the value of information sharing in a supply chain with a seasonal demand process

Computers & Industrial Engineering 65 (2013) 97–108

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Computers & Industrial Engineering

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The value of information sharing in a supply chain with a seasonal demand process

Dong Won Cho, Young Hae Lee ⇑Department of Industrial and Management Engineering, Hanyang University, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Available online 14 December 2011

Keywords:Supply chainBullwhip effectInformation sharingSeasonal demand process

0360-8352/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.cie.2011.12.004

⇑ Corresponding author. Tel.: +82 31 400 5262; faxE-mail addresses: [email protected] (D.W

(Y.H. Lee).

This paper considers a two echelon seasonal supply chain model that consists of one supplier and oneretailer, with the assumption that external demand from the customer follows a seasonal autoregressivemoving average (SARMA) process, including marketing actions that cannot be deduced from the otherparameters of the demand process. In our model, the supplier and the retailer employ order-up-to policyto replenish their inventory. In order to evaluate the value of information sharing in a two echelon sea-sonal supply chain, three levels of information sharing proposed by Yu, Yan, and Cheng (2002) are used.The results for optimal inventory policies under these three levels of information sharing are derived. Weshow that the seasonal effect has an important impact on optimal inventory policies of the supplier underthe three levels of information sharing. Our findings also demonstrate that the replenishment of lead timemust be less than the seasonal period in order to benefit from information sharing. Thus, this result pro-vides managers with managerial insights to improve supply chain performance through informationsharing integration partnerships.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Supply chain management (SCM) is the management of mate-rial and information flows both in and between facilities, such asvendors, manufacturing and assembly plants, and distribution cen-ters (Thomas & Griffin, 1996). The seasonal phenomenon ofdemand, which exists when a series fluctuates according to someseasonal factor, is a common occurrence in many supply chains(Chopra & Meindl, 2007). Seasonal phenomena may stem from fac-tors such as weather, which affects many business and economicactivities like tourism and home building; custom events likeChristmas, which is closely related to sales such as jewelry, toys,cards, and stamps; and graduation ceremonies in the summermonths, which are directly related to the labor force status in thesemonths (Wei, 1990). The phenomenon can potentially cause mis-matching supply with demand in supply chains and increase thecosts of inventory and stock-outs. If the seasonal phenomenonoccurs at a supply chain, appropriate supply chain managementinitiatives need to be implemented for efficient and effective mate-rial and information flows. One key initiative that is commonlymentioned is information sharing between partners in a supplychain (Lee, So, & Tang, 2000).

Information sharing is a collaborative program in which thesupplier obtains and utilizes the demand and inventory status of

ll rights reserved.

: +82 31 602 7730.. Cho), [email protected]

the retailer. Information sharing is the cornerstone of a numberof supply chain management initiatives. These initiatives includecollaborative planning, forecasting, and replenishment (CPFR), effi-cient consumer response (ECR), vendor-managed inventory (VMI),and continuous replenishment programs (CRP). Information shar-ing is a basic element to implement CPFR and ECR. Often times,information sharing is embedded in programs like VMI or CRP(Lee et al., 2000).

Generally, there are three levels of information sharing in a sup-ply chain (Yu et al., 2002). Level 1 is described as decentralizedcontrol recognized as the traditional ordering process. At this level,the supplier and retailer belong to different organizations, and theyoperate in a decentralized fashion. That is, each party makes hisinventory decision according to his own forecast. Level 2 is referredto as coordinated control. The supplier and the retailer employtheir inventory policy. However, the supplier receives the cus-tomer’s demand and ordering information from the retailer. There-fore, the supplier places its order not only according to the orderfrom the retailer but also according to the customer’s demandinformation. Level 3 is described as having centralized control, inwhich case the supplier and the retailer share real-time demandinformation by using information technologies, such as electronicdata exchange (EDI) and the Internet. Therefore, the supplier estab-lishes its inventory policy based on the demand information fromthe retailer directly. As a result, the supplier and retailer canachieve optimal performance to minimize the total inventory costin the supply chain. This is the so-called VMI practice.

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98 D.W. Cho, Y.H. Lee / Computers & Industrial Engineering 65 (2013) 97–108

Information sharing can cause changes in the supply chain oper-ation and improvement in the supply chain performance. Espe-cially, by information sharing, supply chain members can managetheir inventory on the basis of customers’ demands, thus removingor mitigating harmful problems resulting from the bullwhip effect.The bullwhip effect is the phenomenon of demand variabilityamplification when one moves away from the customer to thesupplier in a supply chain. This effect poses very severe problemsfor a supply chain, such as lost revenues, inaccurate demand fore-casts, missed production schedule, ineffective transportation, andexcessive inventory investment. Supply chain partners that sharedemand information can work as a single entity. Therefore, theyhave a greater understanding of the customer and are better ableto react to the changing marketplace. Sharing this informationhas enabled some companies to reduce their cycle times, fulfill or-ders more quickly, cut out millions of dollars in excess inventory,and improve customer service (Stein & Sweat, 1998).

In spite of the importance of managing the seasonal demand,there is a lack of studies concerned with information sharing in aseasonal supply chain. Generally, sharing nonseasonal demandinformation at each stage of the supply chain, i.e., the retailer,warehouse, and supplier, is known to have a beneficial effect.While sharing demand information in a seasonal supply chain isexpected to have a similar effect, the relationship between variousparameters, such as the seasonal parameter, lead time, and corre-lation coefficient, has still not been clearly elucidated. As such, thevalue of demand information sharing in a seasonal supply chainneeds to be evaluated and analyzed in order to provide a betterunderstanding of the impact of the seasonal phenomenon on thesupply chain performance through information sharing.

In this paper, we evaluate the value of information sharing in atwo-echelon seasonal supply chain that consists of one supplierand one retailer. The retailer faces the external customer’s demandprocess. The approach of using a demand process, such as the first-order autoregressive process, AR (1), to capture the behavior of thedemand process may not be good for practical application. Instead,we need to examine a more realistic demand process, such as a sea-sonal autoregressive moving average process, SARMA (1,0) � (0,1)s

with a seasonal period s, s = 1, 2, . . ., to study the value of demandinformation sharing. Many random processes seldom have thecharacteristic of a pure autoregressive process or a pure movingaverage process (Pindyck & Rubinfeld, 1991). And many businessand economic processes contain a seasonal phenomenon that re-peats itself after a regular period of time (Wei, 1990). If a process in-cludes both a seasonal component and nonseasonal components,the seasonal component may be stochastic and correlated withnonseasonal components (Wei, 1990). Therefore, a SARMA processoften fits the demand process better than an only pure and nonsea-sonal process. A SARMA (1,0) � (0,1)s for the case of s = 1 is thesame as the ARMA (1,1) process, whereas, for s > 1, it follows a pro-cess which considers both a seasonal and nonseasonal process.Thus, SARMA (1,0) � (0,1)s is a multiplicative process which in-cludes a nonseasonal process as well as a seasonal process. On theone hand, ARMA process for nonseasonal and stationary time series,denoted as ARMA (p,q), contains both pth order autoregressive andqth order moving average terms (Wei, 1990). On the other hand,ARIMA process for nonseasonal and nonstationary time series, de-noted as ARIMA (p,d,q), includes the degree of homogeneity d,called the dth difference, that is, the number of times that the seriesmust be differenced to produced a stationary series, together with astationary, invertible ARMA process (Wei, 1990). So, taking dth dif-ferences of an ARIMA (p,d,q), produces a stationary ARMA (p,q) pro-cess. We attempt to compare the value of information sharing foreach of the three levels mentioned earlier, and we identify theparameters that affect the magnitude of the benefits derived frominformation sharing in a seasonal supply chain.

