sharing shipment quantity information in the supply chain
TRANSCRIPT
Omega 34 (2006) 427–438www.elsevier.com/locate/omega
Sharing shipment quantity information in the supply chain
Cheng Zhanga,∗, Gek-Woo Tanb, David J. Robbc, Xin Zhengd
aDepartment of Information Management and Information Systems, Management School, Fudan University, Shanghai 200433, PR ChinabDepartment of Information Systems, School of Computing, National University of Singapore, 3 Science Drive 2,
Singapore 117543, SingaporecDepartment of Information Systems and Operations Management, The University of Auckland, Private Bag 92019, Auckland, New Zealand
dGrace Semiconductor Manufacturing Corporation, Zhangjiang Hi-Tech Park, Shanghai 201203, PR China
Received 1 May 2002; accepted 4 December 2004Available online 23 March 2005
Abstract
This paper evaluates the benefit of a strategy of sharing shipment information, where one stage in a supply chain sharesshipment quantity information with its immediate downstream customers—a practice also known as advanced shipping notice.Under a periodic review inventory policy, one supply-chain member places an order on its supplier every period. However,due to supplier’s imperfect service, the supplier cannot always exactly satisfy what the customer orders on time. In particular,shipment quantities arriving at the customer, after a given lead-time, may be less (possibly more) than what the customerexpects—we define this phenomenon as shipment quantity uncertainty. Where shipment quantity information is not shared withcustomers, the only way to respond is through safety stock. However, if the supplier shares such information, i.e. customersare informed every period of the shipment quantity dispatched, the customer may have enough time to adapt and resolvethis uncertainty by adjusting its future order decisions. Our results indicate that in most circumstances this strategy, enabledby information technologies, helps supply-chain members resolve shipment quantity uncertainty well. This study provides anapproach to quantify the value of shared shipment information and to help supply-chain members evaluate the cost-benefittrade-off during information system construction. Numerical examples are provided to indicate the impact of demand/shipmentparameters on strategy implementation. While previous studies mainly focus on the information receiver’s perspective, weevaluate a more general three-tier linear supply chain model via simulation, studying how this strategy affects the wholesupply chain: the information sender, the information receiver and the subsequent downstream tier.� 2005 Elsevier Ltd. All rights reserved.
Keywords: Shipment quantity; Information sharing; Supply chain management; Simulation
1. Introduction
Information flow between member organizations of a sup-ply chain is one of the main concerns in supply chain man-agement (SCM). Information flow, such as point-of-sales
∗ Corresponding author. Tel.: +86 21 65644783.E-mail addresses: [email protected] (C. Zhang),
[email protected] (G.-W. Tan), [email protected](D.J. Robb), [email protected] (X. Zheng).
0305-0483/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.omega.2004.12.005
(POS) data from retailers, and order quantities betweentwo successive stages in the chain directly impact mem-bers’ production and delivery schedules and inventory con-trol. With rapid development of cost-effective informationtechnologies (IT) such as database management systems(DBMS), network communication protocols, electronic datainterchange (EDI), and the Internet, accurate and timely in-formation flow is now possible.
As a result, information sharing in the supply chain canbe viewed as timely IT-enabled feedback on the collabo-rations between members of a supply chain. In a situation
428 C. Zhang et al. / Omega 34 (2006) 427–438
without such communication, the downstream member re-ceives feedback with some time delay. In the worst case thedownstream member only becomes aware of the shipmenttiming and quantity when the goods arrive: feedback beingdelayed for the entire transportation time. Such late feedbackmay not help the company much in calculating and fore-casting; indeed it sometimes even causes miscalculation ofits available future resources. Diehl and Sterman [1] showedthat direct feedback, without any inferring work on the partof receivers, and faster feedback, with shortened delay time,would reduce the dynamic complexity of decision-makingand improve decision quality. On-time information sharing,transmitted through EDI, fax, email and any other IT means,is the solution to supply direct and fast feedback betweena supplier and its customer. It assists companies gain fromdynamic decision-making.
With technological barriers to information sharing be-ing dismantled, the question now shifts to: what type ofinformation is beneficial to share? Although informationsharing among participants may help chain members reduceinformation distortion [2,3], what is not so clear is the cost-benefit trade-off of various information sharing strategiesin the supply chain. Therefore, quantifying and evaluatingthe cost-benefit of information sharing strategies (ISS) mayhelp organizations develop their information systems (IS)capability. On this topic, Tan [4] evaluated the impact ofdemand information sharing on supply chain network struc-tures, product structures and demand mix. Chen et al. [3]studied centralized demand information sharing to reducethe bullwhip effect in supply chains. Li et al. [5] quantifiedthe benefit of demand and inventory information sharing onreducing demand uncertainty. Lee et al. [6] analyzed thebenefits of sharing demand information and identified someof the drivers behind using a two-level supply chain model.While previous studies mostly focus on analyzing demandinformation sharing, we extend the perspective to anotherkind of ISS: shipment information sharing (SIS). If we viewthe demand information as the foreground information, i.e.,the information from the markets members face, then SIS isa kind of background information, i.e. the information frommembers’ supply sources.
Lee and Whang [2] mentioned that a manufacturer coulduse its supplier’s delivery schedule to improve its own pro-duction schedule. Here, we define shipment informationsharing as: in a supply chain network, one stage’s ship-ment information (timing and quantity) is shared with itsimmediate downstream to assist participants with their pro-duction/inventory decisions. For example, a large computerscanner producer in Asia plans to share its shipment in-formation with its direct customers and local sales agentsvia the Internet: direct customers can access the producer’sweb-based information system to check the exact shipmentdates and quantities of each order when the goods are readyto ship out. Since the transportation time between the man-ufacturer and a particular direct customer is quite clear andrelatively constant (lead-time guarantees are becoming more
common), the customer can easily infer the date/quantity ofincoming goods (to know the proportion of demand filledwithin x days) and adjust its future order quantities accord-ingly. Note that if the producer seeks to avoid the risk ofrevealing its production capability to its competitors, it maybe unwilling to share production information (such as whena particular order is scheduled or when the production is car-ried out) with its direct customers. On the customers’ side,they usually do not, and should not, care when the prod-ucts are produced but when and how much of the goodswill appear “on their doorstep”. For them, accuracy on ar-rival quantity is most valuable in improving purchasing de-cisions, which in turn affect its inventory and productionmanagement. This paper evaluates whether sharing such in-formation may help members perform better under shipmentuncertainty.
