0, −1 < ρ < 1 and i ∈ 1, 2, 3, . . . , N. For agiven t , ξ it follows a normal distribution with mean zeroand variance σ 2, and the correlation coefficient between ξ itand ξ jt , i �= j, is ρr , (−1/(N − 1)) ρr < 1. Both σ 2 and ρr areindependent of t and i. The condition −1/(N − 1) ≤ ρr ≤ 1guarantees that the covariance matrix of ξ it is positive semi-definite. For a given i, ξ it are independent and identicallydistributed. We assume further that σ is significantly smallerthan d, so that the probability of a negative demand for anyproduct during any period is negligible. We assume thatthe retailer uses Equation (1) in his/her forecasting andordering process. That is, s/he uses a period’s demand datato forecast demands in the following period.
The manufacturer also uses a AR(1) model to forecastthe order for any product from the retailer. Let Yit be theorder for product i from the retailer to the manufacturer
during period t . The manufacturer’s forecasting model is
Yit = dm + ρrYi(t−1) + δit . (2)
We assume that the manufacturer’s forecast of retailerorder is unbiased. We show later how dm, ρm, and δit arerelated to the parameters in the retailer’s forecasting model.
We consider a periodic-review system in which each sitereviews its inventory level and replenishes its inventory fromthe upstream site at the end of every period using an order-up-to policy. We assume, for convenience, that the replen-ishment lead times are zero.1 At the end of every time periodt , after demand Dit has been realized, the retailer observesthe inventory level of product i and places an order of sizeYit with the manufacturer to meet the customer demand forproduct i during (t + 1). The manufacturer ships the orderquantity in full to the retailer at the end of time period t , im-mediately after receiving the order, and places his/her own
1See Lee et al. (2000) for the impact of manufacturer and retailerlead times on the benefits of information sharing. Our results willnot change qualitatively for any constant lead time.
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1128 Ganesh et al.
order with his/her supplier to meet the retailer’s demandfor the next period.
We assume that no fixed ordering cost is incurred whenplacing the order, and that inventory holding cost rateand shortage cost rate are stationary over time. Further-more, we assume that the holding cost rate and shortage
maxS1t ,S2t ,...,SNt
∫ ∞
−∞
∫ ∞
−∞. . .
∫ ∞
−∞
×
⎛⎜⎜⎝
⎛⎝w
∑Ni=1 Di(t+1) − p
((1 − α)
∑Ni=1 (Di(t+1) − Sit )+ +
(α
∑Ni=1 (Di(t+1) − Sit )+ − ∑N
i=1 (Sit − Di(t+1))+)+ )
−h(∑N
i=1
(Sit − Di(t+1)
)+ − α∑N
i=1
(Di(t+1) − Sit
)+)+
⎞⎠
g(D1(t+1), ..., DN(t+1))dD1(t+1)...dDN(t+1)
⎞⎟⎟⎠,
cost rate are identical across products. This assumption isreasonable because the products in our model are differ-ent varieties of the same underlying basic product. Let hand p, respectively, denote the holding and shortage costper unit per time period for the retailer. Let H and P, re-spectively, denote the holding and shortage cost per unitper time period for the manufacturer. The profit contribu-tion per unit of a product sold is w and W for the retailerand the manufacturer respectively. We assume that w andW are sufficiently large such that both the retailer and themanufacturer realize profits in all periods.
We model substitution among products in the followingmanner. If a product is out of stock, then a proportion α
of the excess demand for that product will buy any of theother available products with equal probability. That is, acustomer who is willing to substitute is indifferent amongother products2. The rest (1 − α) portion of the excess de-mand is backlogged. When α = 0, products are not substi-tutable, and when α = 1, products are perfect substitutes.Neither the retailer nor the manufacturer incurs a shortagecost when demand substitution occurs, but each incurs ashortage cost when its demand is backlogged.
Our model setup assumes that products are identical intheir demand and cost structures. This assumption enablesus to isolate the impact of substitution and to eliminatethe effects of product-specific characteristics on results. Wediscuss the impact of relaxing this assumption in Section 7.
In the absence of information sharing, the manufacturerreceives only the order Yit for product i from the retailer atthe end of period t . The manufacturer does not know the de-mand for this product at the retailer during period t . Whenthe retailer shares its information with the manufacturer,the manufacturer knows the realized demand Dit . Conse-quently, the manufacturer uses this information in his/herforecast under information sharing. Note that in the infor-mation sharing case, the information shared by the retaileris the realized demand before substitution.
2This substitution model is known in the literature as the “randomsubstitution” model (Mahajan and Ryzin, 2001).
4. Ordering decisions
Let Sit denote the retailer’s order-up-to level for producti for period t in order to meet the demand during period(t + 1). Because we assume that the retailer receives his/herorder in full3, s/he uses the following model at the end ofperiod t to find Sit :
where g(D1(t+1), D2(t+1), . . . , DN(t+1)) is the joint prob-ability density function of demands during time pe-riod t + 1, and z+ ≡ max{0, z}. In the above model,w
∑Ni=1 Di(t+1) is the profit contribution from all prod-
ucts, (1 − α)∑N
i=1 (Di(t+1) − Sit )+ is the fraction of demandbacklogged because of those customers unwilling to sub-stitute, (α
∑Ni=1 (Di(t+1) − Sit )+ − ∑N
i=1 (Sit − Di(t+1))+)+ isthe fraction of excess demand from customers willing tosubstitute but unable to find a substitute product and(∑N
i=1 (Sit − Di(t+1))+ − α∑N
i=1 (Di(t+1) − Sit )+)+ is the in-ventory after satisfying demands from all customers. Ra-jaram and Tang (2001) note “there are no known closedform expressions for the optimal order quantity and the cor-responding optimal expected profit when products are sub-stitutable” even for the two-product case for a general valueof α. We derive optimal Sit for the two extreme cases: nosubstitution and perfect substitution. Subsequently we an-alyze the partial substitution case numerically in Section 6.