We make the following original contributions.

(1) The current research quantifies the value of informationsharing in a seasonal supply chain, while most researchersevaluated the value of information sharing under theassumption of nonseasonal demand process.

(2) Most importantly, we identify the parameters that affect themagnitude of benefits derived from information sharing in aseasonal supply chain. Based on the analysis of the observa-tions from numerical experiments, we provide supply chainmanager with managerial insights on information sharing ina seasonal supply chain.

The paper is organized as follows. In Section 2 we review rele-vant literature. Section 3 describes the modeling framework, andSection 4 derives the optimal inventory policies under the threelevels of information sharing. Then, in Section 5, we evaluate theimpact of information sharing on the inventory and the expectedcosts of the supplier. The results of the numerical experimentsare presented in Sections 6, and 7 presents the conclusions andinsights of this study.

2. Literature review

There are many studies on information sharing in a supplychain, and examples of a simple two-stage supply chain with onesupplier and one retailer are as follows. Gavirneni, Kapusciski,and Tayur (1999) studied three pattern cases of information shar-ing between a retailer and a supplier. In their analysis, a retailerfaces i.i.d distributed demands and employs the (s,S) periodic re-view inventory policy. Assuming no delivery lead time, the authorsshowed that the benefit due to information sharing, i.e., the sup-plier’s cost savings, increases with production capacity. Lee et al.(2000) studied the value of information sharing in a two-echelonsupply chain structure. The customer demand process faced bythe retailer is an AR (1). They found that the benefit of sharing de-mand information can greatly increase when demands are signifi-cantly correlated over time. Raghunathan (2001), however, showedthat the value of information sharing can be insignificant in thiscase. Since the retailer orders and replenishes its stock to supplycustomer demand, the retailer’s order history contains informationabout the customer demand. Therefore, the supplier can deducethe customer demand with a high degree of accuracy by usingthe order history. However, if the supplier realizes significant sav-ings from information sharing, the actual demand should not bededuced from other parameters of the demand process. Yu et al.(2002) compared and analyzed three levels of information sharingbetween a retailer and a supplier, as mentioned in Section 1. Theyshowed that the supplier could achieve inventory reduction andcost savings under an information sharing-based supply chainpartnership. Hsiao and Shieh (2006) evaluated the value of infor-mation sharing for cases where the external demand occurring atthe point of the customer is assumed to follow ARIMA (0,1,q). Theyfound that the order variance decreases with information sharingand that the benefit of sharing demand information can increasemarkedly when q increases. Yao and Dresner (2008) comparedthe benefits between the retailer and the supplier realized throughinformation sharing, CRP, or VMI with those of no informationsharing. They showed that information sharing, CRP, and VMI bringvarying benefits in terms of inventory reduction and showed thatthe benefits are not consistently distributed between the retailerand the supplier. They also note that managers may decide theproduct sets and replenishment frequency for improved benefitrealization under CRP and VMI.

Studies on a two-echelon supply chain, with one supplier andmultiple retailers have also been extensive. Aviv and Federgruen

D.W. Cho, Y.H. Lee / Computers & Industrial Engineering 65 (2013) 97–108 99

(1998) considered a supply chain structure with a supplier andmultiple retailers. Customers’ demands are stochastic and occurat the retail locations only. They compared the effectiveness of acentralized system involving information sharing, like with VMI,with that of a traditional, decentralized system without informa-tion sharing between parties, as well as with a decentralizedsystem in which information is shared continuously but decisionsare made individually by different parties. Authors of a numericalstudy reported that the average improvement in supply chain costfrom a decentralized system without information sharing to adecentralized system with information sharing is around 2%, witha range of 0–5%, while VMI reduces cost, relative to a decentralizedsystem with information sharing, by 0.4–9.5% and, on average, by4.7%. They also showed that these savings also accrue to the sup-plier. Cachon and Fisher (2000) evaluated the value of downstreaminventory information in one supplier and multiple identical retail-ers. The periodic customer demands at the retailers are i.i.d., bothacross retailers and across time periods. Each retailer follows an(Rr,nQr) policy. Also, the inventory policy of the supplier employs(Rs,nQs). The replenishment lead time at the supplier is constant,as are the transportation lead times from the supplier to the retail-ers. A linear inventory holding and penalty costs are incurred byboth the supplier and the retailers. They compared the system-wide holding and backorder costs of the traditional system inwhich the supplier only observes the retailers’ orders with thatof a full information sharing system where the supplier has accessto the retailers’ inventory status on a real-time basis. In a numeri-cal study with 768, they reported that information sharing reducessupply chain costs by 2.2% on average, with the maximum at 12%.Raghunathan (2003) analyzed the value of demand informationsharing in a two-stage supply chain with one manufacturer andN retailers. Customers’ demands for a single item occur at theretailers and follow AR (1) processes, and retailers’ demands dur-ing that time period may be correlated. By using the Shapley valueconcept from game theory, Raghunathan (2003) analyzed the ex-pected manufacturer and retailer shares of the surplus generatedfrom information sharing. Also, he showed that a higher correla-tion increases (reduces) the manufacturer (retailer) surplus. How-ever, the declining marginal manufacturer surplus under a high-correlation condition may induce the manufacturer to limit thenumber of retailer partners. Cheng and Wu (2005) analyzed the va-lue of information sharing in terms of inventory and expected costin a two-level supply chain with one supplier and multiple retail-ers. They derived the optimal inventory policy under the threelevels of information sharing mentioned earlier and showed thatboth the inventory level and the expected cost of the manufacturerdecrease with an increase in the level of information sharing.

Fig. 1. Comparison of information flows am

Wu and Cheng (2008) quantified the impact of informationsharing on inventory and the expected cost in a multiple-echelonsupply chain through a general end demand process. They consid-ered a demand process with a retailer’s marketing action thatshould not be deduced from its other parameters. The three levelsof information sharing mentioned earlier in a three-echelon supplychain are given, and the optimal inventory policy under each levelis derived. They also showed that both the inventory level and theexpected cost of the distributor and the manufacturer decreasewith an increase in the level of information sharing.

The literature clearly recognizes that inventory reduction can bereached on the basis of information sharing. This paper contributesto the literature by analyzing the value of information sharing inorder to improve seasonal supply chain performance.

3. Modeling framework

As mentioned earlier, we consider a two-echelon seasonalsupply chain that consists of one supplier and one retailer. Fig. 1presents a descriptive comparison of information flows for eachof the three levels of information sharing.

Both retailer and supplier follow an order-up-to-level periodicreview policy for the review of their inventories and replenish itsinventory from the upstream party every period. The retailer facesexternal demand for a single product from end customers, wherethe underlying demand process is a SARMA (1,0) � (0,1)s demandprocess that includes the retailer’s action. Let Dt, t = 1, 2, . . ., be thedemand process at time period t

Dt ¼ dþ /Dt�1 þ cXt�1 þ et �Het�s; ð3:1Þ

where Dt is the observed demand for the period t, d > 0 is a priorestimate of the average demand, /, c and H is constant coefficientsexpressing the degree of correlation between the demand at thepresent period and the demand at the previous period, the demandat the present period and the retailer’s action taken at the previousperiod, and the demand at the present period and the demand atthe previous seasonal period s, s = 1, 2, . . ., respectively. Also, et isthe error term, which is independent and identically normally dis-tributed with mean 0 and variance r2.