Shipment uncertainty arises primarily from two sources:one is lead-time variability due to uncertain transportation,administrative processing and/or production times [7,8]. Theother uncertainty arises from shipment quantity variability,i.e., shipment quantities arriving at the customer may be less(possibly more) than the customer expects, due to limitedresources (finance, materials, capacity, etc.), making it un-wise for supply-chain members to promise perfect (100%fill-rate) service. Suppliers may backlog the unmet portionof the order and fulfill it in the future (or may simply ig-nore the backlog altogether). While lead-time variability hasdrawn much attention [9–11], and responses such as lead-time guarantees, quantity variability has been ignored. It isthis second area of uncertainty on which this paper focuses.We consider the case where orders can be split, whereas thestudy of lead-time variability relates more to the case whenan order that cannot be fulfilled on time is delayed until itcan be completely fulfilled.
This paper analyzes the impact of sharing delivery infor-mation of shipment quantity, also known as advanced ship-ping notice (ASN), in logistics practice, in a simple linearsupply chain with stochastic demand of a single product.First, we describe the model framework and begin our anal-ysis of the effect of SIS on ordering decisions, inventoryand service levels, and total relevant cost. We then considera special case with two suppliers. We provide numerical ex-amples of the impact in an extensive three-tier chain. De-tailed proofs of propositions are provided in the appendix.
2. A simple two-stage supply chain case
We consider a simple two-stage supply chain consistingof one manufacturer and one retailer. The market demandfor a single product arrives at the retailer, following an i.i.d.distribution of (d, �t ) where d is a constant and �t followsa normal distribution with mean zero and variance �2. “�is significantly smaller than d, so that the probability of anegative demand in period t is negligible” [12]. The excessmarket demand is backlogged at the retailer’s side.
C. Zhang et al. / Omega 34 (2006) 427–438 429
We assume an order-up-to inventory system in which theinventory level is reviewed and replenished during everyperiod. The order process occurring in the supply chain isas follows: at the beginning of period t , t > 0, the retailerreceives the shipment, yt , from its supplier (the manufac-turer), and fulfills the market demand, dt , and then the re-tailer checks its inventory level and places an order, Qt , onthe manufacturer. The shipment, responding to order Qt , issupposed to arrive at the beginning of period t +L, where L
is the constant lead-time, counting from when retailer placesthe order on its supplier until it receives the shipment. Weassume that the lead-time, L, between two tiers is constantand concentrate on analyzing the uncertainty of shipmentquantity.
When it cannot fulfill the retailer’s order on time (dueto limited production/inventory capacity or delayed capac-ity or a planned adjustment to market demand), the manu-facturer could take one of two different actions. Firstly themanufacturer simply ignores the unfulfilled order (we de-fine this as the A1 case, an extreme case of shipping un-certainty, which may appear in a manufacturer-dominantmarket). Secondly, it may backlog the order and satisfy theorder in a future period, i.e. the unfulfilled order at periodt will be produced at the beginning of period t + m, wherem is a constant (this is the A2 case, where the supplierpromises to fulfill backlogged demand in the future, whichmay be due to the supplier’s setup interval or its own pro-duction/inventory economics), and received by the retailerat period t + m + L. These replenishment policies, possi-bly based on contracts between companies or market ori-entation, cause shipment uncertainty. In a later section, weconsider the situation where the retailer has an alternativesupply source, in which case we analyze the trade-off forboth the retailer and the manufacturer (note this issue is be-yond our initial setting of a simple two-tier supply chain, sowe discuss it as a special case in Section 2.4).
In the situation with no SIS (NSIS), at the end of periodt the retailer does not know the exact shipment quantity itwill receive from its supplier in period t +L. Its estimate isbased on the assumption that the order will be totally ful-filled at the beginning of period t + L. In the SIS case, theretailer is informed of the quantity of the incoming ship-ment at the beginning of period t + 1 (or at the end of pe-riod t)—allowing the retailer to adjust its next-period orderdecision). Compared with NSIS, the retailer recognizes theshipment uncertainty, and subsequently adapts its ordering,L-periods faster. Such information is less valuable in theA2 case than in the Al case: in the A2 case, as the suppliermakes up backlogged orders before the retailer observes it,any negative impact of shipment uncertainty on the retaileris reduced. In addition, the shorter the make-up interval (m)
of the supplier, the smaller the shipment uncertainty.Intuitively speaking, if the manufacturer’s fill rate is
100%, there is no shipment forecast error; consequentlySIS has no benefit at all. Meanwhile, it is clear that theupstream fill rate, which is equal to 1−(demand unmet on
time)/(demand), generates the misperception of feedback:the greater the fill rate, the smaller the shipment uncer-tainty. In addition, the longer the lead-time, the longer theinformation delay, and therefore the greater the impact ofsuch misperception. In such situations SIS will prove evenmore worthwhile.
2.1. Order decision
We assume that the retailer has only one source of supplyand that it has no information on the supplier’s fulfillmentpolicy, i.e., service level and make-up scheme (note that theretailer may infer this policy from historic data, but here weassume it cannot). With SIS, the retailer receives informationof the incoming shipment before its next order, which canbe used to adjust its next-period order quantity. Therefore,the only source of error is demand fluctuation. However,with NSIS the retailer has to forecast the arrival quantity(in our case, the retailer assumes that the quantity is exactlywhat it ordered L-periods before) to determine the orderquantity. Therefore, its final ordering decision with NSIShas two error sources: the forecast error on demand and theincoming shipment quantity.
To facilitate our analysis, we assume without loss of gen-erality, that the retailer’s initial stock, including both on-handand in-transit, is equal to the order-up-to level, S, denotedas S = Ld + z�
√Lz = �−1[b/(b + h)], in which �(·) is
the standardized normal distribution, b is the unit backordercost per period and h is the unit inventory holding cost perperiod (note that under different information circumstances,the optimal inventory policy should also be different). How-ever, to facilitate comparison (and also because this paperdoes not focus on optimizing inventory policy but rather onthe impact of “changing the given” [13]), we fix the base in-ventory policy, i.e., S is the same to both NSIS and SIS). Wealso define net inventory as on-hand inventory minus back-orders. This can be negative in the case of a backlog. Thenet inventory at period t is denoted by nt = nt−1 + yt − dt .Therefore, the retailer’s order decision with SIS is
Qt |nt = S −L∑i
yt+i − nt and
Q′t |nt = S −
L∑i
Qt+i − nt
with NSIS (note that the exact order quantity should beexpressed as [0, S −∑L
i Qt+i − n]+t the maximum of 0and the order quantity, to avoid negative order quantities.What we use here is a close approximation).
