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Int. J. Production Economics

Int. J. Production Economics 123 (2010) 118–136

0925-52

doi:10.1

� Cor

E-m

d-klabja

journal homepage: www.elsevier.com/locate/ijpe

Inventory control in serial systems under radiofrequency identification

Jinxiang Pei �, Diego Klabjan

Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA

a r t i c l e i n f o

Article history:

Received 16 January 2009

Accepted 25 July 2009Available online 3 August 2009

Keywords:

Inventory control

Radio frequency identification

73/$ - see front matter & 2009 Elsevier B.V. A

016/j.ijpe.2009.07.011

responding author.

ail addresses: [email protected] (J.

[email protected] (D. Klabjan).

a b s t r a c t

The widely adopted slap-and-ship radio frequency identification strategy provides

valuable information to retailers. On the other hand, suppliers struggle to find benefits

even though they are submerged with new data. Radio frequency identification provides

complete visibility of their shipments, including the time and location of every pallet,

case or even item. We provide a novel model relying on such data that is capable of

producing better inventory and shipping control policies. We first propose a

comprehensive inventory model for serial systems that captures both the supply and

distribution information. We show that the underlying cost-to-go can be decomposed

into two lower dimensional functions. In a special case, the optimal replenishment and

shipping policies are base stock with respect to the underlying positions. In addition, we

also analytically study the value of radio frequency identification in terms of the

expected total minimum cost over a finite time horizon by introducing partial radio

frequency deployment scenarios. Results indicate that additional cost reductions are

possible with broader deployments.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

A basic radio frequency identification (RFID) systemincludes two components: a transponder and an inter-rogator. A transponder is a tiny microcomputer consistingof a microchip, antenna and memory connecting thesetwo. In the simplest form, the so-called passive RFIDtransponders, transponders are idle until woken up by aninterrogator via radio wave signals. Interrogators are,therefore, constantly emitting signals to provide power tothe transponders within their antenna’s field of work.When a transponder receives a signal from an interro-gator, it absorbs some of the radio energy to power itselfand sends back a response, which among other datastored in its memory, includes the transponder’s uniqueidentification number. The interrogator decodes this

ll rights reserved.

Pei),

information and passes it to information systems. In atypical RFID deployment within a supply chain, inter-rogators are mounted at critical locations. Every timegoods affixed with transponders come within the readrange of an interrogator, the location, time, and identifica-tion are recorded. Under normal conditions a typicalinterrogator may interrogate hundreds of transpondersper second.

One of the recent information technology advances isthe adoption of RFID technology in inventory controlsystems. Early benefits earned from RFID deployments areinventory asset tracking, advance shipping notice, real-time order progress information for retailers, and real-time shipping visibility for suppliers. The additional RFIDgenerated information could possibly result in improvedinventory control policies and potential new businessapplications. One of the biggest setbacks to a wider RFIDadoption is the lack of return of investment. Many entitiesin supply chains are overwhelmed with data generatedfrom RFID deployment, yet this data is seldom used toenhance business intelligence.

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As an emerging auto data-capture enabler, RFIDtechnology intrigues supply chain researchers and practi-tioners. Companies have rushed to develop RFID solutionswithout having a clear idea about the potential value ofRFID to their business. One of the values of RFID in supplychains is derived from better supply chain visibility. AnRFID deployment improves supply chain visibility; how-ever, many benefits in inventory control are still untapped.The value of RFID obtained from labor cost reductions andsimilar basic benefits can be satisfactorily assessed by casestudies. Empirical studies and proofs of concepts are oflimited scope since they have to rely on existing processesand data. It is not clear how RFID can further reducesupply chain costs via improved visibility. Educatedguesses are currently driving such estimates. Analyticallymodeling inventory control systems with an RFID deploy-ment is critical to enhance our understanding of the valueof RFID.