In order to be stationary and invertible for the SARMA(1,0) � (0,1)s process, we should have �1 < / < 1, �1 < H < 1, and�1 < c < 1, respectively. Like Lee et al. (2000), it is assumed thatr is significantly smaller than d, so that the probability of a nega-tive demand is negligible. As with the introduction of Wu andCheng (2008), the demand process includes Xt�1 to represent theretailer’s actions, such as a promotion, price reduction, and adver-tising taken during time period t � 1 that may affect the demand

ong three levels of information sharing.

Table 1Natural logarithms of monthly passenger totals (measured in thousands) in international air travel.

January February March April May June July August September October November December

1949 4.718 4.771 4.883 4.860 4.796 4.905 4.997 4.997 4.913 4.779 4.644 4.7711950 4.745 4.836 4.949 4.905 4.828 5.004 5.136 5.136 5.063 4.890 4.736 4.9421951 4.977 5.011 5.182 5.094 5.147 5.182 5.293 5.293 5.215 5.088 4.984 5.1121952 5.142 5.193 5.263 5.199 5.209 5.384 5.438 5.489 5.342 5.252 5.147 5.2681953 5.278 5.278 5.464 5.460 5.434 5.493 5.576 5.606 5.468 5.352 5.193 5.3031954 5.318 5.236 5.460 5.425 5.455 5.576 5.710 5.680 5.557 5.434 5.313 5.4341955 5.489 5.451 5.587 5.595 5.598 5.753 5.897 5.849 5.743 5.613 5.468 5.6281956 5.649 5.624 5.759 5.746 5.762 5.924 6.023 6.004 5.872 5.724 5.602 5.7241957 5.753 5.707 5.875 5.852 5.872 6.045 6.146 6.146 6.001 5.849 5.720 5.8171958 5.829 5.762 5.892 5.852 5.894 6.075 6.196 6.225 6.001 5.883 5.735 5.8201959 5.886 5.835 6.006 5.981 6.040 6.157 6.306 6.326 6.138 6.009 5.892 6.0041960 6.033 5.969 6.038 6.133 6.157 6.282 6.433 6.407 6.230 6.133 5.966 6.068


during the next period. According to Wu and Cheng (2008), it is as-sumed that Xt is normally distributed with mean lx and variancer2

x because there are many such actions whose effects on the de-mand are small and independent of one another. So Xt(t = 1,2, . . .)is independent of one another, and the retailer shares with the sup-plier the value of Xt at the ends of time period t in the informationsharing level. Thus, since Xt cannot be inferred by the supplierusing the retailer’s order history, information sharing is valuableto the supplier.

As a representation of real world data to show the similar con-text of the above SARMA demand process, Table 1 shows naturallogarithms of monthly passenger totals (measured in thousands)in international air travel from 1949 to 1960 quoted by Brown(1962). The series shows a marked seasonal pattern since travelis at its highest in the late summer months, while a secondary peakoccurs in the spring. The seasonal effect implies that an observa-tion for a particular month, say April, is related to the observationsfor previous Aprils. However, the error term would not in generalbe uncorrelated. For example, the total of airline passengers inApril, 1960, while related to previous April totals, would also berelated to totals in March of 1960, February of 1960, January of1960, etc. So, in the example above, the basic time interval t isone month and the seasonal period is s = 12 months. In result,the seasonal effect occurs per 12 months. And in the basic timeinterval, the retailer’s action such as air travel promotion and airtravel advertising taken at the previous period can has the impacton the total of airline passengers’ at the present period. However,the retailer’s action taken at each period is independent of oneanother. And the retailer shares the retailer’s action with the sup-plier at the ends of time period in the information sharing level.Thus, the supplier cannot know the value of the retailer’s actionand be prepare for the retailer’s order without information sharingbetween them.

Within each period, the retailer’s ordering process occurs in thefollowing sequences. Before the end of time period t, t = 1, 2, . . ., theretailer first observes the arriving demands, then decides an ordersize to bring his inventory position to order-up level Sr

t and placesan order of size Yt. This order placed by the retailer is received atthe beginning of time period t + lr + 1. Any orders for the retailerthat are not fulfilled immediately due to excess demand are back-ordered with penalty cost.

Next, the supplier operates his ordering process as follows. Atthe end of time period t, the supplier receives and delivers the re-quired order size Yt to the retailer. After the supplier observes Yt,the supplier places an order with his upstream party at the endof period t to bring his inventory position to order-up level Sd

t . Thisorder will arrive at the beginning of t + ld + 1. If the supplier doesnot have enough stock for this order, it will replenish its inventoryfrom the upstream party at the beginning of time period t + ld + 1with an additional cost representing the penalty cost of this

stock-out. Like Lee et al. (2000), we further assume that no fixedordering cost is incurred when placing an order and that unit hold-ing costs and shortage costs per time period for each the retailerand the supplier, denoted as hr, hd, pr and pd, are stationary overtime.

4. Optimal inventory policies

We derive optimal inventory policies of the retailer and the sup-plier under each level of information sharing to find that theretailer’s optimal inventory policy remains the same under thethree information sharing scenarios. Like Yu et al. (2002), this out-come is based on two assumptions. One is that the retailer knowsthe perfect information about the customer’s demand and cannotobtain additional customer demand information, except for theinformation shared with the supplier. The other is that the retai-ler’s lead time lr is a fixed amount of time between the arrival ofthe replenishment from the supplier and the time at which the re-tailer’s order was placed to the supplier. Thus, without any infor-mation sharing between the supplier and the retailer, the retailermust estimate its lead time considering the supplier’s reliability.Under existing information sharing scenarios, the supplier sharesthe lead time with the retailer or even makes the supplier shortenthe retailer’s lead time. The retailer can also achieve more accuratelead time information with information sharing-based partner-ships. As such, the retailer can obtain the benefits from a shortand accurate lead time. However, we do not consider this situationfor the retailer.

Therefore, there is no change in the way the retailer places itsorders at the three levels of information sharing. Instead, we eval-uate the supplier’s optimal inventory policies under the three dif-ferent information sharing scenarios and analyze the inventoryreduction and cost savings associated with information sharing.

4.1. Optimal inventory policy of the retailer

Consider the first optimal inventory policy of the retailer. Sincethe retailer follows an order-up-to periodic inventory policy, thegoal of this ordering policy is to bring the actual inventory towardsthe desired inventory position. The retailer’s order size Yt is givenby

Yt ¼ Dt þ Srt � Sr

t�1; ð4:1Þ

where Srt is the desired order-up-to level of period t, Sr

t�1 is the or-der-up-to level of period t � 1, and Dt is the customer’s demand.We need to decide the optimal order-up-to level S�rt that minimizesthe total expected holding and shortage costs in period t + lr + 1. Wedenote the total demand over the lead time as

Plrþ1i¼1 Dtþi. By using

the recursive relationship of Dt given in (3.1), we have

s:


Xlrþ1

i¼1

Dtþi ¼1

1� /

dPlrþ1

i¼1ð1� /iÞ þ ð1� /lrþ1Þ/Dt þ c

Plrþ1

i¼1ð1� /iÞXtþlrþ1�i

þPlrþ1

i¼1ð1� /iÞetþlrþ2�i �H

Plrþ1

i¼1ð1� /iÞet�sþlrþ2�i

0BBB@

1CCCA:

ð4:2Þ

Let mrt ¼ E

Plrþ1i¼1 DtþijDt

h iand vr

t ¼ VarPlrþ1

i¼1 DtþijDt

h ibe the con-

ditional expectation and the conditional variance ofPlrþ1

i¼1 Dtþi,respectively. Then,

mrt ¼

11� /

dXlrþ1

i¼1

ð1� /iÞ þ ð1� /lrþ1Þ/Dt þ cð1� /lrþ1ÞXt

þc lr �ð1� /lr Þ/

1� /

!lx

!ð4:3Þ

v rt ¼

re1�/

� �2 Plrþ1

i¼1ð1� /iÞ2ð1þH2Þ � 2H

Plr�sþ1

i¼1ð1� /iÞð1� /sþiÞ

�

þ crx1�/

� �2Plri¼1ð1� /iÞ2

�; s 6 lr

re1�/

� �2 Plrþ1

i¼1ð1� /iÞ2ð1þH2Þ þ crx

1�/

� �2Plri¼1ð1� /iÞ2

� �; s > lr

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

:

ð4:4ÞTherefore, the optimal order-up-to level S�rt of the retailer at timeperiod t is

S�rt ¼ mrt þ kr

ffiffiffiffiffiv r

t

p; ð4:5Þ

where kr = U�1[pr/(pr + hr)] for the standard normal distributionfunction U.