We first analyze the Al case. When the demand d1 occurs,the retailer should order d1 from its supplier (Q1 = d1). Itssupplier fulfills this order with �′
1Q1, where �′t is the manu-
facturer’s fill rate at period t , with mean and standard devia-tion (�′, �′)�′ follows 0 < �′ �1. We assume that �′ is signif-icantly smaller than �′ and the two probability processes, �′
t
430 C. Zhang et al. / Omega 34 (2006) 427–438
and dt , are independent. (When we assume that the demandfluctuation is far away from the upper bound of the man-ufacturer’s production/supply capability, the manufacturer’sfill rate is mostly affected by its own production/inventoryscheduling and the correlation between the demand and thefill rate to the manufacturer is not likely to be significant.This assumption allows us to concentrate on the retailer’sbehavior when facing uncertain shipment.) With SIS, theretailer gets this information (feedback) before placing itsnext order. Therefore Q2 is adjusted to meet the shortage:Q2=d2+(1−�′
1)Q1. When the retailer observes the on-timefeedback of �′
2Q2, it determines Q3 = d3 + (1 −�′2)Q2, so
on and so forth. Therefore, its expected order decision canbe approximately denoted as E(Qt ) = [1 − (1 − �′)t /�′]d∗(see appendix) and its long-term average order is
limt→∞
1
t
∑E(Qt ) = 1
�′ d(∗).
With NSIS, the retailer receives feedback of its order ful-fillment with an L-period delay. Therefore the timing of theorder adjustment is always L periods later than that occur-ring with SIS: during the first L periods, since the supplier’sfulfillment information is unknown, the retailer retains itsorder quantity as Q′
t = dt . It is only when the shipmentarrives, at period L + 1 that the retailer discovers the realfulfillment information. Therefore, in the next l periods, i.e.t = L + 1, . . . , 2L, it adjusts its order to Q′
t = dt + (1 −�′t−l )Q
′t−l
, etc. As a result, its expected order decision andlong-term average quantity are approximately, respectively;
E(Q′t ) = [1 − (1 − �′)[t/L]/�′]d(∗)
and
limt→∞
1
t
∑E(Q′
t ) = 1
�′ d .
Several observations arise from the above. First, shipmentuncertainty aggravates demand fluctuations. Second, the de-cision based on SIS is more flexible than that of NSIS: SISassists the downstream organization to adapt to shipment un-certainty. The adjustment speed with NSIS is much slower,depending on the feedback delay time, i.e. the length of thelead-time, between the supplier and itself. The impact ofsuch a slower response is discussed in Sections 2.2 and 2.3.
In the A2 case, the supplier ships the retailer’s period-torder Qt in two parts: a first shipment of �′
tQt at time t ,and the remainder (1 −�′
t )Qt at period t +m, where m is aconstant, 1�m�L. Therefore, the retailer’s order decisionwith SIS is as follows: during the first m-periods, it ordersQt = dt + (1 − �′
t−1)Qt−1, with the supplier fulfillmentbeing �′
tQt . From the m+1th period, since the retailer getsthe information of the complementing shipment, its orderchanges to
Qt = dt + (1 − �′t−1)Qt−1 − (1 − �′
t−m)Qt−m.
Therefore we get
limt→∞
1
t
∑E(Qt ) = d(∗).
In NSIS, the retailer receives feedback on its order fulfill-ment with a delay of L-periods, i.e. it begins to adjust itsorder quantity from the L + m + 1th period, with an L-period delay compared to that of SIS: from period 1 to L
and from period L + 1 to L + m, its orders are Q′t = dt
and Q′t = dt + (1 − �′
t−L)Q′t−L
, respectively, as describedbefore. From period L + m + 1, its order changes to
Qt = dt + (1 − �′t−L)Qt−L − (1 − �′
t−m)Qt−m.
Similarly, its long-term average quantity is
limt→∞
1
t
∑E(Q′
t ) = d.
Although the expectation equation of ordering shows thedifference of the ordering decisions under different infor-mation environments, it does not explicitly express how theorganization’s performance is affected by its responsivenessto shipment uncertainty. In the next sections, we providevarious cost-benefit indices.
2.2. Inventory level and fill rate
Here, inventory level is the net inventory, and service levelrefers to the retailer’s actual fill rate to its customers. Basedon our initial setting, the expected net inventory at the endof period L is equal to SS, the safety stock, z�
√L.
With SIS in Al, the incoming shipment at period L+1, iswhat the retailer requested at period 1, namely �′
1Q1, andtherefore the net inventory at period L + 1 is nL+1 = SS +�′
1Q1 − dL+1. Similarly nL+2 = SS + �′1Q1 − dL+1 +
�′2Q2 − dL+2 and so on. A more general formula is
E(nL+t ) = SS +(
1 − 1
�′ + (1 − �′)t+1
�′
)d(∗) and
limt→∞
1
t
∑E(nt ) = SS +
(1 − 1
�′)
d.
Similarly, we obtain the retailer’s expected net inventory andlong-term net inventory under NSIS as
E(n′L+t ) = SS +
(1 − 1
�′ + (1 − �′)[t/L]+1
�′
)Ld and
limt→∞
1
t
∑E(n′
t ) = SS +(
1 − 1
�′)
Ld.
Since E (Net stock)�E (On hand) [7], the expected netstock is the safety stock (used to maintain a certain servicelevel against demand forecast error). Due to shipment uncer-tainty, the value of the safety factor is actually smaller thanthe expected one, z (Defining zt as the actual safety factorin the Al case with SIS, z1 =z−(1/�′ −1)d/�
√L�z when
0 < �′ �1, which means the actual service level, �(z1), issmaller than that expected, �(z). So when the shipment from
C. Zhang et al. / Omega 34 (2006) 427–438 431
upstream is uncertain, downstream customer service deteri-orates, even with SIS. However, the situation with SIS is bet-ter than that with NSIS: �(z′
1)=�[z−(1/�′ −1)Ld/�√
L],with �(z′
1)��(z1)��(z) when L�1, indicating that SIShelps organizations improve its service level in the A1 case.