4.1. No substitution case (α = 0)
First-order conditions for the retailer’s maximization modelyield the familiar newsvendor solution for each productbecause the marginal distribution of a multivariate normaldistribution is a univariate normal distribution. Thus, theoptimal order-up-to level for product i for period t underno substitution is given by the following:
Sit = d + ρDit + kσ, (3)
where k = φ−1p/(p + h) for the standard normal distribu-tion function φ. Note that ρr does not play any role in theretailer’s ordering decision when products are not substi-tutable. This is intuitive because even though the demandsof different products during a time period may be corre-lated, the demand in a time period is sufficient to determinethe demand distribution for the product in the following
3If the retailer’s order is not fully satisfied, then the retailer shouldtake expectation with respect to the quantity s/he will receive fromthe manufacturer while determining the order-up-to levels.
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Value of information sharing 1129
period. At the end of period t , the retailer’s order for prod-uct i,Yit , is given by the following:
Yit = Dit + (Sit − Si(t−1)). (4)
Notice that the order quantity replenishes the demand dur-ing period t plus the change being made in the order-up-tolevel from period (t − 1) to t . Immediately after receivingthe retailer’s order for a product, the manufacturer shipsthe retailer’s order and places its own order to bring itsinventory to its order-up-to level,Tit . The manufacturer’sorder-up-to level depends on the information available tothe manufacturer at the time it computes the order-up-tolevel, which, in turn, depends on whether the retailer shareshis/her information with the manufacturer.
4.1.1. No information sharingHere the manufacturer receives only the retailer’s order foreach product; the retailer does not share the actual demandfor the product during any time period. Using Equations(1), (3) and (4), we obtain:
Yi(t+1) = d + ρYit + (1 + ρ)ξi(t+1) − ρξit . (5)
Comparing Equation (4) with Equation (2), we find thatparameters in the manufacturer’s forecasting model underno information sharing and those in the retailer’s forecast-ing model are related in the following way: dm = d, ρm =ρ and δit = (1 + ρ)ξit − ρξi(t−1). Although ξit is knownto the retailer at the end of period t , it is not sharedwith the manufacturer when the retailer orders Yit . Fromthe manufacturer’s perspective, Yi(t+1) has a variance ofσ 2((1 + ρ)2 + ρ2) in the no information sharing case. LetTnis
it be the manufacturer’s order-up-to level for product iat the end of period t to meet the retailer’s demand duringthe period (t + 1) under no information sharing. Then, fol-lowing an analysis similar to the one for the retailer, we getTnis
it as the following:
Tnisit = d + ρYit + Kσ
√((1 + ρ)2 + ρ2, (6)
where
K = φ−1(
PP + H
).
4.1.2. Information sharingIn the information sharing case, we assume that, at the endof a period, the retailer shares its demand data for thatperiod for 0 < n ≤ N products. When n = N, we have fullinformation sharing. The setting where partial informationsharing occurs could be the result of factors related to themanufacturer or the retailer. Acquiring, and perhaps, moreimportantly processing each piece of information from theretailer costs the manufacturer. Consequently, the manu-facturer will be interested in determining the incrementalvalue derived from information from each additional prod-uct. In the same vein, from the retailer side, a retailer wouldnot voluntarily share information unless it could extract a
share of the benefit the manufacturer realizes from the in-formation. Lee et al. (2000) noted that manufacturers usea variety of incentive mechanisms, such as price discount,smaller lead time and financial credit, to induce retailersto share their information. Thus, a retailer will be inter-ested in the incremental value of additional information tothe manufacturer in order to design an appropriate “price”structure for the information. Similarly, a manufacturer willalso be interested in the incremental value of additional in-formation to evaluate incentive mechanisms to induce in-formation sharing. Without loss of generality, we assumethat the retailer shares its demand data for the first n (i.e.,1 ≤ i ≤ n) products. In this scenario, at the end of periodt , the manufacturer knows ξit for 1 ≤ i ≤ n. Now, basedon this information, the manufacturer can update the dis-tributions of ξjt , n + 1 ≤ j ≤ N. The following property ofMultivariate Normal (MN) distributions is used to derivethe distribution of ξjt , n + 1 ≤ j ≤ N given ξit , i ≤ n.
Property 1. If X ∼ MN(μ, ϑ), where μ is a N-dimensionalmean vector and ϑ is a N × N covariance matrix, then(X(2)|X(1) = x(1)) ∼ MN(μ2.1, ϑ2.1), where
X =(
X(1)
X(2)
),μ =
(μ(1)
μ(2)
),∑
=(
ϑ11 ϑ12
ϑ21 ϑ22
), ϑ21 = ϑT
12,
covariance(X(1)) = ϑ11, covariance(X(2)) = ϑ22, covariance(X(1), X(2)) = ϑ12, μ2.1 = μ (2) + ϑ2.1ϑ
−111 (x(1) − μ (1) ), and
ϑ2.1 = ϑ22 − ϑ21ϑ–111 ϑ12.
In our model of partial information sharing, we have thefollowing:
ξNx1 = [ ξ1t .. ξnt ξ(n+1)t .. ξNt ]TμNx1 =[
0 .. 0]T
∑NxN
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
σ 2 ρrσ2 .. .. .. ρrσ
2
ρrσ2 σ 2 ..
.. ..
.. ..
.. σ 2 ρrσ2
ρrσ2 .. .. .. ρrσ
2 σ 2
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
=[
ϑ11nxn ϑ12nx(N−n)
ϑ21(N−n)xn ϑ22(N−n)x(N−n)
].