4.2. Optimal inventory policy of the supplier

The supplier faces and meets the retailer’s order. If there is notenough stock, it will replenish its inventory from its upstreamparty and it will receive the order at time t + ld + 1. Then, from(4.1)–(4.5), we have

Yt ¼ Dt þ S�rt � S�rt�1

¼ ð1� /lrþ2Þ1� /

Dt�ð1� /lrþ1Þ/

1� /Dt�1þ

cð1� /lrþ1Þ1� /

ðXt � Xt�1Þ: ð4:6Þ

We denote the total retailer’s order size over the lead time asPldþ1i¼1 Yr

tþi. By using the recursive relationship of Yt given in (4.6),we have

Ytþi ¼ð1� /iÞd

1� /þ /iYt þ c

Xi�1

j¼0

ð1� /lrþ1Þ/j

1� /Xtþi�j

� cXi�1

j¼0

ð1� /lr Þ/jþ1

1� /Xtþi�j�1 þ

Xi�1

j¼0

ð1� /lrþ2Þ/j

1� /etþi�j

�Xi�1

j¼0

ð1� /lrþ2ÞH/j

1� /et�sþi�j �

Xi�1

j¼0

ð1� /lrþ1Þ/jþ1

1� /etþi�j�1

þXi�1

j¼0

ð1� /lrþ1ÞH/jþ1

1� /et�sþi�j�1 ð4:7Þ

Xldþ1

i¼1

Ytþi ¼ ld þ 1� ð1� /ldþ1Þ/1� /

!d

1� /þ ð1� /ldþ1Þ/

1� /Yt

þ cXld

i¼0

ð1� /lrþiþ1Þ1� /

Xtþldþ1�i �cð1� /lr Þð1� /ldþ1Þ/

ð1� /Þ2Xt

þXld

i¼0

ð1� /lrþiþ2Þ1� /

etþldþ1�i �ð1� /lrþ1Þð1� /ldþ1Þ/

ð1� /Þ2et

�Xld

i¼0

ð1� /lrþiþ2ÞH1� /

et�sþldþ1�iþð1�/lrþ1Þð1� /ldþ1ÞH/

ð1� /Þ2et�

ð4:8Þ

Level 1. Since there is neither information sharing nor any order-ing coordination between each party of the supply chain at this le-vel, the supplier knows nothing except the retailer’s order size Yt

when the supplier determines its optimal order-up-to level S�dt j1.Hence, Yt is looked at as a known variable and et+i (i = 0,1, . . . ,ld + 1), et�s+i(i = 0,1, . . . , ld + 1), and Xt+i(i = 0,1, . . . , ld + 1) are three

stochastic variables. Let mdt j1 ¼ E

Pldþ1i¼1 YtþijYt

h iand vd

t j1 ¼ VarPldþ1i¼1 YtþijYt

h ibe the conditional expectation and conditional

variance ofPldþ1

i¼1 Ytþi, respectively. Then,

mdt j1 ¼ ld þ 1� ð1� /ldþ1Þ/

1� /

!d

1� /þ ð1� /ldþ1Þ/

1� /Yt

þ c ld þ 1� ð1� /ldþ1Þ/1� /

!lx ð4:9Þ

s 6 ld;vdt j1 ¼

re

1� /

� �2 Pldi¼0ð1� /lrþiþ2Þ2 þ ð1�/lrþ1Þð1�/ldþ1Þ/

1�/

� �2 !

� ð1þH2Þ � 2HXld�s

i¼0

ð1� /lrþiþ2Þð1� /lrþsþiþ2Þ

þ2H/ð1� /lrþ1Þð1�/ldþ1Þð1�/lrþld�sþ3Þ1�/

!þ crx

1� /

� �2

�Xld

i¼0

ð1� /lrþiþ1Þ2 þ ð1� /lr Þð1� /ldþ1Þ/1� /

!20@

1Að4:10Þ

s> ld;vdt j1 ¼

re

1�/

� �2 Pldi¼0ð1�/lrþiþ2Þ2 þ ð1�/lrþ1Þð1�/ldþ1Þ/

1�/

� �2 !

� ð1þH2Þþ 2H/ð1�/lrþ1Þð1�/ldþ1Þð1�/lrþ2Þ1�/

ca

!

þ crx

1�/

� �2 Xld

i¼0

ð1�/lrþiþ1Þ2 þ ð1�/lr Þð1�/ldþ1Þ/1�/

!2!;

ð4:11Þ

where ca ¼1; s ¼ ld þ 10; o:w

�.

Therefore, according to Lee et al. (2000), the optimal order-up-to level S�dt j1 of the supplier at level 1 is

S�dt j1 ¼ mdt j1 þ kd

ffiffiffiffiffiffiffiffiffivd

t j1q

; ð4:12Þ

where kd = U�1[pd/(pd + hd)] for the standard normal distributionfunction U.

Level 2. While the supplier and the retailer employ their inven-tory policy, the supplier receives the customer’s demand andordering information from the retailer with coordinated control.Thus, the supplier places his order not only according to the retai-ler’s order Yt, but also the customer’s demand Dt. Then, the supplierknows Yt, Xt, et, and et�s+i(i = 0,1, . . . , ld + 1) for s P i. Howeveret+i(i = 1,2, . . . , ld + 1), et�s+i(i = 0,1, . . . , ld + 1) for s < i, andXt+i(i = 1,2, . . . , ld + 1) are stochastic variables. As a result, we have

s 6 ld;mdt j2

¼ ld þ 1� ð1� /ldþ1Þ/1� /

!d

1� /þ ð1� /ldþ1Þ/

1� /Yt

� cð1� /lr Þð1� /ldþ1Þ/ð1� /Þ2

Xt þclx

1� /ld þ 1� /lrþ1ð1� /ldþ1Þ

1� /

!

� ð1� /lrþ1Þð1� /ldþ1Þ/ð1� /Þ2

et þð1� /lrþ1Þð1� /ldþ1ÞH/

ð1� /Þ2et�s

�Xs�1

i¼0

ð1� /lrþld�sþiþ3ÞH1� /

et�i ð4:13Þ


s > ld;mdt j2

¼ ld þ 1� ð1� /ldþ1Þ/1� /

!d

1� /þ ð1� /ldþ1Þ/

1� /Yt

� cð1� /lr Þð1� /ldþ1Þ/ð1� /Þ2

Xt þclx

1� /ld þ 1� /lrþ1ð1� /ldþ1Þ

1� /

!

� ð1� /lrþ1Þð1� /ldþ1Þ/ð1� /Þ2

et þð1� /lrþ1Þð1� /ldþ1ÞH/

ð1� /Þ2et�s

�Xld

i¼0


et�sþld�iþ1 ð4:14Þ

vdt j2 ¼

re1�/

� �2 Pldi¼0ð1� /lrþiþ2Þ2 þH2 Pld�s

i¼0ð1� /lrþiþ2Þ2 � 2H

Pld�s

i¼0ð1� /lrþiþ2Þð1� /lrþsþiþ2Þ

!