Let
limt→∞ E(�nt ) = lim
t→∞1
t
[∑E(nt ) −
∑E(n′
t )]
be the difference of safety stock reduction under differentinformation environments—indicating the safety stock re-duction using SIS rather than NSIS. The result is
limt→∞ E(�nt ) = (L − 1)
(1
�′ − 1
)d .
Since �[limt→∞ E(�nt )]/�l = (1/�′ −1)d �0, as the lead-time increases, the benefit of SIS increases. This result fitswith the observation of Diehl and Sterman [1] that the longerthe feedback delay, the greater the degree of misinterpre-tation. As information transit delays reduce under variousIS technologies, this benefit is becoming available to moresupply chain members.
Since �[limt→∞(E(�nt )]/��′ = −(1/�′)2(L − 1)d �0,limt→∞(E(�nt ) decreases with the up-stream member’sfill rate, �′. This indicates that better upstream performancegenerates lower feedback misperception. This observationis quite valuable. First, it indicates the degree to which up-stream supply performance directly impacts the welfare ofdownstream customers. Second, the performance of the in-dividual member in a supply chain is affected by its part-ners. This external impact can be exploited by close supplychain collaboration.
Considering the A2 case with SIS, the upcoming shipmentfor period L + 1, (what the retailer requested at period 1),should be �′
1Q1. Therefore, the net inventory at period L+1is nL+1 = SS + �′
1Q1 − dL+1. From period L + m + 1onwards, its inventory becomes
nL+m+1 = nL+m + �′mQm − dL+m+1 + (1 − �′
1)Q1
and a more general formula is
nL+m+t = nL+m+t−1 + �′m+t−1Qm+t−1
− dL+m+t + (1 − �′t )Qt .
Therefore, we obtain limt→∞ 1/t∑
E(nt ) = SS(∗). Sim-ilarly limt→∞ 1/t
∑E(n′
t ) = SS in case A2 without SIS.In the A2 case, the impact of shipment uncertainty is
small because the manufacturer satisfies its backorder withina fixed time period, which makes the shipment process be-come more steady and predictable. Therefore, the retailer’sordering decision can adapt well, no matter whether the in-formation feedback is on time or not. In such circumstances,SIS, compared with NSIS, still assists organizations becomemore sensitive to shipment fluctuations, but this impact only
provides slight improvement to an organization’s perfor-mance, such as decision quality, inventory and fulfillmentlevel, and inventory-related costs.
2.3. Inventory cost
In this section, we study the impact of SIS on relatedinventory costs in the Al case. Ignoring ordering cost andthe material cost, the total relevant cost is the sum of hold-ing cost and backorder cost and can be expressed as T C =�√
L[(b+h)f (k)+hk] [6], where f (x) is the right loss func-tion for the standard normal distribution as f (x)= ∫∞
c (k −x)d�(k), where �(k) is the standard normal probabilitydistribution.
Setting F(x)=(b+h)f (x)+hx, it is easy to show F(x)’sfirst derivative is zero when x =�−1(b/(b+h)) and its sec-ond derivative is always greater than or equal to zero. ThusMin[F(x)] = (b + h)f (z) + hz and F(x1) > F(x2) > F(z)
if x1 < x2 < z. Since the actual safety factors under SIS andno-SIS for the Al case are z1 and z′
1, respectively, wherez′
1 �z1 �z, we obtain the relationship of total cost of thedifferent information environments as, T C′
1 for NSIS andT C1 for SIS, with T C′
1 �T C1. One can conclude that SIShelps the retailer reduce its total relevant inventory cost. Thisreduction can be viewed as the benefit of SIS and the ba-sis on which one can determine whether to implement thisinformation policy in IS construction.
2.4. Special case with alternative supply source
Previous Al and A2 cases assume that the retailer has onlyone supplier to place orders, therefore it has to adjust itsorder quantity to the supplier to resolve such shipment un-certainty. In reality, chain members usually have more thanone supply source. So here we consider another situationin the two-tier supply chain: the retailer has an alternativesource to order: this alternative can provide a 100%, supplywithin a shorter lead-time but demands a higher purchaseprice than does the retailer’s regular manufacturer. Due tothe cost issue, the retailer considers this alternative sourceas a supplemental source and only orders the least possiblequantity to maintain the actual stock level at the expectedvalue (i.e. this urgent order is used to maintain the com-pany’s target service level, not a 100% service level). The al-ternative source could be a third-party supplier, the originalmanufacturer (the manufacturer may work overtime to ful-fill these urgent order, based on a pre-negotiated contract),or a same-tier partner, for example, the company may pur-chase from a neighboring organization that belongs to thesame supply chain channel or with which it has a close col-laboration. These urgent orders could be delivered via air orsome other mode, but only at higher purchasing and/or trans-portation costs. The more urgent the order, the higher costthe purchaser should pay (we use “urgent” and “normal” todistinguish these two types of order). Since in this case theretailer’s order can be completely fulfilled on time (the sum
432 C. Zhang et al. / Omega 34 (2006) 427–438
of shipment from the supplier and the alternative source ex-actly equals what the retailer expects), the value of the re-tailer’s order decision, inventory level and service level areexactly the theoretical value of what the typical inventorymodel specifies, except for the total cost—since the pur-chase cost at the two supply sources is different. Therefore,in this section we focus on analyzing the total cost changesunder different information circumstances.
Most of the system parameters follow our initial setting inthe Al case except for the normal purchasing cost (i.e. C.I.F.(Cost, Insurance and Freight), the sum of transportation costand purchasing cost) from the manufacturer, which is c. Inthe NSIS situation, the retailer places its urgent order tomeet the period-t shortage at period t −1 with a higher unitcost of c + c1. The shipment arrives in period t . In the SISsituation, the urgent order is placed L−1 periods before, i.e.at period t − L − 1, to meet the period-t shortage since theretailer knows the real shipment quantity arriving in periodt from SIS. The unit cost this time is c2, higher than c butsmaller than c + c1, as this order is less urgent than that forNSIS (c1 > c2 > 0).
We assume that the additional cost continuously increasesas the required fulfillment time for the urgent order, t , de-creases. We obtain c1 −c2 =[k(1)−k(L)]c0 where c0 is thebase unit cost for the urgent-order quantity and is greaterthan zero. k(t) is the coefficient of this additional cost andis a decreasing function of t .