Now applying property 1 to our distributions defined above,we get the conditional mean and conditional variance ofξjt , n + 1 ≤ j ≤ N , given ξit , i ≤ n, as the following:
μ2.1 =[
ρr
n∑ι=1
ξit
n... ρr
n∑ι=1
ξit
n
]T
, ϑ2.1(N−n)x(N−n)
=[σ 2
((1 − ρr )(nρr + 1)(1 + (n − 1)ρr )
)
. . . σ 2(
(1 − ρr )(nρr + 1)(1 + (n − 1)ρr )
)]T
.
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1130 Ganesh et al.
Thus, the manufacturer’s order-up-to levels under informa-tion sharing are given by the following:
T isit = d + ρYit − ρξit + Kσ (1 + ρ) for i ≤ n,
T isjt = d + ρYjt − ρρr
n
n∑ι=1
ξit (7)
+ Kσ
√(1 + ρ)2+ρ2
((1 − ρr )(nρr + 1)(1 + (n − 1)ρr )
)for j > n
4.2. Perfect substitution case (α =1)
When products are perfect substitutes, though a customermay have a preference for a specific product, s/he will bewilling to buy an alternate product when his/her preferredproduct is out of stock. This is likely to occur when either thecustomers’ 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 re-tailer is indifferent between stocking just one or any numberof product varieties. Though the perfect substitution caseis unrealistic, we analyze this case to derive a bound on theimpact of substitution on the value of information sharing.
Substituting α =1 in the retailer’s maximization model,we find that its cost is a function of only the total demandand the total stocking level for all products. Consequently,it is sufficient for the retailer to determine the optimal to-tal stocking level based on the total demand. That is, theretailer can treat the set of all products as a single entityand determine the order-up-to level for this entity as awhole. Once the total order (for the entire set of products)for a period is computed, the total order can be dividedinto orders for individual products based on any allocationscheme. Notice that the allocation scheme does not affecteither the manufacturer’s or the retailer’s profit because ofour assumption that all products are perfect substitutes ofeach other and the assumption that neither the retailer northe manufacturer incurs any substitution-specific cost. Weconjecture that substitution costs will simply force a mini-mum order quantity for each product, while still maintain-ing the same overall order quantity. We assume that theretailer will allocate the total order equally among all prod-ucts in our analysis. A similar reasoning also applies to themanufacturer.
The total demand for all products for the retailer is givenby the following:
N∑i=1
Dit = Nd + ρ
N∑i=1
Di(t−1) +N∑
i=1
ξit .
Notice that Var(∑N
i=1 ξit ) = σ 2(N(1 + (N − 1)ρr )). Thus,the sum of the order-up-to levels for all products for theretailer can be derived using newsvendor analysis as in the
following:
N∑i=1
Sit = Nd + ρ
N∑i=1
Dit + kσ√
(N(1 + (N − 1)ρr )). (8)
Note that, unlike the no substitution case, the retailer’sorder-up-to level in the substitution case depends on thecorrelation between product demands. This is because whiledemand correlation does not affect the variance of demandfor a single product, it affects the variance of the total de-mand; the variance of total demand is increasing in thedemand correlation.
We derive the sum of the order-up-to levels for all prod-ucts for the manufacturer in a manner similar to that in theno substitution case.
4.2.1. No information sharingThe total retailer order is given by
N∑i=1
Yi(t+1) = Nd + ρ
N∑i=1
Yit + (1 + ρ)N∑
i=1
ξi(t+1)−ρ
N∑i=1
ξit .
(9)Thus, we have:
N∑i=1
T′nisit = Nd + ρ
N∑i=1
Yit
+ Kσ√
((1 + ρ)2 + ρ2)(N(1 + (N − 1)ρr )). (10)
4.2.2. Information sharingUsing an analysis similar to that in the no substitution case,we get the following for the total order-up-to level for themanufacturer:
N∑i=1
T′isit = Nd + ρ
N∑i=1
Yit − ρ
n∑i=1
ξit
− ρ(N − n)ρr
(∑n
i=1ξit
/n)
+ Kσ√{(1 + ρ)2(N(1 + (N − 1)ρr ))} + ρ2{(N − n)
((1 − ρr )2
+N(ρr − ρ2
pr
))/(1 + (n − 1)ρr )} .
(11)
5. The value of information sharing
Because the retailer’s ordering decisions are unaffected byinformation sharing, only the manufacturer could benefitfrom information sharing. Furthermore, because informa-tion sharing does not affect mean demands from the retailerto the manufacturer, the benefit from information sharingarises only from a reduction in the manufacturer’s inven-tory holding and shortage costs. If the manufacturer’s ser-vice level, as indicated by K, is sufficiently high, then thecost reduction comes primarily from inventory reduction
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Value of information sharing 1131
because the probability of shortage as well as the expectedamount of shortage will be, though not zero, very small. Weassume that K is high for our theoretical analysis becauseincluding the shortage cost makes the analysis intractable.4
In our numerical simulation, we consider both inventoryholding and shortage costs. For any order-up-to-Tt systemwith Yt being the demand in period t , the inventory levelat the end of the period is given by the following:
Tt − E(Yt+l). (12)
Since information sharing does not change E(Yt+l), thesaving in the average inventory because of informationsharing is equal to the reduction in the order-up-to levelwhen information is shared. The expected total demandduring period t is
∑Ni=1 Dit . Furthermore, we define Q ≡
limat→∞ E [Dit ] = d/(1 − ρ) in order to compute the long-
term value of information sharing as done by Lee et al.(2000). We define the value of information sharing as thepercentage increase in the manufacturer’s profit because ofinformation sharing, as given below:
Value of information sharing
= Profit w/information sharing − Profit w/o information sharingProfit w/o information sharing
×100. (13)Using Equations (6) and (7), we get the value of informationsharing under no substitution, Vno−subs, as
Vno−subs = N√
((1 + ρ)2 + ρ2) − {n(1 + ρ) + (N − n)√
(1 + ρ)2 + ρ2((1 − ρr )(nρr + 1)/((1 + (n − 1)ρr ))}N(WQ/HKσ −
√((1 + ρ)2 + ρ2))
(14)
and using Equation (10) and (11), we get the value of infor-mation sharing under perfect substitution, Vperf−subs, as
Vperf−subs =√
((1 + ρ)2 + ρ2)N −√
((1 + ρ)2)N + ρ2{(N − n)(1 − ρr )/((1 + (n − 1)ρr )}(NWQ/HKσ
√(1 + (N − 1)ρr )) −
√((1 + ρ)2 + ρ2)N
(15)
We first perform a comparative static analysis of thevalue of information sharing in the no-substitution andin the perfect substitution case to understand the impactof model parameters, such as the demand correlation, onthe value of information sharing. Then, we compare thevalue of information sharing in no substitution and per-fect substitution cases to gain insight into the impact ofsubstitution on the value. Now, we show the followingresult.