Beyond replenishment inventory control on the supplyside, the distribution side deals with shipping decisions.As is the case with the supply, RFID deployments yield realtime visibility of shipments. To achieve such capability, itis typical to install portals with readers at importantlocations (e.g., in and/or outbound docks) and tag thecorresponding goods. Consider an RFID mandate imposedby a retailer. A supplier places transponders on theproducts and ships them to the retailer’s distributioncenters, thus complying with the corresponding RFIDmandate. Since typically transponders are affixed justbefore leaving the final facility of the supplier, thisstrategy is known as slap-and-ship. Clearly, to obtainfurther benefits from RFID, it is advised to push thetagging process further upstream in the supplier’s ownsupply chain. The main drawback of slap-and-ship is theinability to produce return on investment. As alreadydiscussed, the retailer could benefit from continuouslymonitoring the inventory levels and outstanding orderprogress in its own chain. However, it is not clear how thesupplier can benefit from the mandate even if real-timedistribution information is provided by the retailer. This isa typical quote about such suppliers: ‘‘They (an applesupplier to Wal-Mart) know exactly what day and timethe container was scanned through its portal, when itentered a distribution center and what day and time itwent to the store. The company has yet to determine howbest to use this data.’’, Inbound Logistics, June 2006. Themain objective of this research is to show how supplierscan benefit under such circumstances even under slap-and-ship. We assume a decentralized system where eachentity in the chain acts independently. The main entity isan installation somewhere in the middle of the entirechain. The firm makes two decisions: (1) the replenish-ment decision from its own supplier and (2) the shipmentdecision how much to ship downstream.

We study a single-product, multi-echelon serial supplychain system, in which the supplier streamlines both itsreplenishment and distribution processes by using theRFID data. We propose a dynamic programming model tocapture the real time inventory information generatedfrom an RFID deployment across the entire supply chain.The paper is organized as follows. A comprehensive

inventory model is first presented in Section 2. In addition,we provide a special case with an instantaneous replen-ishment process. We apply multi-echelon techniques todecompose the proposed model into two sub-problems.The optimal control policy under certain conditions ischaracterized as the echelon base stock policy. The valueof RFID in a serial distribution process is clearly identifiedand rigorously proved in Section 3 through discussions ofpartial RFID deployment scenarios. We conclude theintroduction with a brief literature review.

1.1. Literature review

Our models assume stochastic lead times. In standardsingle-stage stochastic inventory models, the lead time isconsidered either as a known deterministic constant or arandom variable with known distribution. In these models(Kaplan, 1970; Nahmias, 1979; Ehrhardt, 1984) the leadtime is assumed to be time-independent with knowndistribution. The non-crossover property is also assumedin order to make the study tractable. In our analysis, weborrow concepts from multi-echelon systems. The semi-nal work on the serial multi-echelon inventory problemwas conducted by Clark and Scarf (1960). In their research,the global system is decomposed into separate sub-systems. At each echelon, it is optimal to follow the basestock policy with respect to the echelon inventory.

RFID is the most promising technology providingcomplete and comprehensive supply chain visibility. It isa surprisingly simple computing and communicationarchitecture since only two basic building blocks areneeded—a tag and a reader (AIM Inc., 1999; Clampittet al., 2006). We have already argued that RFID is anenabling technology for visibility that is assumed by ourmodel. There are estimates about the value of RFID insupply chain management, including labor cost savings,reduced inventory holding costs, and stock-outs (Lee andOzer, 2007; Hardgrave, 2005). Most studies regardingRFID in inventory control concentrate on supply chainsimulations (Lee et al., 2004; Fleisch and Tellkamp, 2005;Kang and Stanley, 2005).

Bottani and Rizzi (2008) describe profitability ofdeploying RFID in a three-tier supply chain. They showby a real world case in a fast-moving consumer goodsmarket the benefits of pallet-level tagging, and muchmore lenient results of case- and item-level tagging.Ustundag and Tanyas (2009) investigate impacts ofdifferent factors, such as product value, lead time, anduncertain demand on the supply chain cost performanceat echelon levels in conjunction with RFID tagging.