Under this urgent ordering policy, the retailer places anurgent order, equal to the amount that the manufacturer can-not fulfill on time, on the alternative source—to maintainthe expected inventory level and service level. Therefore thepurchasing cost must be included in the total relevant costmodel: in the NSIS situation, the average cost per period forthe retailer is
E(T C1) = �√
l[(b + h)f (z) + hz] + cd + c1(1 − �′)d ,
where z=�−1[b/(b+h)], in which �(·) is the standardizednormal distribution, b is the unit backorder cost per periodand h is the unit inventory holding cost per period (thefirst term on the right-hand side is the sum of the averageinventory carrying cost and backorder cost per period, thesecond and the third terms represent, respectively, the sumof purchasing cost (C.I.F., i.e. the sum of material cost andtransportation cost) per period of both the “normal” orderand the “urgent” order, coming from c�′d + (c + c1)(1 −�′)d); in the SIS situation, this is
E(T C2) = �√
l[(b + h)f (z) + hz] + cd + c2(1 − �′)d.
The retailer gains benefit from implementing SIS:
E(�T C) = (c1 − c2)(1 − �′)d= [k(1) − k(L)](1 − �′)c0d > 0.
Since �E(�T C0)/��′ = −[k(1) − k(L)]c0d < 0 and�E(�T C0)/�l′ = −k′(L)(1 − �′)c0d > 0, it is clear that
the benefit for the retailer is increasing with lead-time, L,and decreasing with �′, indicating that the longer feedbackdelay and worse supplier performance is an incentive forthe retailer to require SIS.
The benefit to the supplier providing its shipment infor-mation may come from two sources (note that such a benefitis not certain but a function of inter-organizational negotia-tion, etc.). One is from the reduced penalty cost of unfulfilledorders: it can be proved that limt→∞ 1/t
∑E(Qt ) = d,
where Qt is the order quantity the retailer places on the man-ufacturer. Therefore the supplier’s average cost reduction isE(�T C′) = (b′
1 − b′2)(1 − �′)d, where b′
1 b′2 (b′
1 �b′2) are
the supplier’s penalty cost (for example, the credit to the re-tailer) under the NSIS and the SIS situations, respectively.Therefore, the total cost reduction for the entire supply chainis
E(�T C0) = [k(1) − k(L)](1 − �′)c0d
+ (b′1 − b′
2)(1 − �′)d,
which is decreasing with �′ as
�E(�T C0)/��′ = −[k(1) − k(L)]c0d − (b′1 − b′
2)d < 0,
but increasing with L as
�E(�T C0)/�l′ = −k′(L)(1 − �′)c0d > 0.
This result indicates that longer feedback delay and worsepartner performance motivate the retailer to request SISwhile the manufacturer is encouraged to share the informa-tion of its shipment quantity when it can reduce its penaltycost.
A second benefit for the supplier comes from gainingstrategic advantage in competition to maintain its marketshare, i.e. the retailer’s ordering quantity, when it does notshare its shipment quantity with the retailer, the suppliermust satisfy the condition that
�√
L[(b + h)f (z) + hz] + cd + c1(1 − �′)d��
√L − 1[(b + h)f (z) + hz] + (c + c2)d,
i.e. the total cost of choosing the manufacturer as the mainsupplier should not be greater than choosing the alternativesource as the main supplier. To prevent the retailer consid-ering the alternative source as its primary supplier, the sup-plier must ensure �′
1 �1 − c2/c1 + �/c1, where � = (√
L −√L − 1)[(b + h)f (z) + hz]�/d. The above equation is the
lower bound of service level the supplier should maintain.As the competitor reduces its delivery cost, the supplier mustimprove its service level to retain its market. (Note that ��c2is required, otherwise the customer will turn to the alter-native source for regular ordering. Blackburn [14] providesrelated discussion of the value of lead-time reduction.)
When sharing the information of its shipment quantitieswith the retailer, the supplier should ensure that
�√
L[(b + h)f (z) + hz] + cd + c2(1 − �′)d��
√L − 1[(b + h)f (z) + hz] + (c + c2)d,
C. Zhang et al. / Omega 34 (2006) 427–438 433
from which we obtain the condition equation �′2 ��/c2. It
can be shown that �′2 ��′
1 (the lower bound of �′1 and �′
2),which indicates that by implementing SIS it is easier for thesupplier to satisfy the retailer’s requirement and to compete.
Note that the retailer may choose to order more from thealternative source when SIS is available; as the retailer’s ur-gent order cost is reduced by using on-time shipment infor-mation, the retailer may find a way to lower cost to balancethe proportion of order quantity on the manufacturer andthe quantity on the alternative source. However, this issueis beyond the scope of our current research.
3. Numerical examples
We now show some numerical examples of the impactsof demand/shipment parameters on a retailer’s performance,i.e. total relevant cost and fill rate. In particular, we considerthe impact of three variables, �′, L and �, on inventory leveland the subsequent impact on the retailer’s performance. Inthis example, we set d = 100, b = 50 and h= 2. The retailersets its safety factor as �−1(b/(b +h)) in which �(·) is thestandardized normal distribution, which is 1.77 in this case.When analyzing L, we set �′ = 0.8, �= 30 and vary L as 3,5, 7 and 9. With �′, we set L = 5, � = 30 and vary �′ as 1,0.95, 0.9, 0.8 and 0.7, each of which, except 1, follows theuniform distribution of [0.9,1], [0.8, 1], [0.6,1] and [0.4, 1],respectively. With �, we set �′ =0.8, L=5 and vary � as 10,30 and 50. We set m, the replenishment interval in A2, as1. Each situation is simulated for 15,000 time periods andreplicated 10 times, using the same initial conditions, but adifferent random number generator seed for each replicate.Average values from the simulations are used as estimatesof average system performance.
3.1. Results
Fig. 1 suggests that in the Al case (i.e. the supplier sim-ply ignores the unfulfilled order) the retailer’s inventory costand fill rate deteriorates when lead-time increases. This ag-gregate negative effect is directly caused by the length ofthe feedback delay. SIS helps a company eliminate the feed-back delay. In the more reliable supply behavior of A2 (i.e.the supplier promises to fulfill backlogged demand in thenext period) the retailer’s performance becomes insensitiveto such feedback delay.
Fig. 2 demonstrates the impact of the external environ-ment, i.e. the supplier’s fill rate, on a company. When thesupplier’s fill rate deteriorates, the retailer’s performancedrops sharply in Al, while SIS mitigates this impact. In thesituation where the supplier promises to fulfill the backlogwithin a fixed time period, the retailer’s performance is notinfluenced much by its poor behavior in the past, in the caseof both SIS and NSIS. This result indicates that the sup-plier’s fill rate is a credit to its customer: a good fill rate
and a steady fulfillment policy reduce customers’ costs andengender more trust.