Proposition 1.
(i)∂Vno−subs
∂ρr≥ 0.
∂Vno−subs
∂n≥ 0.
∂Vno−subs
∂N≤ 0.
(ii)∂Vperf−subs
∂ρr≥ 0,
∂Vperf−subs
∂n≥ 0.
∂Vperf−subs
∂N≤ 0.
4The expected shortage can be expressed as an integral whosevalue can only be numerically computed. Further, Lee et al. (2000)showed that approximating the total cost by only the inventoryholding cost does not result in a significant error.
Proof. A proof is provided in the Appendix. �Proposition 1(i, ii) shows that, in both no substitution
and perfect substitution cases, the value of informationsharing is increasing in the demand correlation and thenumber of products whose demands are shared. The valueof information sharing increases with demand correlationbecause a higher demand correlation enables the manufac-turer to more accurately forecast the demand of a prod-uct from the demand information of other products. In thesame vein, a higher forecast accuracy results when demandinformation of a larger number of products is shared. Weshow analytically that the value of information sharing isdecreasing in the total number of products. It is surprisingwhy the value of information should decrease with an in-crease in N. The reason is that when n is kept constant, anincrease in N means that the fraction of products whoseinformation is shared decreases, which suggests that theextent of reduction in demand uncertainty because of in-formation sharing relative to the absolute demand certaintyis lower when N is larger. In summary, our analysis showsthat the impacts of demand correlation, the total numberof products and the number of products whose informationis shared are qualitatively similar in the no substitution andperfect substitution cases.
By comparing Equations (14) and (15), we can showthat, in general, the value of information sharing can belarger or smaller under perfect substitution than under nosubstitution. However, an analysis of special cases of thegeneral model provides some interesting insights. We findthat if information about all products are shared, i.e., underfull information sharing (n = N), then the value of infor-mation sharing is less under perfect substitution than un-der no substitution. Second, if demands of products dur-ing a period are perfectly correlated, then the value of in-formation sharing under perfect substitution is identicalto that under no substitution. Third, when demands ofproducts during a period are independent, the value of in-formation sharing is less under perfect substitution thanunder no substitution. For other cases, the value of infor-mation sharing under perfect substitution is less than thatunder no substitution only when the following conditionholds:
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1132 Ganesh et al.
WQHKσ
>
⎛⎜⎜⎝
√((1 + ρ)2 + ρ2)
{√((1 + ρ)2 + ρ2)N −
√(1 + ρ)2N + ρ2
((N−n)(1−ρr )
1+(n−1)ρr
)}
−√
((1+ρ)2+ρ2)N
{N
√((1 + ρ)2 + ρ2) −
{n (1 + ρ) + (N − n)
√(1 + ρ)2 + ρ2
((1−ρr )(nρr +1)(1+(n−1)ρr )
)}}⎞⎟⎟⎠
⎛⎜⎜⎝
√((1 + ρ)2 + ρ2)N −
√(1 + ρ)2N + ρ2
((N−n)(1−ρr )
1+(n−1)ρr
)−
1√(1+(N−1)ρr )
{N
√((1 + ρ)2 + ρ2) −
{n (1 + ρ) + (N − n)
√(1 + ρ)2 + ρ2
((1−ρr )(nρr +1)(1+(n−1)ρr )
)}}⎞⎟⎟⎠
(16)
Numerical computations confirm the above condition.For example, Table 3 shows the value of information shar-ing for different values of r and ρr when H = 1, W = 1.0,N = 25, Q = 200, ρ = 0.7 and σ = 2. The first and secondvalues in each cell show, respectively, the value under nosubstitution and perfect substitution. The cells indicatedin bold show the regions in which the value under perfectsubstitution exceeds that under no substitution.
Condition (16) suggests that for the value of informationsharing to be higher under perfect substitution than underno substitution, the profit contribution has to be signifi-cantly larger than the sum of inventory holding and short-age costs. For example, for the parameter values used togenerate Table 3, we found that when r = 0.2 and ρr = 0.8,WQ should be at least 16 for the value of information shar-ing to be greater under perfect substitution than under nosubstitution, and the same should be at least 55 when r =0.2 and ρr = 0.2. In terms of the inventory holding andshortage costs as a fraction of the profit contribution, wefind that the cost should be less than 11% of the profit con-tribution when r = 0.2 and ρr = 0.8 and less than 3% whenr = 0.2 and ρr = 0.2. These numbers indicate that when in-ventory holding and shortage costs are significant, the valueof information sharing is always lower when products areperfectly substitutable than when they are not substitutable.Given that the value of information sharing arises primar-ily from a reduction in the inventory holding and short-age costs, information sharing becomes valuable only when
Table 3. Value information sharing (percentage increase in manufacturer‘s profit) under no substitution and under perfect substitution
Correlation coefficient ρr
r = (n/N) Scenario −0.01 0 0.2 0.4 0.6 0.8 1
0.2 No substitution 0.05 0.05 0.07 0.11 0.16 0.20 0.25Perfect substitution 0.006 0.01 0.08 0.13 0.18 0.22 0.25
0.4 No substitution 0.1 0.10 0.12 0.15 0.19 0.22 0.25Perfect substitution 0.01 0.02 0.10 0.15 0.19 0.22 0.25
0.6 No substitution 0.15 0.15 0.17 0.19 0.21 0.23 0.25Perfect substitution 0.02 0.03 0.11 0.16 0.19 0.23 0.25
0.8 No substitution 0.2 0.20 0.21 0.22 0.23 0.24 0.25Perfect substitution 0.03 0.04 0.12 0.16 0.20 0.23 0.25
1 No substitution 0.25 0.25 0.25 0.25 0.25 0.25 0.25Perfect substitution 0.04 0.05 0.12 0.16 0.20 0.23 0.25
inventory holding and shortage costs are significant. Conse-quently, we conclude that substitution reduces the value ofinformation sharing in situations when information sharingis more valuable.