þ crx1�/

� �2Pldi¼0ð1� /lrþiþ1Þ2; s 6 ld

re1�/

� �2Pldi¼0ð1� /lrþiþ2Þ2 þ crx

1�/

� �2Pldi¼0ð1� /lrþiþ1Þ2; s > ld

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

ð4:15Þ




t j2q

; ð4:16Þ

where kd = U�1[pd/(pd + hd)] for the standard normal distributionfunction U.

Level 3. Since the supplier and the retailer share real-timedemand information through information technology, the suppliercan obtain the customer’s demand information directly. Thesupplier needs to meet the retailer’s order Yt, but Yt should eventu-ally satisfy the customer’s demand Dt. Thus, we need to deduce therelationship between

Pldþ1i¼1 Ytþi and Dt, not the relationship

betweenPldþ1

i¼1 Ytþi and Yt as at other levels. From (4.6), we have

Xldþ1

i¼1

Ytþi ¼Xldþ1

i¼1

ð1� /lrþ2Þ1� /

Dtþi �ð1� /lrþ1Þ

1� /Dtþi�1

þ cð1� /lrþ1Þ1� /

ðXt � Xt�1Þ!

¼ dXldþ1

i¼1

ð1� /iÞ1� /

þ ð1� /lrþ1Þð1� /ldþ1Þ/ð1� /Þ2

!

þ ð1� /ldþ1Þ/lrþ2

1� /Dt þ

cð1� /ldþ1Þ/lrþ1

1� /Xt

þXld

i¼0

cð1� /lrþiþ1Þ1� /

Xtþldþ1�i þXld

i¼0


etþldþ1�i

�Xld

i¼0


et�sþldþ1�i: ð4:17Þ

At this level, Dt, Xt, et, and et�s+i(i = 0,1, . . . , ld + 1) for s P i areknown variables. et+i(i = 1,2, . . . , ld + 1), et�s+i(i = 0,1, . . . , ld + 1) fors < i, and Xt+i(i = 1,2, . . . , ld + 1) are stochastic variables. Hence, wehave

mdt j3 ¼

dPldþ1

i¼1

ð1�/iÞ1�/ þ

ð1�/lrþ1Þð1�/ldþ1Þ/ð1�/Þ2

!þ ð1�/ldþ1Þ/lrþ2

1�/ Dt

þ cð1�/ldþ1Þ/lrþ1

1�/ Xt þ clx1�/ 1þ ld � /lrþ1ð1�/ldþ1Þ

ð1�/Þ2

� �

�Ps�1

i¼0

ð1�/lrþld�sþiþ3ÞH1�/ et�i; s 6 ld

dPldþ1

i¼1

ð1�/iÞ1�/ þ

ð1�/lrþ1Þð1�/ldþ1Þ/ð1�/Þ2

!þ ð1�/ldþ1Þ/lrþ2

1�/ Dt

þ cð1�/ldþ1Þ/lrþ1

1�/ Xt þ clx1�/ 1þ ld � /lrþ1ð1�/ldþ1Þ

ð1�/Þ2

� �

�Pldi¼0

ð1�/lrþiþ2ÞH1�/ et�sþldþ1�i; s > ld

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

ð4:18Þ

vdt j3 ¼

re1�/

� �2 Pldi¼0ð1� /lrþiþ2Þ2 þH2 Pld�s

i¼0ð1� /lrþiþ2Þ2

�2HPld�s

i¼0ð1� /lrþiþ2Þð1� /lrþsþiþ2Þ

!

þ crx1�/

� �2Pldi¼0ð1� /lrþiþ1Þ2; s 6 ld

re1�/

� �2Pldi¼0ð1� /lrþiþ2Þ2 þ crx

1�/

� �2Pldi¼0ð1� /lrþiþ1Þ2; s > ld

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

ð4:19Þ




t j3q

; ð4:20Þwhere kd = U�1[pd/(pd + hd)] for the standard normal distributionfunction U.

5. The value of information sharing

From Section 4, we assume that the levels of information shar-ing have no impact on the way the retailer places its orders. There-fore, we compute the means and the variances of the supplier’sorder quantity for the three levels of information sharing. Wenow utilize these to evaluate the value of information sharing forboth the inventory level and the expected cost to the supplier. Also,we derive the characteristic properties of information sharingusing the variances of the supplier. Then, we shall make use ofsome of following properties to see the benefits of informationsharing.

Proposition 1. vdt does not depend on the period t, but does depend

on the seasonal period s for all three levels of information sharing.From Proposition 1, we must consider the seasonal effect on effec-

tive and efficient operation in a seasonal supply chain.

Proposition 2. For cases of level 2 and level 3, which share informa-tion between the retailer and the supplier, if�1 < H 6 0; vd

t for s 6 ldis always larger than vd

t for s > ld.

Proof. We denote vdt for s 6 ld by vd

t ð1Þ and ld for s 6 ld by ld(1).Then, we denote vd

t for s > ld by vdt ð2Þ and ld for s > ld by ld(2).


vdt ð1Þ � vd

t ð2Þ ¼re

1� /

� �2

þ crx

1� /

� �2 !

�Xldð1Þi¼0

ð1� /lrþiþ2Þ2 �Xldð2Þi¼0

ð1� /lrþiþ2Þ2 !

þ re

1� /

� �2

� H2Xldð1Þ�s

i¼0

ð1� /lrþiþ2Þ2 � 2H

�Xldð1Þ�s

i¼0

ð1� /lrþiþ2Þð1� /lrþsþiþ2Þ!: ð5:1Þ

Hence, vdt ð1Þ � vd

t ð2Þ P 0 for � 1 < H 6 0.From Proposition 2, we can know that vd

t is different from thecondition between seasonal period and lead time of the supplierwhen the retailer and the supplier share information. Especially,in order to maintain a small value of vd

t in a seasonal supply chain,a lead time of supplier must be smaller than the seasonal period.Thus, in order to obtain the benefit of information sharing in aseasonal supply chain, the supply chain members need tomaintain a lead time smaller than the seasonal period throughcollaboration.

On the one hand, it is well known in the literature thatenhancing information sharing will reduce the total demandvariance over the lead time of the supplier, thus removing ormitigating the bullwhip effect (Hsiao & Shieh, 2006; Lee et al.,2000; Yu et al., 2002). Thus, it is necessary to investigate how theeffect of information sharing has an impact on the total demandvariance over the lead time of the supplier in a seasonal supplychain. h

Proposition 3. For any �1 < / < 1, �1 < H < 1, and �1 < c < 1;vd

t j1 P vdt j2 ¼ vd

t j3.

Proof. From 4.10, 4.11, 4.15, and 4.19, we know

vdt j1 � vd

t j2 ¼

re1�/

� �2ð1�/lrþ1Þð1�/ldþ1Þ/

1�/

� �2ð1þH2ÞþH2 Pld

i¼ld�sþ1ð1�/lrþiþ2Þ2

þ 2H/ð1�/lrþ1Þð1�/ldþ1Þð1�/lrþld�sþ3Þ1�/

0BBB@

1CCCA

þ crx1�/

� �2 ð1�/lr Þð1�/ldþ1Þ/1�/

� �2; s 6 ld

re1�/

� �2ð1�/lrþ1Þð1�/ldþ1Þ/

1�/

� �2ð1þH2Þ þH2 Pld

i¼0ð1� /lrþiþ2Þ2

þ 2H/ð1�/lrþ1Þð1�/ldþ1Þð1�/lrþ2Þ1�/ ca

0BB@

1CCA

þ crx1�/

� �2 ð1�/lr Þð1�/ldþ1Þ/1�/

� �2; s > ld;

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

ð5:2Þ

where ca ¼1; s ¼ ld þ 10; o:w

�. Hence, vd

t j1 � vdt j2 P 0.