Fig. 3 shows that for Al, as � increases, the retailer’s per-formance improves. Based on previous equations for z1 andz′
1, this phenomenon is caused by the safety stock amount,i.e. the larger �, the larger the safety stock and z1(z′
1), andconsequently, a greater counteraction to the combined ef-fect of demand and shipment uncertainty in Al. Althoughthis correlation of uncertainties has not been analyzed, it isclearly displayed in the simulation result. Such correlationis not obvious for A2.
4. Extension to a three-tier linear supply chain
In previous sections, we have analyzed how a supplier’sfill rate affects its customer’s performance with the assump-tion that the supplier’s service level is independent of itscustomer’s ordering decision. Here we extend the model topresent the chain-wide impact of one stage’s service leveland the value of SIS to the whole chain.
4.1. Experiment setting
Consider a three-tier linear supply chain consisting ofone global wholesaler, one local agent and one local retailer.The channel manages a single product. Customers cometo the retailer and make their purchases for that productregularly. The retailer sells the goods to the customer andplaces orders on its supplier, the local agent, based on itsforecast of future need and its own inventory managementpolicy. The agent fulfills the retailer’s order and places neworders on its supplier, the wholesaler, based on the forecastsof the retailer’s demand and its own inventory policy, alsoseeking to ensure continuous fulfillment to the retailer. Thewholesaler behaves exactly as the retailer and agent. Previ-ous backlogs are prioritized for fulfillment at the beginningof each cycle. We assume that the wholesaler’s supplierhas infinite capacity to supply whatever the wholesalerorders.
Assume the transportation time between each member inthe chain is 3 periods and all members use simple movingaverages to forecast their direct customer’s future demand.Chen et al. [3] argued that the variance of the orders, placedby the downstream to its supplier, satisfies a lower boundas 1 + (2L/p + 2L2/p2)(1 − �p), where L is the lead-time between two successive stages in a supply chain, p
is the number of demand observations in the simple mov-ing average forecast and � is the correlation parameter ofthe demand process. To provide a smooth enough demandforecast and to avoid unexpected increases in variability ofdemand information passing through the supply chain, weset a high forecasting window, i.e. 20 times greater than thelead-time—60 periods in this case.
What supply chain members are most concerned aboutin this case is how to set a proper target fill rate to satisfy
434 C. Zhang et al. / Omega 34 (2006) 427–438
3 5 7 9LT
0500
1000150020002500300035004000450050005500
TC
A1-SIS
A1-NSIS
A2-SIS
A2-NSIS
3 5 7 9LT
0.200
0.3000.4000.500
0.600
0.700
0.800
0.900
1.000
reta
iler's
fill
rate
A1-SIS
A1-NSIS
A2-SIS
A2-NSIS
LT TC A1-SIS A1-NSIS A2-SIS A2-NSIS
15 7 9
288 335 381 423
879 1745 2844 4138
241 304 355 400
241 304 354 400
LT FR A1-SIS A1-NSIS A2-SIS A2-NSIS
35 7 9
0.970 0.972 0.972 0.971
0.840 0.693 0.551 0.426
0.989 0.987 0.986 0.984
0.989 0.987 0.986 0.984
Fig. 1. Impact of lead-time (LT) on retailer’s TC and FR.
A1-SISA1-NSISA2-SISA2-NSIS
1.000
0.900
0.800
0.700
0.600
0.500
0.400
0.300
0.200
reta
iler's
fill
rate
01 0.95 0.9 0.8 0.7
FR FR1 0.95 0.9 0.8 0.7
5001000150020002500300035004000450050005500
TC
A1-SIS
A1-NSIS
A2-SIS
A2-NSIS
FR TC A1-SIS A1-NSIS A2-SIS A2-NSIS
10.95 0.9 0.8 0.7
293294 300 335 453
293 326 482
1745 5145
293 293 295 304 319
293 294 296 304 318
FR FR A1-SIS A1-NSIS A2-SIS A2-NSIS
10.95 0.9 0.8 0.7
0.989 0.987 0.984 0.972 0.943
0.989 0.973 0.932 0.693 0.302
0.989 0.989 0.989 0.987 0.984
0.989 0.989 0.989 0.987 0.985
Fig. 2. Impact of supplier’s fill rate (FR) on retailer’s TC and FR.
0500
1000150020002500300035004000450050005500
10 30 50
STD
TC
A1-SISA1-NSISA2-SISA2-NSIS
10 30 50
STD
reta
iler's
fill
rate
A1-SISA1-NSISA2-SISA2-NSIS
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
STD TC A1-SIS A1-NSIS A2-SIS A2-NSIS
10 30 50
293 335 513
4283 1745 1191
125 304 498
124 304 498
STD FR A1-SIS A1-NSIS A2-SIS A2-NSIS
10 30 50
0.949 0.972 0.969
0.238 0.693 0.826
0.991 0.987 0.981
0.991 0.987 0.981
Fig. 3. Impact of demand fluctuation (STD) on retailer’s TC and FR.
C. Zhang et al. / Omega 34 (2006) 427–438 435
Table 195% confidence interval of the differences of SIS vs. NSIS system performances (same variance not assumed)
FRW0 FRR FRA FRW STOCKR STOCKA STOCKW
0.5 0 +∗ + 0 − +0.6 0 0 0 0 − +0.7 0 +∗ − 0 − 00.8 0 +∗ − 0 − −0.9 0 0 − 0 − −0.95 0 0 − 0 −∗ −FRW0: target fill rate of wholesaler; FRR: actual fill rate level of retailer; FRA: actual fill rate of agent; FRW: actual fill rate of wholesaler;0: the interval covers zero; +: the difference is greater than zero; −: the difference is less than zero; *: p < 0.1 (90% confidence).
its customer while gaining maximum benefit for itself. Weassume end customers who require at least a 90% servicelevel. To attain this goal, the retailer not only has to setits target service level as 90%, but also has an incentive torequire its supplier to provide a high service level.