Furthermore, another interesting finding from our ex-tensive numerical analysis is that only when the number ofproducts for which information is shared is low, is the valueof information sharing under substitution higher than un-der no substitution under certain conditions. We find thatthis is more likely to occur when the demand correlation ishigh, but not perfect. This is somewhat counter-intuitive be-cause one would expect substitution to mitigate the variancebecause of the demand-pooling effect and hence to resultin a smaller value of information sharing. The seeminglycounter-intuitive finding is the outcome of the interplay oftwo effects of demand correlation that impact the value ofinformation sharing. First, demand correlation enables in-formation about the demand of one product to be used toreduce the uncertainty about the demand of other products.This effect of correlation is present under substitution andno substitution cases. The second effect, which is presentonly under the substitution case, is that demand correla-tion increases the variance of the total or pooled demand.When the number of products whose information is sharedis small, the latter effect dominates the former, resulting ina higher value of information sharing under substitution.As the number of products whose demand information is
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Value of information sharing 1133
shared increases, the impact of the second effect reduces rel-ative to that of pooling under product substitution. Whendemand correlation is high, the second effect weakens onlywhen the number of products whose demand is shared isalso high.
In a different context, Huang and Iravani (2005) showedthat in a single manufacturer and two retailer supply chain,under certain conditions, the manufacturer can capturemost of the savings from information sharing from infor-mation from a single retailer. Furthermore, they showedthe information-pooling effect and economies of scale withrespect to information sharing. In our context, we find thatwhen the manufacturer acquires information only for someproducts, demand-pooling effects of the substitution mayactually increase the value of information sharing depend-ing on the correlation between demands. That is, demandpooling caused by substitution can enhance the benefit frominformation if only information about a subset of productsis shared by the retailer.
6. Simulation results
We imposed two restrictions for our theoretical analysis.First, we considered only the two extreme cases of substitu-tion: no substitution and perfect substitution. Second, weconsidered only the reduction in the inventory holding costto compute the value of information sharing. The savingsin shortage cost were excluded. These restrictions were im-posed for analytical tractability. In the simulation analysis,we relaxed both these restrictions and analyzed the “true”value of information sharing that includes both inventoryholding and shortage costs. Furthermore, we also analyzedthe impact of partial substitution using simulation. The ob-jectives of our simulation were two-fold. First, we wantedto verify that results from our theoretical analysis of theapproximate value of information sharing also hold for thetrue value of information sharing. Second, we wanted toobtain a sense of the magnitude of the effect of substitu-tion on the value of information sharing. For our simula-tion, we fixed the following parameters for all simulationscenarios:
σ = 2, ρ = 0.7, h = 1, H= 1,
p = 25, P = 25, d = 40, and, W = 0.1.
We varied the following parameters within the respectiveranges shown:
N ∈ {5, 10, 25, 50}r = n
N∈ {0, 0.2, 0.4, 0.6, 0.8, 1.0}
ρr ∈ {−0.01, 0, 0.2, 0.4, 0.6, 0.8, 1.0}α ∈ {0, 0.2, 0.4, 0.6, 0.8, 1.0}
The negative value for ρr was restricted to only −0.01, be-cause the condition that −1/(N − 1) < ρr stated in Section
Table 4. Summary of simulation results
Value of information sharing as thepercentage increase in manufacturer’s profit
The degree ofsubstitution α Minimum value Maximum value Mean value
0.0 2.57 14.03 8.300.2 2.30 13.20 7.750.4 1.93 12.96 7.440.6 1.51 12.96 7.240.8 0.86 12.96 6.911.0 0.09 12.96 6.53
3 severely restricted the range of negative values that couldbe used. For each scenario, we conducted the simulationfor 10 000 periods.
For the partial substitution case, we computed the opti-mal order-up-to level through exhaustive enumeration. Un-like Rajaram and Tang (2001), who developed a heuristicto determine an approximate order-up-to level, we used ex-haustive search to determine the true optimal order-up-tolevel in order to compute the exact value of informationsharing. Table 4 provides a summary of the results, i.e.,the value of information sharing as a percentage increasein profit computed using Equation (13), obtained in oursimulation.
Table 4 shows that the value of information sharingranged from 0.09% to 14.03%. The minimum occurredwhen products were perfect substitutes and the maximumoccurred when products were not substitutes. Furthermore,the mean, minimum and the maximum value of informationsharing showed the same trend with respect to the degree ofsubstitution. For the parameter values we discuss in detailin this section, the value of information sharing was alwayslower under substitution than under no substitution. How-ever, consistent with our theoretical results, we did find thatthe value of information sharing was higher under substitu-tion than under no substitution for some parameter values.For instance, when the value of p and P were changed to two(and other parameters were kept the same as those indicatedearlier), the value of information sharing was higher undersubstitution than under no substitution under the follow-ing conditions: r = 0.2 and ρr ∈ {0.4, 0.6, 0.8}, and r = 0.4and ρr ∈ {0.6, 0.8}. Furthermore, we also found that themagnitude of the value of information sharing under theseconditions ranged from 0.46 to 1.14, confirming our the-oretical prediction that substitution reduces the value ofinformation sharing in situations when information shar-ing is more valuable.