From Proposition 3, we see that vdt j1 P vd

t j2 ¼ vdt j3. This shows

that the total demand variance over the lead time of the supplierdecreases with an increase in the level of information sharing. h

Proposition 4. vdt j1 � vd

t j2(or vdt j3Þ has the following properties.

(a) For 0 6 / < 1, 0 6H < 1 and �1 < c < 1; vdt j1 � vd

t j2 increaseas /, re, rx, lr, or ld increase.

(b) For 0 6 / < 1, 0 6H < 1 and �1 < c 6 0;vdt j1 � vd

t j2decreases as c increases.

(c) For 0 6 / < 1, 0 6H < 1 and 0 < c 6 1;vdt j1 � vd

t j2 increasesas c increases.

(d) For 0 6 / < 1, �1 < c < 1,s > ld + 1, and �1 < H 6 0;vdt j1�

vdt j2 decreases as H increases.

(e) For 0 6 / < 1, �1 < c < 1,s > ld + 1, and 0 6 H < 1; vdt j1 � vd

t j2increases as H decreases.This proof is clearly given from (5.2).From Proposition 4, we see the impact of the parameter onthe effect of information sharing.

5.1. Inventory reduction

Since the supplier employs an order-up-to inventory policy, weuse the average inventory level discussed in Silver and Petersen(1985) for any order-up-to system with St being the order-up-to le-vel, Yt being the demand at period t, and

Plþ1i¼1Ytþi being the total

demand from period t + 1 to period t + l + 1. The average inventorylevel can be approximated by

It ¼ St � EXlþ1

i¼1

Ytþi

" #þ E½Yt�

2¼ St �mt þ

E½Yt �2

; ð5:3Þ

where mt is the mean ofPlþ1

i¼1Ytþi.Thus, the average inventory level of the supplier can be approx-

imated by

Idt j1 ¼ S�dt j1 �md

t j1 þE½Yt�

2¼ kd


t j1q

þ dþ c2ð1� /Þ ; ð5:4Þ


t j2 þE½Yt�

2¼ kd


t j2q

þ dþ c2ð1� /Þ ; ð5:5Þ


t j3 þE½Yt�

2¼ kd


t j3q

þ dþ c2ð1� /Þ : ð5:6Þ

From Proposition 3, we see that Idt j1 P Id

t j2 ¼ Idt j3. This shows

that the inventory level of the supplier decreases with an increasein the level of information sharing. And from Proposition 4, we seehow to change the average inventory level of the supplier when theparameter of information sharing changes.

5.2. Expected cost reduction

In order to develop the supplier’s expected cost, we denote L(x)as the right loss function for the standard normal distribution,where

LðxÞ ¼Z 1

xðz� xÞdUðzÞ ð5:7Þ

and U(z) is the standard normal probability distribution.According to Lee et al. (2000), for the case of St ¼ mt þ k

ffiffiffiffiffiv tp

being the order-up-to level, the supplier’s expected holding andshortage costs at time period t + l + 1 is

Ct ¼ E pZ 1

St

ðx� StÞdFtðxÞ þ hZ St

�1ðSt � xÞdFtðxÞ

� �¼

ffiffiffiffiffiv tp½ðhþ pÞLðkÞ þ hk�; ð5:8Þ

where p and h are the unit holding cost and unit shortage cost of thesupplier, respectively, k = U�1[p/(p + h)], and Ft (x) � N(mt,vt).

Hence, the supplier’s expected holding and shortage costs aregiven by

Cdt j1 ¼


t j1q

½ðhd þ pdÞLðkdÞ þ hdkd�; ð5:9Þ

Cdt j2 ¼


t j2q

½ðhd þ pdÞLðkdÞ þ hdkd�; ð5:10Þ

Cdt j3 ¼


t j3q

½ðhd þ pdÞLðkdÞ þ hdkd�: ð5:11Þ

From Proposition 3, we see that Cdt j1 P Cd

t j2 ¼ Cdt j3. Thus, the

supplier can decrease the expected costs by increasing the levelof information sharing. From Proposition 4, we see how to changethe supplier’s expected costs when the parameter of informationsharing changes.

20

30

40

Ave

rage

cos

t

Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, l =4


From Section 4, we see that there is no difference in the totaldemand variance over the lead time of the supplier between infor-mation sharing at level 2 and level 3. As a result, the average inven-tories and costs derived for the supplier at level 2 and level 3 areequal. Thus, we only execute numerical experiments of level 1and level 2 to analyze the impacts of the parameters included ineach level of information sharing.

φ0

10

-0.9 -0.6 -0.3 0.3 0.6 0.9

d

Level 2, =3, ld =2

Fig. 3. Impact of / on average supplier’s cost.

s

0

5

10

15

20

25

5 10 15 20 25 30

Ave

rage

inv

ento

ry

Level 1, ld =30

Level 1, ld =4

Level 2, ld =30

Level 2, ld =4

Fig. 4. Impact of s on average supplier’s on inventory.

6. Numerical analysis

In this section, we verify our analysis and illustrate the impactsof the demand and inventory parameters on the value of level 1and level 2 of information sharing in a seasonal supply chain usingnumerical experiments. The results of these numerical experi-ments highlight the managerial application of our model andfindings.

We vary the autoregressive coefficient / to examine how theparameter affects the magnitude of inventory reductions and costsavings for the supplier in level 1 and level 2 of information shar-ing. And we specify fixed values for all other parameters and vary /from �1 to 1. Average supplier’s on inventory derived from level 1and level 2 of information sharing is given by Eqs. (5.4) and (5.5),respectively. And average supplier cost derived from level 1 and le-vel 2 of information sharing is given by Eqs. (5.9) and (5.10),respectively. To analyze the impact of the autoregressive coeffi-cient / on average supplier’s on inventory and average supplier’scost, the supplier’s cost parameter values are given as hd = 1 andpd = 19. And we set d = 10, c = 0.1, H = 0.3, re = 1, rx = 1, andlr = 2. Also, we set s = 3 and ld = 4 for s > ld, or s = 3 and ld = 2 fors 6 ld in both level 1 and level 2 of information sharing.

Figs. 2 and 3 illustrate the results of the impact of / on the sup-plier’s inventory reductions and cost savings. We observe that theaverage inventory and cost of the supplier are increasing when / isincreasing. Also, Fig. 2 and 3 suggest that the inventory reductionsand cost savings of the supplier increase with an increase in the le-vel of information sharing, and these inventory reductions and costsavings are greater when / is larger. The phenomenon depicted inFigs. 2 and 3 are consistent with the analytical findings presentedin Section 5 (see Propositions 3 and 4a). When / is large, the degreeof autocorrelated demand is high, hence, the current demand infor-mation is very valuable for predicting future demands. Also, thecollaboration between the supplier and the retailer using informa-tion sharing provides greater inventory reduction and cost saving.