However, it may not be necessary for all upstream sup-pliers in the supply chain to keep high service levels, as thesafety stock maintained by the agent, the retailer’s succes-sive supplier, resolves shipment uncertainty from the whole-saler side so that this uncertainty may not heavily affectthe retailer’s fulfillment performance. But this shipment un-certainty, if unknown or delayed to customers, may causeimproper ordering from its customer, which leads to unsat-isfactory performance. From this perspective, a supplier’shigher fill rate is a credit to its customer. Meanwhile, if cus-tomers receive such shipment uncertainty information ontime, they may adjust their order decision, possible reduc-ing the customers’ loss due to this uncertainty. In summary,to collaborate with other members in the supply chain, thewholesaler may choose either a high fill rate or SIS strategy.Thus, in this section, we analyze the impact of shipment un-certainty on customers and the whole supply chain—to seewhether it may help the wholesaler maintain low costs with-out reducing the benefit of other members. Two cases willbe considered: the wholesaler sharing its shipment informa-tion with the agent (SIS case) and not sharing information(NSIS case).
We now show simulation results of the impact of thewholesaler’s fill rate on the supply chain. In this example,we set the mean demand as d = 100, � = 30, and assumea normal distribution. The transportation time between eachmember is 3 periods. The agent and retailer both set theirtarget service level at 90%, i.e., set the safety factor as 1.27,while the wholesaler may set its service level at 50%, 60%,70%, 80%, 90% and 95%.
4.2. Data analysis
To examine the impact of shipment uncertainty and thebenefits of SIS we study two system performances in thesupply chain: the chain members’ service level and averagestock holding. The cost measurements, such as backlog cost,
inventory holding cost and total relevant inventory cost canbe inferred from these two indices.
Firstly,we study the differences of these two system per-formances (service level and holding stock) under SIS andNSIS at each tier in the supply chain via independent-samples t-tests. If the confidence interval of the differencebetween the estimated mean of SIS and the estimated meanof NSIS covers zero, it stands that at the given confidencelevel, the difference of the member’s performance under SISand NSIS is not significant, i.e. it may be zero. If the con-fidence interval of the difference with a given confidencecannot cover zero but is greater (lower) than zero, it standsthat the member’s performance under SIS is greater (lower)than that under NSIS. This statistical analysis gives us abrief, but clear, picture of how the SIS affects the supplychain performances.
Several observations can be drawn from the above re-sult. Firstly, the retailer, the “bystander” of SIS collabora-tion, does not benefit from the wholesaler sharing the ship-ment quantity information with the agent. In this case, theretailer’s performances are almost the same under SIS andNSIS. This can be attributed to the retailer and the agent’sbuffer effect, which partially helps the retailer resist uncer-tainties caused by the external environment, i.e. unreliableshipments from upper tiers in the supply chain. Secondly, theon-hand stock level of the agent is reduced while its servicelevel is preserved, or even improved. As a result, the agent’stotal relevant cost, including the holding cost and backordercost will decrease, so the SIS is a pure benefit to the agent.Thirdly, it is not clear whether the information provider (thewholesaler) may gain net benefit from the SIS, since inmost cases, the service level and the stock level changes inthe same direction, i.e., shows the typical trade-off. There-fore, whether the total relevant inventory cost reduces orincreases under SIS depends on the specific environment(Table 1).
4.3. Extended analysis: the impact of demand variance
In previous sections we have focused on how the sup-plier’s target service level affects SIS performance. Here, weextend the scope to the impact of demand variance in this
436 C. Zhang et al. / Omega 34 (2006) 427–438
Table 295% confidence interval of the differences of SIS vs. NSIS system performances (same variance not assumed)
FRW0 STD FRR FRA FRW STOCKR STOCKA STOCKW
0.5 10 0 0 + 0 − 030 0 +∗ + 0 − +50 0 + + 0 − +
0.6 10 0 0 − 0 − +30 0 0 0 0 − +50 0 +∗ + 0 − +
0.7 10 0 +∗ − 0 − −30 0 +∗ − 0 − 050 0 +∗ 0 0 − +
0.8 10 0 + − 0 − −30 0 +∗ − 0 − −50 0 0 − 0 − −
0.9 10 0 0 − 0 0 −30 0 0 − 0 − −50 0 0 − 0 − −
0.95 10 0 0 − 0 0 −30 0 0 − 0 −∗ −50 0 0 − 0 − −
FRW0: target fill rate of wholesaler; STD: the standard deviation of market demand; FRR: actual fill rate level of retailer; FRA: actual fillrate of agent; FRW: actual fill rate of wholesaler; STOCKR: the actual on-hand stock of the retailer; STOCA: the actual on-hand stock ofthe agent; STOCKW: the actual on-hand stock of the wholesaler; 0: the interval covers zero; +: the difference is greater than zero; −: thedifference is less than zero; *: p < 0.1 (90% confidence).
Table 395% confidence of the mean differences of the system performances ratio: SIS vs. NSIS (same variance not assumed)
FRW0 FRA FRW STOCKA STOCKW
0.5 1, 2 < 3 1 < 2 < 3 1 < 2, 30.6 1 < 2, 3(1 < 2 < 3∗) 1 < 2, 30.7 1, 2 < 3(1 < 2 < 3∗) (1 > 2∗) 1 < 2, 30.8 1 < 2 < 3 1 > 3(1 > 2, 3∗) 1 < 2, 30.9 1 < 3 1 > 2 > 3 1 < 3(1 < 2, 3∗)
0.95 1 > 2 > 3 1 < 3
FRW0: target fill rate of wholesaler; FRA: Ratios of SIS to NSIS on agent’s actual fill rate; FRW: Ratios of SIS to NSIS on wholesaler’sactual fill rate; STOCKA: Ratios of SIS to NSIS on agent’s on-hand stock level; STOCKW: Ratios of SIS to NSIS on wholesaler’s on-handstock level; 1: in the case that the demand variance is 10; 2: in the case that the demand variance is 30; 3: in the case that the demandvariance is 50; *: p < 0.1 (90% confidence).
information environment, for a given wholesaler’s fill rate.All the experiment settings follow our last experiment exceptthat we vary the demand standard deviation � = 10, 30, 50.To examine the impact of demand variance on SIS, we stillchoose the chain members’ service level and average stockholding as the measurements. Independent-samples t-testsare used in SPSS [15].