Next, we analyze the impact of substitution and otherparameters on the value of information sharing. Oursimulation study included 864 scenarios. We present ourresults for some representative scenarios to save space. Theresults discussed for these scenarios are qualitatively iden-tical to those for other parameter values.
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1134 Ganesh et al.
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
0 0.2 0.4 0.6 0.8 1
Degree of Substitution α
Val
ue o
f In
form
atio
n Sh
arin
g
ρr = 1.0
ρr = 0.8
ρr = 0.6
ρr = 0.4
ρr = 0.2
ρr = 0.0
ρr = -0.01
ρr= Correlation Coefficient
Fig. 1. The impact of the degree of substitution on the value of information sharing.
6.1. Impact of the degree of substitution
We discuss the impact of the degree of substitution on thevalue of information sharing using Fig. 1, which showsthe value of information sharing for N = 25 and r = 0.8.The value of information sharing ranges from 12.96% to aslow as 0.59%. We find that the value of information sharingis decreasing in the degree of substitution for all values ofρr except when ρr = 1. The rate of decrease is higher whenthe degree of substitution is higher. The reasons for thedecrease in the value of information sharing under substi-tution are two-fold. The first is the demand-pooling effectof substitution. When products are more substitutable, themanufacturer assigns more weight to the pooled or total de-mand of products. It is well known that the pooled demandhas a lower standard deviation than the sum of standarddeviations of individual demands. Thus, the safety stockas well as inventory holding and shortage costs are lowerwhen products are more substitutable. The pooling effect ispresent even when information is not shared. Thus, becausethe effective demand uncertainty faced by the manufacturerwhen there is no information sharing is lower, informationsharing reduces the manufacturer’s demand uncertainty lesswhen the degree of substitution is higher. Hence, the valueof information sharing is lower when the degree of substitu-tion is higher. Second, because demand pooling caused bysubstitution reduces inventory holding and shortage costseven when information is not shared, the base level profit ishigher when the degree of substitution is higher. A higherbase level also reduces the value of information sharing,which is expressed as a percentage increase in profit.
The most important implication of our observationabout the impact of substitution on the value of informa-tion sharing relates to the extent by which the value of in-formation sharing can be overstated if substitution effectsare ignored. When demands are independent, the value ofinformation sharing is overestimated by more than 92%
(the value of information sharing is 10.45 and 0.59 respec-tively when α = 0 and α = 1, and (10.45 − 0.59)/10.45 =0.94 if products are perfect substitutes but substitution isignored.
6.2. Impact of demand correlation on the valueof information sharing
Figure 2 shows the impact of correlation between prod-uct demands during a period on the value of informationsharing for N = 25 and r = 0.8. The figure confirms ourtheoretical result that the value of information sharing in-creases when the demand correlation increases under nosubstitution and perfect substitution cases. Furthermore,the figure shows that the same trend also holds for inter-mediate levels of substitution. The result that the value ofinformation sharing is higher when the demand correlationis higher for all degrees of substitution is expected becausea higher demand correlation increases the demand uncer-tainty, which means that a higher value could be gainedfrom information sharing. The rate of increase in the valueof information sharing is higher when the degree of sub-stitution is higher. This is because correlation plays a rolein the manufacturer’s decisions only when the manufac-turer uses the pooled demand to make decisions. A greaterdegree of substitution results in a higher degree of pool-ing of demands, and thus enhances the role of demandcorrelation.
While ignoring substitution effects can lead to a signifi-cant overestimation of the value of information sharing, wefind that a higher correlation among demands of these sub-stitutable products, which is highly likely, can mitigate theoverestimation. In the extreme case of perfect correlation,the manufacturer can safely ignore substitution effectsin computing the value it can obtain from the retailer’sinformation.
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Value of information sharing 1135
0
2
4
6
8
10
12
14
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
Correlation Coefficient ρr
Val
ue o
f In
form
atio
n Sh
arin
g
α = 0.0
α = 0.2
α = 0.4
α = 0.6
α = 0.8
α = 1.0
α = Degree of substitution
Fig. 2. The impact of demand correlation on the value of information sharing.
6.3. Impact of the number of products on the valueof information sharing
Figure 3 shows the impact of number of products onthe value of information sharing. For this figure, we useρr = 0.6 and n = 4. Our simulation results are consistentwith our theoretical result for the no substitution and per-fect substitution scenarios. Furthermore, simulation resultsshow that our theoretical results hold for any degree ofsubstitution. We find that the value of information sharingranged from 11.89% (for the no substitution scenario) to7.82% (for the perfect substitution scenario) when N = 5,but the same range was from 5.24% to 4.62% when N = 50.This observation suggests that the value of informationsharing is less affected by substitution as the number ofsubstitutable products increases. An interesting observa-tion from Fig. 3 is that the marginal decrease in value ishigher when the products are less substitutable, suggestingthat an increase in the number of substitutable products
has a more pronounced effect on the value of informationsharing when products are less substitutable.