Among the few studies that analytically deal with RFIDin inventory control, Song and Zipkin (1996) provide amodeling framework for the inventory control problemwith supply information. While this study dates back topre-era of modern RFID, it requires data available onlythrough today’s RFID deployments. The replenishmentlead time is time-dependent and evolves over time. Weborrow their modeling technique and enrich their studyby focusing on the distribution side, thus dealing with twoconcurrent decisions. We also present results addressing

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J. Pei, D. Klabjan / Int. J. Production Economics 123 (2010) 118–136120

the value of RFID in such distribution systems. Gaukleret al. (2008) quantify the benefits of RFID in the supplysystem of a retailer who faces uncertain demand and theoption of emergency orders. They develop an orderprogress information model to study the optimal policiesfor both regular and emergency orders, under theassumption of a single outstanding order. Szmerekovskyand Zhang (2008) discuss impacts of item-level RFIDtagging in a two-tier system under vendor managedinventory. First, the demand processes are characterizedin both RFID and non-RFID systems. Then they study thecontrol policies for each entity. Second, they studychannel coordination efforts through sharing of the RFIDcosts by comparing centralized and decentralized sys-tems. Atali et al. (2006) analytically study inventoryinaccuracy, which is a joint effect of transaction errors,shrinkage, and misplacements. RFID yields more accurateinventory records and easier audits. Bottani et al. (2009)examine the impact of RFID on out-of-stocks of promo-tional items in the fast-moving consumer goods context.They show that by reducing the main causes of unavail-ability of sales through RFID can yield substantial savings.In addition, a reengineering process is exploited tocompare the reduction of stock-outs. Results of anexperiment suggest that RFID has the potential to reducelosses and improve profits in fast-moving consumergoods.

New processes hinging on RFID are of particularinterest. Expediting on the supply side is one such processthat can benefit from RFID. Kim et al. (2006) addressexpediting strategies based on RFID data. Pricing strate-gies based on added value on goods are discussed inSchneider (2007). RFID is used as a technology to allowdynamic pricing.

2. Comprehensive inventory model

Most of prior inventory models deal with supplyinformation, i.e., the focus is on the procurement side ofsupply chains. Such models are appropriate for retailersand the endpoints of supply chains. We extend thesemodels to capture both supply and distribution sides. Inthis section, a comprehensive inventory control model isfirst proposed. It contains both the supply and distributionsides of the entire chain. Next, we give the state reductionresult for the pure distribution setting (upstream installa-tions are neglected). This result is then extended byincluding upstream installations. Finally, optimal controlpolicies are studied.

Given several installations as shown in Fig. 1, let ui

denote the ith upstream installation and let dj denote the

u1

Xt

• • •

It, u1 It, un−1Y

IOt

un-1 un/d0u0

Fig. 1. The comprehensiv

jth downstream installation. There are two specialinstallations: dn represents the point-of-sale installation,and un ¼ d0 represents the main decision making installa-tion. It will be called the main installation. We can identifythe main installation as a supplier to Wal-Mart andd1; . . . ; dn�1 as installations owned by Wal-Mart. The maininstallation ships orders to Wal-Mart by directly shippingthrough all these installations. Upstream installationsu0; . . . ;un�1 represent the supply side of the maininstallation. The on-hand inventory IOt at the maininstallation, net inventory It;dn

at the point-of-sale, andthe net inventory It;k for every k 2 fu1; . . . ;un�1g [

fd1; . . . ;dn�1g describe the system. There are two impor-tant differences between standard multi-echelon inven-tory systems and our model. We focus on both the supplyand distribution sides of a decentralized system. Tradi-tional multi-echelon models deal only with the supplyside of a centralized system. In a centralized system,holding cost is accounted for at each installation. Mean-while, in our decentralized system, it is accounted for onlyat a single installation. We consider such a firm with itsphysical location corresponding to installation d0, Fig. 1.Downstream Installations d1; . . . ;dn represent anotherplayer, namely the retailer and upstream installationsu0; . . . ;un�1 attribute to different players as well. If RFID isdeployed end to end in this serial supply chain, then atany point in time, the location of all outstanding ordersplaced with the supplier corresponding to installation u0

are known. Likewise, the location of outstanding ship-ments is also known at any point in time. RFID is anenabling technology for cost effective tracking of goodsand as such it provides real-time visibility, includingrecording quantities of outstanding orders and shipments.The arcs in Fig. 1 represent possible order/shipmentmovements in a time period. There is no order/shipmentmoving into an installation if no incoming arcs attached tothe installation appear. Each installation has a singleoutgoing arc, which could also be a self-loop representingthe scenario of the order staying put. The outgoing arcfrom installation dn represents the exogenous end custo-mer demand Dt .