The statistical result indicates that the influence of demandvariance on each tier’s performance is as expected, i.e., theretailer cannot gain significant benefit from the wholesalersharing the shipment quantity information with the agent;the on-hand stock level of the agent is usually reduced whileits service level is preserved, or even improved; it is not clear
whether the wholesaler may gain net benefit from the SIS.We now examine the significance of the performance
changes under SIS vs. NSIS, i.e., the ratio of performancewith SIS and NSIS, when facing different demand variance.This will help us understand the impact of demand varianceon sharing shipment quantity information. As we have al-ready shown (Table 2) that the performances of the retailerdo not have significant changes when implementing SISwith different demand variance, here we focus on analyzingthe impact, and its trend, on the agent and the wholesaler.Table 3 shows the results of one-way ANOVA test viaSPSS.
The result shows that the agent’s fill rate improvement
C. Zhang et al. / Omega 34 (2006) 427–438 437
is not significantly influenced by demand variance in mostcases. When the wholesaler’s target fill rate is relatively high,i.e. above 60%, the reduction of stock level by implement-ing SIS is increasing in demand variance increases (as inTable 3, the agent’s stock level is reduced by SIS, so the ra-tios of SIS to NSIS at this tier are all below 1, which meansthe larger number of the ratio, the less reduction). This resultproves, from another perspective, the positive correlationof demand uncertainty and shipment uncertainty: demandfluctuating reduces the reliability of shipment of each tier inthe supply chain. This concurs with observations in supplychain management that on-time delivery is more difficult tomaintain when demand variability increases, and the tiersin the chain naturally increase their safety stock to resistsuch uncertainty. Therefore, as the shipment uncertaintyfrom the agent’s supplier, i.e. the wholesaler, increases,due to the increasing demand variance, SIS may encouragethe agent, which facilitates such information, to reduceits safety stock level and maintain a stable delivery to theretailer. When the wholesaler’s target fill rate is relativelylow, e.g., 0.5, the agent is willing to store more stock as de-mand variance increases. At the wholesaler’s, it is still thetypical trade-off: as demand variance increases, it naturallyincreases the safety stock level in order to meet a promisedservice level.
5. Discussion
We have quantified the impact of SIS in the supplychain. The result shows that SIS helps downstream organi-zations adapt and resolve shipment uncertainty well, whilethe adjustment speed in the NSIS environment is muchslower—depending on the feedback delay time between itssupplier and itself. The more uncertain the shipments, themore benefit the organization can gain from SIS. When thebacklog is eventually fulfilled within a fixed period, SIShas little value. Therefore, if a supplier has a good enoughorder fulfillment track record SIS may not be needed.Otherwise, faster collaboration and sharing of shipmentinformation among supply chain members via IT may helpimprove supply chain performance. SIS always benefitsthe information receiver but may not do the same to theinformation sender. This uneven benefit to IS participantsmay cause barriers to information collaboration, but mayalso motivate participants to negotiate a satisfactory shareof the IS implementation costs.
This research can be extended in several ways. Firstly, weassume that a downstream member trusts that its order canbe fulfilled completely on time (and it retains safety stockto resist demand uncertainty). However, it may be reason-able to adjust its order based on shipment history, such asaverage upstream fill rate, to reduce shipment forecast er-ror. This may require a new order decision model. Secondly,the shipment quantity uncertainty has a close relation withthe issue of lead-time uncertainty and may be solved by in-
creasing safety stock.
Appendix
Proof 1.
E(Qt ) = 1 − (1 − �′)t�′ d (case A1)
= E[dt + (1 − �′t−1)(dt−1 + (1 − �′
t−2)
× (......(d2 + (1 − �′1)d1)))]
= d + (1 − �′)d + · · · + (1 − �′)t−1d
= 1 − (1 − �′)t�′ d.
Proof 2.
E(Q′t ) = 1 − (1 − �′)[t/L]
�′
× d(t = m∗L and [t/L] = m + 1) (case A1)
= E[dt + (1 − �′t−L)(dt−L + (1 − �′
t−2L)
× (......(dL+n + (1 − �′n)dn)))]
= d + (1 − �′)d + · · · + (1 − �′)md
= 1 − (1 − �′)[t/L]�′ d.
Proof 3. limt→∞ 1t
∑E(Qt ) = 1
�′ d (case A1).
Lemma 1. limt→∞ xt = A ⇒ limt→∞ 1t
∑xt = A.
(When |xt − A| < �2 ∀t > N1 (N1 is a nature number and
� is positive), there is | (x1−A)+......(xN1−A)
t | < �2 ∀t > N2
(N2 is nature). Let N = max{N1, N2}, so∣∣∣∣x1 + ......xt
t− A
∣∣∣∣�∣∣∣∣ (x1 − A) + ......(xN1 − A)
t
∣∣∣∣+∣∣∣∣ (xN1+1 − A) + ......(xt − A)
t
∣∣∣∣< �
2+ t�
2t
= � ∀t > N).
Therefore limt→∞ 1t
∑E(Qt ) = limt→∞ E(Qt ) = d
�′ .
438 C. Zhang et al. / Omega 34 (2006) 427–438
Proof 4. limt→∞ 1t
∑E(Qt ) = d (case A2) (as m and Q
are finite numbers)
= limt→∞
1
m + t
m+t∑1
E(Qt ) = limt→∞
1
m + t
× E
[m+t∑
1
di +(
m+t∑2
(1 − �i−1)Qi−1
)
−(
t∑1
(1 − �i )Qi
)]
= limt→∞
1
m + tE
⎡⎣m+t∑
1
di +⎛⎝m−1∑
1
(1 − �t+i−1)Qt+i−1
⎞⎠⎤⎦
= limt→∞
1
m + t
∑E
(m+t∑
1
di
)= d .
Proof 5.
E(nL+t ) = SS + d
�′ (�′ − 1 + (1 − �′)t+1) (case A1)
= E(nL+t−1 + �′tQt − dl+t )
= E
⎛⎝nl +
t∑i=1
(�′iQi − dl+i )
⎞⎠
= SS +(
1 − 1 − (1 − �′)t+1
�′
)d .
Proof 6. limt→∞ E(nl+t+m) = SS (case A2)
= limt→∞ E[nl+m+t−1 + �′
m+tQm+t
− dl+m+t + (1 − �′t )Qt ]
= limt→∞ E
⎡⎣nl +
t+m∑i=1
(�′iQi − dl+i ) +
t∑i=1
(1 − �′i )Qi
⎤⎦
= limt→∞ E
⎡⎣SS +
t∑i=1
(Qi − dl+i )
+m∑
i=1
(�′t+iQt+i − dl+t+i )
⎤⎦
= SS.
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