6.4. Impact of n
Figure 4 shows the impact of increasing the number ofshared products on the value of information sharing. Forthis figure, we use ρr = 0.6 and N = 25. The finding thatthe value of information sharing increases as the number ofproducts for which information is shared increases irrespec-tive of the degree of substitution. The finding is intuitive;more information reduces demand variance more, whichleads to a higher value. We also found that the impact ofsubstitution is more significant when n is large. In this figure,when n = 5, the value of information sharing ranged from7.63%, which was obtained for the no substitution case, to5.73%, which was obtained for the perfect substitution case.The same range was 12.94% to 7.2% when n = 25. Thus,
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
5 10 25 50
Number of Products N
Val
ue o
f In
form
atio
n Sh
arin
g
α = 0.0
α = 0.2
α = 0.4
α = 0.6
α = 0.8
α = 1.0
α = Degree of substitution
Fig. 3. Impact of number of products on the value of information sharing.
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1136 Ganesh et al.
0
2
4
6
8
10
12
14
5 10 15 20 25
Number of shared products n
Val
ue o
f In
form
atio
n Sh
arin
g
α = 0.0
α = 0.2
α = 0.4
α = 0.6
α = 0.8
α = 1.0
α = Degree of substitution
Fig. 4. Impact of the number of shared products on the value of information sharing.
we conclude that ignoring substitution can lead to severeoverestimation when the retailer shares information aboutmore products.
6.5. Joint impact of n and demand correlation
Table 5 shows the joint impact of n and the demand corre-lation on the value of information sharing when α = 0.6.The qualitative behavior of the impact was also identicalfor other values of α.
We can make the following observations from the datain Table 5. The incremental value of information sharingwhen information about an additional product is shared in-creases initially when the demand correlation increases, butwhen the demand correlation exceeds a threshold value, theincremental value starts to decrease when demand correla-tion increases. Furthermore, this threshold value is loweror the same when n is higher. The results imply that higherdemand correlation enhances the value obtained from eachadditional piece of information as long as the demands arenot highly correlated. The enhancement effect vanishes ifthe demand correlation is sufficiently high because high
demand correlation enables the manufacturer to infer theunknown demands more accurately, thus reducing the in-cremental value from the additional information.
In summary, our simulation results confirmed that ourtheoretical results about the impact of model parameters onthe value of information sharing hold not only for the nosubstitution and perfect substitution cases, but for all val-ues of intermediate levels of substitution. Furthermore, thesimulation confirmed that the value of information sharingobtained by considering the total cost behaves qualitativelysimilar to that obtained by considering only the inventoryholding cost. The simulation analysis also showed that, un-der certain conditions, the extent of overestimation can besignificant if substitution effects are ignored.
7. Conclusions
In this paper, we analyzed the impact of consumer productsubstitution on the value of information sharing in sup-ply chains. We showed that substitutability among prod-ucts generally reduces the value of information sharing.Furthermore, the reduction in the value of information
Table 5. Joint impact of demand correlation and the number of shared products on the value of information sharing
Change in value of information as thepercentage increase in manufacturer’s profit
Correlationcoefficient ρr
When nincreases from
zero to one
When nincreases from
one to two
When nincreases from
two to three
When nincreases fromthree to four
When nincreases from
four to five
−0.01 1.56 1.47 1.78 1.65 1.680.0 1.55 1.52 1.8 1.71 1.730.2 2.26 1.65 1.78 1.59 1.650.4 3.55 1.93 1.69 1.37 1.480.6 5.76 1.97 1.31 0.93 1.020.8 8.97 1.4 0.83 0.45 0.581.0 12.96 0 0 0 0
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Value of information sharing 1137
sharing because of substitutability is higher when the degreeof substitution is higher, the demands of products are lesscorrelated, the number of substitutable products is smallerand the number of products whose information is sharedis higher. We also showed that there are instances in whichthe value of information sharing is higher under substitu-tion than under no substitution. However, we also foundthat the inventory holding and shortage costs will have tobe significantly smaller than the profit contribution for thisto occur.
The key implication of our findings is that if substitu-tion effects are ignored, there is a significant risk of over-estimating the value of information sharing. Consequently,managers assessing returns from supply chain managementand collaboration software that aim to improve visibilityamong supply chain partners through information sharingshould be cautious about the claims made based on resultsfrom single-product simulations. The risk can be especiallysignificant when the degree of substitution is large and de-mands of these products are independent or not stronglycorrelated.
We believe that this paper is the first, in the growing litera-ture on supply chain information sharing, to investigate theeffect of consumer substitution on the value of supply chaininformation sharing. Being the first study, we made severalsimplifying assumptions in our analysis. For instance, weassumed that the lead time for the manufacturer to sup-ply to the retailer is zero. We also considered a two-levelsupply chain. We made these assumptions because the im-pacts of lead time and the number of levels in the supplychain have been considered by prior literature that doesnot consider substitution. We do not anticipate that leadtime and number of supply chain levels will have a differ-ent impact when substitution is considered. We assumed anAR(1) model for the demand forecasting process. However,the qualitative nature of our results is not dictated by thespecific demand model. While the magnitude of the valueof information sharing under substitution and no substi-tution will depend critically on the demand model and thelevel of demand uncertainty, the impact of substitution isnot likely to change.
Future research should address other demand forecast-ing models with substitution. Our model assumes that allproducts are symmetric; that is, demand models and costsare identical across products. This was done to isolate theeffects of substitution so that factors such as demand sizesand costs do not contaminate our findings. However, prod-ucts could be different in their inventory holding and short-age costs, production costs, mean demands and other pa-rameters. These differences will have two effects on ourresults. First, they change the magnitude of the value ofinformation sharing. Second, and more importantly, thevalue of information sharing now depends on which de-mand information is shared by the retailer. The value ofinformation sharing will be higher if the retailer shares in-formation about products that have higher inventory hold-
ing and shortage costs. When all products are identical, thevalue of information sharing is affected by the number of(rather than which specific) products whose information isshared. Finally, we also assumed a model in which excessdemand is backlogged. In many scenarios, excess demandmight be lost. When the demand is backlogged, the meannumber of units sold is unaffected by either substitution orinformation sharing. Thus, the benefit of information shar-ing can be quantified solely by savings in inventory andholding costs. However, in a lost-sale model, the expectednumber of units sold is a function of both information shar-ing and substitution. A higher degree of substitution willincrease the number of units sold, and information sharingwill decrease the number of units sold because informationsharing will reduce the safety stock. Consequently, the im-pact of substitution on the value of information sharing in alost-sales model cannot be ascertained without performinga detailed analysis. Future research can look into relaxingthe above limitations and identify the exact impact of theselimitations.