In order to model downstream shipment transitions,we introduce a random variable W. For each realization w

of W there is a transition vector MðwÞ that specifies wherethe current outstanding shipments at each stage movenext. This vector has n elements and it encodes thetransitions in downstream shipments. In Fig. 2 we showtwo such possible transitions. Self-loops correspond to theevents that the outstanding shipments stay put at theinstallation. For example, under the top realization wehave Itþ1;dn�1

¼ It;d3þ It;d4

and under the bottom realization

Dt

• • •t

It, d1It,dn−1

It, dn

d1 dn-1 dn

e inventory model.

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Yt

...

Enter Arrival

Yt

Enter ArrivalM (w) for a different realization w

M (w) for realization w of W

dn-1 dnd1 d2d3 d4un / d0 ...

dn-1 dnd1 d2 d3 d4un / d0 ...

It, d1

It, dn−1 It, dnIt,d2

It, d3It, d4

It, d1It, dn−1

It, dn

It, d2It, d4

It, d3

Dt

Dt

Fig. 2. Two possible outstanding shipment transitions.


we obtain Itþ1;d4¼ It;d3

þ It;d4. Similarly, we introduce a

random variable R to encode the upstream stochasticorder transitions. The corresponding transition function isdenoted by Q ðrÞ, where r is a realization of R.

2.1. Model

Throughout the document, we use the followingnotation.

Ct p
rocurement cost function in time t
St s
hipping cost function in time t
ht p
er unit holding cost in time t at the main installation un
pt p
er unit shortage cost in time t occurred at the point-of-sale
installation dn

Q ðuk;RÞ o
ne step upstream transition at installation uk , i.e., the
coordinate of Q ðRÞ corresponding to installation uk

Qpðuk;RÞ p
-step upstream transition at installation uk , i.e., transition in
p time periods corresponding to installation uk

Mðdk ;WÞ o
ne step downstream transition at installation dk , i.e., the
coordinate of MðWÞ corresponding to installation dk

Mpðdk ;WÞ p
-step downstream transition at installation dk , i.e., transition
in p time periods corresponding to installation dk

Luu
pstream stochastic lead time Lu
¼Minfl41 : Q lðu0 ;RÞ ¼ ung

Ld d
ownstream stochastic lead time
Ld¼Minfl41 : Ml

ðd0 ;WÞ ¼ dng

It s
tate vector in time t:
It ¼ ðIt;u1; It;u2

; . . . ; It;un�1; IOt ; It;d1

; It;d2; . . . ; It;dn

Þ

Dt s
tochastic demand in time period t
Dl�1t

c
umulative demand over l periods starting from time
t : Dl�1t ¼

Ptþl�1k¼t Dk

We make the following assumptions.

�
We assume that installations u0 and d0, respectively,denote the entering point for the replenishment anddistribution process and installation dn is the point-of-sale location. In addition, un and d0 are physically thesame installation. � We assume that the unmet demand is backlogged at
the point-of-sale installation dn.
� Both upstream and downstream stochastic lead times
are at least equal to or 42. Formally, we require thatu0oQ ðu0;RÞoun and d0oMðd0;WÞodn.

�
We assume that the outstanding orders and shipmentsnever cross in time. The current order and shipmentsequence is preserved, or at least not reversed in bothsupply and distribution networks. Formally,
Q ðuk; rÞ � Q ðuk�1; rÞ for every k ¼ 1;2; . . . ;n and any

realization r of R,

Mðdk;wÞ � Mðdk�1;wÞ for every k ¼ 1;2; . . . ;n and any

realization w of W .
� Orders and shipments cannot be broken apart. Once an
order or shipment is placed, all corresponding unitstravel together.