We shift the seasonal period s to investigate how the parameteraffects the magnitude of inventory reductions and cost savings forthe supplier in level 1 and level 2 of information sharing. And wespecify fixed values for all other parameters and vary s from 5 to30. Average supplier’s on inventory obtained from level 1 and level2 of information sharing is given by Eqs. (5.4) and (5.5), respec-tively. And average supplier cost obtained from level 1 and level2 of information sharing is given by Eqs. (5.9) and (5.10), respec-

φ

0

20

40

60

80

-0.9 -0.6 -0.3 0.3 0.6 0.9

Ave

rage

inv

ento

ry

Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 2. Impact of / on average supplier’s on inventory.

tively. To investigate the impact of the seasonal period s on averagesupplier’s on inventory and average supplier’s cost, the supplier’scost parameter values are given as hd = 1 and pd = 19. And we setd = 10, c = 0.1, / = 0.3, H = 0.3, re = 1, rx = 1, and lr = 2. Also, weset ld = 4 for s > ld, or ld = 30 for s 6 ld in both level 1 and level 2of information sharing.

Figs. 4 and 5 report the results of the impact of s on the sup-plier’s inventory reductions and cost savings. We observe that fors > ld, s does not have an impact on the supplier’s inventory levelor cost at any level of information sharing, or for s 6 ld, we observethat the supplier’s inventory and cost are increasing when s isincreasing. Hence, Figs. 4 and 5 suggest that the inventory reduc-tions and cost savings of the supplier decrease with an increasein the level of information sharing. The phenomena depicted inFigs. 4 and 5 are consistent with the analytical findings presentedin Section 5 (see Propositions 3 and 4a). Thus, both the supplierand its upstream member need to collaborate to have a smallerld than s to remove the impact of s on the supplier performancefor each level of information sharing.

We vary the seasonal moving average coefficient H to investi-gate how the parameter affects the magnitude of inventory reduc-tions and cost savings for the supplier in level 1 and level 2 ofinformation sharing. And we specify fixed values for all otherparameters and vary H from �1 to 1. Average supplier’s on inven-tory in level 1 and level 2 of information sharing is given by Eqs.(5.4) and (5.5), respectively. And average supplier cost in level 1and level 2 of information sharing is given by Eqs. (5.9) and(5.10), respectively. To examine the impact of the seasonal periodH on average supplier’s on inventory and average supplier’s cost,the supplier’s cost parameter values are given as hd = 1 andpd = 19. And we set d = 10, c = 0.1, / = 0.3, re = 1, rx = 1, and lr = 2.Also, we set s = 3 and ld = 4 for s > ld, or s = 3 and ld = 2 for s 6 ldin both level 1 and level 2 of information sharing.

Figs. 6 and 7 show the results of the impact of H on the sup-plier’s inventory reductions and cost savings. We observe that for

0

5

10

15

20

5 10 15 20 25 30

Ave

rage

cos

t

s

Level 1, ld =30

Level 1, ld =4

Level 2, ld =30

Level 2, ld =4

Fig. 5. Impact of s on average supplier’s cost.

Θ

0

5

10

15

20

-0.9 -0.6 -0.3 0.3 0.6 0.9

Ave

rage

inve

ntor

y

Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =2

Level 2, =3, ld =4

Fig. 6. Impact of H on average supplier’s on inventory.

Θ

0

5

10

15

-0.9 -0.6 -0.3 0.3 0.6 0.9

Ave

rage

cos

t Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 7. Impact of H on average supplier’s cost.

lr

0

50

100

150

5 10 15 20 25 30A

vera

ge in

vent

ory Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 8. Impact of lr on average supplier’s on inventory.

l r

0

50

100

150

5 10 15 20 25 30

Ave

rage

cos

t

Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 9. Impact of lr on average supplier’s cost.


0 6 / < 1, �1 < c < 1, s > ld + 1 and �1 < H 6 0, the supplier’s inven-tory reductions and cost savings decreases as H increases, or for0 6 / < 1, �1 < c < 1, s > ld + 1, and 0 6H < 1, the supplier’s inven-tory reductions increases and cost savings as H decreases. Thephenomenon depicted in Figs. 6 and 7 are consistent with the ana-lytical findings presented in Section 5 (see Proposition 4d and e).And for cases of level 2, which shares information between the re-tailer and the supplier, if �1 < H 6 0; vd

t for s 6 ld is always largerthan vd

t for s > ld (see Proposition 2). Thus, the collaboration be-tween the supplier and its upstream party for a case with smallerld than s in level 2 of information sharing can remove the impact ofH on the total demand variance over the lead time of the supplier.

We shift the retailer’s lead time lr to examine how the parame-ter affects the magnitude of inventory reductions and cost savingsfor the supplier in level 1 and level 2 of information sharing. Andwe specify fixed values for all other parameters and vary lr from5 to 30. Average supplier’s on inventory derived from level 1 andlevel 2 of information sharing is given by Eqs. (5.4) and (5.5),respectively. And average supplier cost derived from level 1 and le-vel 2 of information sharing is given by Eqs. (5.9) and (5.10),respectively. To examine the impact of lr on average supplier’s oninventory and average supplier’s cost, the supplier’s cost parametervalues are given as hd = 1 and pd = 19. And we set d = 10, c = 0.1, /= 0.3, H = 0.3, re = 1, and rx = 1. Also, we set s = 3 and ld = 4 fors > ld, or s = 3 and ld = 2 for s 6 ld in both level 1 and level 2 of infor-mation sharing.

Figs. 8 and 9 illustrate the results of the impact of lr on the sup-plier’s inventory reductions and cost savings. We observe that theaverage inventory and cost for the supplier are increasing when lris increasing. Also, Figs. 8 and 9 suggest that the inventory reduc-tions and cost savings of the supplier increase with a higher level ofinformation sharing. These inventory reductions and cost savingsare greater when lr are larger. The phenomena depicted in Figs. 8and 9 are consistent with the analytical findings presented in Sec-tion 5 (see Propositions 3 and 4a). Thus, although there is in the

case of ld 6 s, supply chain members need to collaborate to reduceld.

We shift the supplier’s lead time ld to examine how the param-eter affects the magnitude of inventory reductions and cost savingsfor the supplier in level 1 and level 2 of information sharing. Andwe specify fixed values for all other parameters and vary ld from5 to 30. Average supplier’s on inventory derived from level 1 andlevel 2 of information sharing is given by Eqs. (5.4) and (5.5),respectively. And average supplier cost derived from level 1 and le-vel 2 of information sharing is given by Eqs. (5.9) and (5.10),respectively. To examine the impact of ld on average supplier’son inventory and average supplier’s cost, the supplier’s cost param-eter values are given as hd = 1 and pd = 19. And we set d = 10,c = 0.1, / = 0.3, H = 0.3, re = 1, rx = 1 and lr = 2. Also, we set s = 5for s > ld, or s = 30 for s 6 ld in both level 1 and level 2 of informa-tion sharing.

Figs. 10 and 11 illustrate the results of the impact of ld on thesupplier’s inventory reductions and cost savings. Like the case oflr, we observe that the average inventory and cost for the supplierare increasing when ld is increasing. Also, Figs. 10 and 11 suggest

l d

0

5

10

15

20

25

5 10 15 20 25 30

Ave

rage

inve

ntor

y

Level 1, =5

Level 1, =30

Level 2, =5

Level 2, =30

Fig. 10. Impact of ld on average supplier’s on inventory.

ld

0

5

10

15

20

5 10 15 20 25 30

Ave

rage

cos

t

Level 1, =5 Level 1, =30

Level 2, =5 Level 2, =30

Fig. 11. Impact of ld on average supplier’s cost.

σe

0

50

100

150

200

250

5 10 15 20 25 30

Ave

rage

inve

ntor

y

Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 12. Impact of re on average supplier’s on inventory.

σe

0

50

100

150

200

250

5 10 15 20 25 30

Ave

rage

cos

t

Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 13. Impact of re on average supplier’s cost.