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Value of information sharing 1139
Appendix
Proof of Proposition 1
(i)∂Vno−subs
∂ρr
= ρ2 (N − n) ρr (2n + nρr (n + 1))
2N(1 + (n − 1)ρr )2(
(WQ)/(HKσ ) −√
((1 + ρ)2 + ρ2)) √
(1 + ρ)2 + ρ2 (((1 − ρr ) (nρr + 1))/(1 + (n − 1) ρr ))
∂Vno−subs
∂N= −
√(1 + ρ)2 + ρ2
((1 − ρr ) (nρr + 1)(1 + (n − 1) ρr )
) ((N − n)
√((1 + ρ)2 + ρ2) + n((WQ)/(HKσ )) − N
)+√
((1 + ρ)2 + ρ2)((N − 1)((WQ)/(HKσ )) + n(1 + ρ)) − N((WQ)/(HKσ )) (1 + ρ)
N2(((WQ)/(HKσ )) −
√((1 + ρ)2 + ρ2)
)2 < 0.
∂Vno−subs
∂n= 1
N(
(WQ)/(HKσ ) −√
((1 + ρ)2 + ρ2))
×{ √
(1 + ρ)2 + ρ2 ((1 − ρr ) (1 + nρr )/(1 + (n − 1) ρr )) − (1 + ρ) + ρ2 (N − n) ρ2r (1 − ρr )
2 (1 + (n − 1)ρr )2√
(1 + ρ)2 + ρ2 ((1 − ρr ) (1 + nρr )/(1 + (n − 1) ρr ))
}> 0.
(ii) Vperf−subs =√
((1 + ρ)2 + ρ2)N −√
((1 + ρ)2)N + ρ2{(N − n) (1 − ρr )
/(1 + (n − 1) ρr )
}(
NWQ/HKσ√
(1 + (N − 1) ρr ))
−√
((1 + ρ)2 + ρ2)N
− ((1 + ρ)2) + ρ2 {(1 − ρr )/1 + (n − 1)ρr }2√
((1 + ρ)2)N + ρ2{(N − n) (1 − ρr )/1 + (n − 1) ρr }
⎛⎜⎜⎝
NWQ
HKσ√
(1 + (N − 1) ρr )−
√((1 + ρ)2 + ρ2)N
+ ((1 + ρ)2)N + ρ2{(N − n) (1 − ρr )/(1 + (n − 1)ρr )}((1 + ρ)2) + ρ2{(1 − ρr )/(1 + (n − 1)ρr )}
⎞⎟⎟⎠
∂Vperf−subs
∂N=
⎛⎝−
WQ√
((1 + ρ)2 + ρ2)N
HKσ
(2 (1 + (N − 1) ρr )3/2 − 2 (1 + (N − 1) ρr ) − Nρr
2 (1 + (N − 1) ρr )3/2
)⎞⎠
/(NWQ
HKσ√
1 + (N − 1) ρr−
√((1 + ρ)2 + ρ2)N
)2
< 0.
∂Vperf −subs
∂n= ρ2 (1 − ρr )
√((1 + ρ)2)N + ρ2
{(N − n) (1 − ρr )
1 + (n − 1) ρr
}/(
2 (1 + (n − 1) ρr )
(NWQ
HKσ√
1 + (N − 1) ρr−
√((1 + ρ)2 + ρ2)N
) )> 0.
− ρ2 (N − n) n
2√
((1 + ρ)2)N + ρ2 {(N − n) (1 − ρr )/(1 + (n − 1) ρr )}
((NWQ/HKσ
√1 + (N − 1) ρr
)−
√((1 + ρ)2 + ρ2)N
)
∂Vperf−subs
∂ρr=
(N − 1)2
(1 + (N − 1) ρr )3/2
(√((1 + ρ)2 + ρ2)N −
√((1 + ρ)2)N + ρ2
{(N − n) (1 − ρr )
1 + (n − 1) ρr
})((
NWQ/(HKσ√
1 + (N − 1) ρr ))
−√
((1 + ρ)2 + ρ2)N)2 > 0
�
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1140 Ganesh et al.
Biographies
Chandrasekharan Rajendran is a Professor of Operations Managementin the Indian Institute of Technology Madras. His areas of interest in-clude scheduling, simulation, meta-heuristics, total quality managementand supply chain management. He has published several articles in inter-national journals. He has publications in Naval Research Logistics, Euro-pean Journal of Operational Research, International Journal of ProductionResearch, Journal of the Operational Research Society and InternationalJournal of Production Economics. He serves as referee for many journals.He is a recipient of the Alexander von Humboldt Fellowship of Germany.
Muthusamy Ganesh is a PhD student in the area of Operations Man-agement in the Indian Institute of Technology Madras. His areas of
interest include supply chain management, inventory management andsimulation.
Srinivasan Raghunathan is a Professor of Information Systems in theSchool of Management, The University of Texas at Dallas. He obtaineda B.Tech degree in Electrical Engineering from IIT, Madras, a Post Gradu-ate Diploma in Management from IIM, Calcutta and a Ph.D. in BusinessAdministration from the University of Pittsburgh. His current research in-terests are in the economics of information security and collaboration andsecurity in supply chains. His papers have been published in leading jour-nals such as Management Science, Information Systems Research, JMIS,
various IEEE Transactions, IIE Transactions, and others. He serves asan associate editor of ISR and Journal of Information Technology andManagement.
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