Note that the lead times of 41 are just a technicalcondition to rule out the trivial case. It requires that bothinbound and outbound orders must go through at leastone intermediate installation before arriving to the finaldestinations. In addition, it is easy to see that the non-crossover property implies that Qp

ðuk; rÞ � Qpðuk�1; rÞ for

every realization r of R, k ¼ 1;2; . . . ;n, and p ¼ 1;2; . . . ; Lu,and Mp

ðdk;wÞ � Mpðdk�1;wÞ for every realization w of W,

k ¼ 1;2; . . . ;n, and p ¼ 1;2; . . . ; Ld.Within time period t, events occur as follows:

�
the state vector It is observed at the beginning of timeperiod t, next; � the shipping decision is made, then; � the replenishment decision is made; � replenishment orders due in time period t arrive and � shipment orders due in time period t arrive; � demand Dt in time period t occurs; and finally � holding and penalty costs are assessed.
It is important to point that at the end of each timeperiod the holding and penalty costs in the proposedmodel are evaluated differently from standard inventorymodels. In our proposed model, the holding cost isassessed at the main installation at the end of each timeperiod. On the other hand, the penalty cost is evaluated atthe point-of-sale installation where the customer demand

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might not have been satisfied. It is crucial to note that weonly account for the costs incurred at the main installation(the supplier) even though there are other costs incurredby the remaining entities, for example the retailer. As forthe replenishment orders and shipments, we charge theprocurement and shipping costs when they are placed. Wenote that all these costs are actually incurred by the maininstallation (even though the penalty cost is accounted forat the point-of-sale installation). This reflects our initialassumption of a decentralized system focusing on themain installation. This also implies that there is noholding cost at other installations but un ¼ d0. Let Xt � 0denote the procurement quantity and Yt � 0 the shippingquantity in time period t, Fig. 1. For ease of notation, weuse It;u0

¼ Xt and It;d0¼ Yt in Section 2.

The underlying system dynamics are

IOtþ1 ¼ IOt þXun�1

uk :Q ðuk ;RÞ¼un1�k�n�1

It;uk� It;d0

and for u1 � ui � un�1 we have

Itþ1;ui¼

Xuk :Q ðuk ;RÞ¼ui

0�k�n�1

It;uk.

On the distribution side, for d1 � dj � dn�1 we obtain

Itþ1;dj¼

Xdk :Mðdk ;WÞ¼dj

0�k�n�1

It;dk

and

Itþ1;dn¼ It;dn

þX

dk :Mðdk ;WÞ¼dn1�k�n�1

It;dk� Dt .

Note that due to the non-crossover property, ifQ ðuk;RÞ ¼ un for every k; 1 � k � n� 1; then Q ðukþ1;RÞ

¼ Q ðukþ2;RÞ ¼ � � � ¼ Q ðun�1;RÞ ¼ un. Similar propertyholds for M. The on-hand inventory IOtþ1 in time t+1equals the on-hand inventory in time t plus the out-standing orders which have just arrived at the maininstallation, minus the shipment Yt made in time t. Theon-hand inventory at installation ui equals the arrivedoutstanding orders and potential Xt . Downstream instal-lation inventory Itþ1;dj

in time t+1 is equal to theoutstanding shipments which move to installation dj plusshipment Yt if transition Mðd0;WÞ ¼ dj occurs. Finally, thenet inventory at the point-of-sale installation in time t+1equals the net inventory in time t plus the outstandingshipments, which arrive at the point-of-sale, minus thecustomer demand in time t.

Let bVt be the expected total minimum system coststarting from time t up to T under an optimal policy.The optimality equation over finite time horizon [0, T]reads

bVtðItÞ ¼ MinIt;u0�0

It;d0�0

It;d0�IOt

CtðIt;u0Þ þ StðIt;d0

Þ�

þ ht � ER IOt þ

Xun�1


It;uk� It;d0

0BB@1CCA

þ pt � ED;W Dt � It;dn

�X

dk :Mðdk ;WÞ¼dn1�k�ðn�1Þ

It;dk

0BB@1CCAþ26643775

26643775

þED;W ;R½bVtþ1ðItþ1Þ�

o. (1)

The action space constraint It;d0� IOt guarantees that we

do not ship more than we have on hand.This comprehensive model is complex. We instead first

analyze a simplified version and tackle the comprehensivemodel after learning from the analysis in such a setting. Inthe following section, we study a special case of thecomprehensive model where we neglect the replenish-ment stochastic lead time, i.e., the replenishment order isinstantaneous. After better understanding such a distribu-tion model, we decompose the comprehensive model intotwo problems, which can be handled separately. Thisdecomposition is shown in Section 2.3.