σx

0

20

40

60

80

5 10 15 20 25 30

Ave

rage

inv

ento

ry Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 14. Impact of rx on average supplier’s on inventory.


that the inventory reductions and cost savings of the supplier in-crease with a higher level of information sharing. However, ld hasa larger impact than lr on the inventory reductions and cost savingsof the supplier. We also see that the average supplier’s inventoryand cost for ld > s are larger than those for ld 6 s for each level ofinformation sharing in Figs. 10 and 11. This is because s does nothave an impact on the total demand variance over the lead timeof the supplier in the case of ld 6 s for all levels of information shar-ing. The phenomena depicted in Figs. 10 and 11 are consistent withthe analytical findings presented in Section 5 (see Propositions 3,and 4a). Like the case of lr, although there is in the case of ld 6 s,supply chain members need to collaborate to reduce ld.

We move re to examine how the parameter affects the magni-tude of inventory reductions and cost savings for the supplier inlevel 1 and level 2 of information sharing. And we specify fixedvalues for all other parameters and vary re from 5 to 30. Averagesupplier’s on inventory in level 1 and level 2 of information sharingis given by Eqs. (5.4) and (5.5), respectively. And average suppliercost in level 1 and level 2 of information sharing is given by Eqs.(5.9) and (5.10), respectively. To examine the impact of re on aver-age supplier’s on inventory and average supplier’s cost, the sup-plier’s cost parameter values are given as hd = 1 and pd = 19. Andwe set d = 10, c = 0.1, / = 0.3, H = 0.3, rx = 1 and lr = 2. Also, weset s = 3 and ld = 4 for s > ld, or s = 3 and ld = 2 for s 6 ld in both level1 and level 2 of information sharing.

Figs. 12 and 13 show the results of the impact of re on the sup-plier’s inventory reductions and cost savings. We observe that theaverage inventory and cost for the supplier increase when re in-crease. Also, Figs. 12 and 13 suggest that the inventory reductionsand cost savings of the supplier increase with a higher level ofinformation sharing, and these inventory reductions and costsavings are greater when re are larger. The phenomena depictedin Figs. 12 and 13 are consistent with the analytical findings pre-sented in Section 5 (see Propositions 3 and 4a). This result suggeststhat, as re gets larger, the benefits from a higher level of informa-tion sharing increase.

We shift rx to examine how the parameter affects the magni-tude of inventory reductions and cost savings for the supplier in le-vel 1 and level 2 of information sharing. And we specify fixedvalues for all other parameters and vary rx from 5 to 30. Averagesupplier’s on inventory in level 1 and level 2 of information sharingis given by Eqs. (5.4) and (5.5), respectively. And average suppliercost in level 1 and level 2 of information sharing is given by Eqs.(5.9) and (5.10), respectively. To examine the impact of rx on aver-age supplier’s on inventory and average supplier’s cost, the sup-plier’s cost parameter values are given as hd = 1 and pd = 19. Andwe set d = 10, c = 0.1, / = 0.3, H = 0.3, re = 1 and lr = 2. Also, weset s = 3 and ld = 4 for s > ld, or s = 3 and ld = 2 for s 6 ld in both level1 and level 2 of information sharing.

Figs. 14 and 15 illustrate the results of the impact of rx on thesupplier’s inventory reductions and cost savings. Like the case ofre, we observe that the average inventory and cost for the supplierincrease when rx increase. Also, Figs. 14 and 15 suggest that theinventory reductions and cost savings of the supplier increase witha higher level of information sharing, and these inventory reduc-tions and cost savings are greater when rx are larger. The phenom-ena depicted in Figs. 14 and 15 are consistent with the analytical

0

20

40

60

80

5 10 15 20 25 30

Ave

rage

cos

t

σx

Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 15. Impact of rx on average supplier’s cost.

0

20

40

60

-0.9 -0.6 -0.3 0.3 0.6 0.9

Ave

rage

inv

ento

ry

γ

Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 16. Impact of c on average supplier’s on inventory.

0

20

40

60

-0.9 -0.6 -0.3 0.3 0.6 0.9

Ave

rage

cos

t

γ

Level 1, =3, ld =4

Level 1, =3, ld =2

Level 2, =3, ld =4

Level 2, =3, ld =2

Fig. 17. Impact of c on average supplier’s cost.


findings presented in Section 5 (see Propositions 3 and 4a). This re-sult suggests that, as rx gets larger, the benefits from a higher levelof information sharing increase.

We shift c to examine how the parameter affects the magnitudeof inventory reductions and cost savings for the supplier in level 1and level 2 of information sharing. And we specify fixed values forall other parameters and vary c from �1 to 1. Average supplier’s oninventory in level 1 and level 2 of information sharing is given byEqs. (5.4) and (5.5), respectively. And average supplier cost in level1 and level 2 of information sharing is given by Eqs. (5.9) and(5.10), respectively. To examine the impact of c on average sup-plier’s on inventory and average supplier’s cost, the supplier’s costparameter values are given as hd = 1 and pd = 19. And we set d = 10,/ = 0.3, H = 0.3, re = 1, rx = 1 and lr = 2. Also, we set s = 3 and ld = 4for s > ld, or s = 3 and ld = 2 for s 6 ld in both level 1 and level 2 ofinformation sharing.

Figs. 16 and 17 illustrate the results of the impact of c on theaverage supplier’s inventory and cost. We observe that the averageinventory and cost for the supplier increase or decrease with therange of c. However, supplier performance in the case of ld 6 sfor level 2 of information sharing is the smallest among the other

cases. The phenomena depicted in Figs. 16 and 17 are consistentwith the analytical findings presented in Section 5 (see Proposition4b and c). This result suggests that the supplier always obtains thebenefits in the case of ld 6 s for the level of information sharing.

These numerical experiments show the sensitivity of the aver-age inventory and cost to the supplier of changes in the parameterspotentially related to the value of information sharing. The resultssuggest that a higher level of information sharing may be morebeneficial for the supplier in some situations, such as when theseasonal period is large and/or autocorrelation is high, than in oth-ers. In particular, our results provide managers with insights onhow the seasonal effect has an impact on the value of informationsharing.

7. Conclusion and insight

In this paper, we evaluate the value of information sharing in atwo-echelon seasonal supply chain. We derive the optimal inven-tory policies, considering the means and variances of the supplier’sorder quantity for three levels of information sharing. We alsoshow that the seasonal effect has a significant influence on theoptimal inventory policies of the supplier for the three levels ofinformation sharing. Thus, the seasonal effect also influences theaverage inventory levels and the expected costs of the supplier.Finally, we show that the average inventory level and expectedcost of the supplier, in most situations, decrease when the levelof information sharing increases.

Our research provides practitioners with managerial insights inthe following aspects.

(1) Our results clearly indicate that the value of informationsharing is sensitive to the change of seasonal phenomenon.So, a focus on seasonal phenomenon may lead to a growthin overall supply chain profits.

(2) The average inventory level and expected cost of the sup-plier at levels 2 and 3 are equal. So, this result is that thesupplier should take the initiative to establish informationsharing-based partnerships and also give the retailer someincentives to induce the retailer’s cooperation.

(3) Both the supplier and its upstream member need to collab-orate to have a smaller supplier’s lead time than seasonalperiod to remove the impact of seasonal period on the sup-plier performance for each level of information sharing.

For a more meaningful investigation of the value of informationsharing in a seasonal supply chain, one may need to extend thisresearch to the cases of more complicated seasonal demand pro-cesses and a multi-echelon supply chain. In addition, more generalinventory policies adequate for seasonal demand and inventoryneed to be studied.

Acknowledgement

This work was supported by the research fund of Hanyang Uni-versity (HY-2011-P).

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the value of information sharing in a supply chain with a seasonal demand process

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