2.2. Distribution side

This section demonstrates how the supplier couldmake use of the real-time information to improve thedistribution operations. Imagine that a supplier to Wal-Mart has access to the distribution information of itsgoods via Retailink (http://retaillink.wal-mart.com). In atraditional setting, only point-of-sale information isavailable. Soon after first RFID enabled pallets startedarriving to the distribution centers, suppliers had com-plete visibility of their shipments through the samesystem.

To recapture, each shipment goes through a stream ofinstallations labeled as d0; d1; . . . ;dn. Each installationcorresponds to a transportation node or a specific stock-piling point. Installation d0 stands for the supplier itself,i.e., the main decision-making installation and, dn repre-sents the point-of-sale. Furthermore, each of the nodes isdeployed with RFID. Without RFID, it would not beaccessible to observe the installation inventories and theexogenous stochastic information W. To better understandthe distribution process, instantaneous replenishment isassumed.

Two models are discussed in this section. The fullmodel captures all installation inventories, while thereduced model only includes the inventory on-hand andthe downstream echelon inventory with respect to thepoint-of-sale. The full model serves as the starting pointand it models slap-and-ship. The reduced model issimpler, nevertheless, it is equivalent in a certain senseas shown here. Quantities Vt represent the expected totalminimum cost over time periods t; t þ 1; . . . ; T in the fullmodel or the cost-to-go. Similarly, Vt represent theexpected total minimum cost over time periods t; t þ

1; . . . ; T in the reduced model. Since we do not considerthe costs incurred beyond time point T, the terminal costsVTþ1 and VTþ1 are assumed to be zero.

An illustration of the full model is shown in Fig. 3. Inaddition to previously defined terms, we depict that theorder replenishment is instantaneous. The rest of the

http://retaillink.wal-mart.com

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model configuration is identical to the comprehensivemodel.

The optimality equation of the full model reads

VtðIt;dÞ ¼ MinIt;u0�0

It;d0�0

It;d0�IOt


Þ þ htðIOt þ It;u0� It;d0

Þ

þpt � ED;W½ðDt � It;dn

�P


It;dkÞþ� þ ED;W

½Vtþ1ðItþ1;dÞ�

8>><>>:9>>=>>;,

(2)

where the state vector is It;d ¼ ðIOt ; It;d1; It;d2

; . . . ; It;dnÞ. The

underlying system dynamics equations are

IOtþ1 ¼ IOt þ It;u0� It;d0

,

Itþ1;dj¼

Xdk:Mðdk ;WÞ¼dj

0�k�n�1

It;dkd1 � dj � dn�1,

Itþ1;dn¼ It;dn

þX


It;dk� Dt .

The first equation above is different from the correspondingequation in the comprehensive model since there is no leadtime on the supply side. The inventory on-hand in time t+1equals the inventory on-hand in time t plus instantaneousreplenishment order It;u0

, minus the shipment It;d0in time t.

The remaining two equations are identical.The full model contains many state variables. The

multi-dimensional state vector results in difficulties whencomputing inventory control policies. We next introducethe reduced model with distribution information asshown in Fig. 4. The reduced model has only twovariables: the inventory on-hand IOt and downstreamechelon inventory rt;1 with respect to the point-of-sale.Formally, we define rt;1 ¼

Pnk¼1It;dk

.The optimality equation for the reduced model reads

VtðIOt ; rt;1Þ ¼ MinIt;u0�0

It;d0�0

It;d0�IOt



Þ�

þ ELd

ðptþld�1

� ED½ðDLd

�1t � It;d0

� rt;1Þþ�Þ

þ ED Vtþ1ðIOtþ1; rtþ1;1Þ� ��

, (3)

where the underlying system dynamics equations are

IOtþ1 ¼ IOt þ It;u0� It;d0

,

rtþ1;1 ¼Xn

k¼1

Itþ1;dk

¼ Itþ1;dnþXn�1

k¼1

Itþ1;dk

¼ It;dnþ

Xdk :Mðdk ;WÞ¼dn

1�k�n�1

It;dk� Dt þ

Xn�1

k¼1

Xdg :Mðdg ;WÞ¼dk

0�g�n�1

It;dg(4)

d0 d1

• • •It, d0

IOt It, d1

It, u0

Fig. 3. Full model with dist

¼ It;dn� Dt þ

Xn

k¼1

Xdg :Mðdg ;WÞ¼dk

0�g�n�1

It;dg(5)

¼ It;dn� Dt þ It;d0

þXn�1

g¼1

It;dg

¼ rt;1 þ It;d0� Dt .

In (4) we use system dynamics equations forItþ1;di

for every i ¼ 1; . . . ;n. The third term in (5) coversall the possibilities of shipment movements in time t. As aresult, this term corresponds to the summation of alloutstanding shipments, plus shipment It;d0

; which has justbeen made in time t.

Next, we state and prove the main result in thissection. The following theorem, proved in Appendix A,shows the relationship between the full and reducedmodels in the distribution process.

Theorem 1. We have

VtðIt;dÞ ¼ VtðIOt ; rt;1Þ þ at , (6)

where

at ¼ ELd XLd�2

j¼0

ptþj � ED;W Dj

t � It;dn�Xjþ1

p¼1

Xdk :M

pðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@1CCAþ26643775

0BB@1CCA

represents the expected total penalty cost over the shipping

lead time.

By means of Theorem 1, we show that in a distributionprocess the original value function is divided into two parts:the reduced value function and a penalty cost term. Thisdecomposition proves to be crucial in order to study theinventory control policies in Section 2.3. Accessing theechelon-level inventory statuses and the current inventoryon-hand suffices to characterize the optimal policies since thepenalty cost term only depends on the exogenous demand,shipment movements, and the current inventory. Further-more, the penalty term is an explicit, closed form function.

2.3. Decomposition and analysis of the comprehensive

model

In the previous section, we discussed in detail thedistribution process with a complete RFID deployment. Bya system-wise deployment of RFID both supply anddistribution sides can be tracked. The comprehensiveinventory model with both supply and distributioninformation has already been presented in Section 2.1.In this section, the system is decomposed into two

dn-1 dn

Dt

It, dn−1It, dn

ribution information.

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dn-1 dn

Dt

• • •

IOt

rt, 1

d0 d1

It, d0

It, u0

Fig. 4. Reduced model with distribution information.


sub-problems in terms of the value function. Under someassumptions the inventory control policy of each sub-problem is determined separately. The optimal systemcontrol policy is the echelon base stock policy with twothreshold numbers, as shown later in this section.

By applying the same techniques as those presented inSection 2.2 with respect to (1), we obtain the followingresult.

Theorem 2. We have

bVtðItÞ ¼ at þ MinIt;u0�0

It;d0�0

It;d0�IOt


Þ

þht � ERðIOt þ

Pun�1


It;uk� It;d0

Þ

þELd

ðptþLd�1 � E

D½ðDLd

�1t � It;d0

� rt;1Þþ�Þ þ ED;W ;R

½eVtþ1ðItþ1Þ�

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

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

(7)

where

at ¼ ELd XLd�2

j¼0

ptþj � ED;W Dj

t � It;dn�Xjþ1

p¼1

Xdk:M

pðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@1CCAþ26643775

0BB@1CCA

and rt;1 is defined in Section 2.2.

To derive an optimal policy, we focus on the minimiza-tion part in (7). We first define NðuiÞ ¼ minfuk : Q ðuk;RÞ ¼

ui; 0 � k � n� 1g for every i ¼ 1;2; . . . ;n. We denoteIOe

t ¼ IOt þ rt;1, and Iet;ui¼Pn�1

k¼i It;ukþ IOe

t for every i ¼

0;1;2; . . . ; n� 1. We represent the system state asIet ¼ ðI

et;u1; . . . ; Ie

t;un�1; IOe

t ; rt;1Þ. Next, we rewrite (7) in theechelon version as

V tðIet Þ ¼ Min

It;u0�0

It;d0�0

It;d0�IOe

t�rt;1


Þ þ ht � ERðIe

t;NðunÞ� rt;1

�

� It;d0Þ þ ELd

ptþLd�1 � E

D DLd�1

t � It;d0� rt;1

� �þ � �þ ED;W ;R

½V tþ1ðIetþ1Þ�

. (8)

In (8), the system dynamics are expressed as

rtþ1;1 ¼ rt;1 þ It;d0� Dt ,

IOetþ1 ¼ Ie

t;NðunÞ� Dt ,

and for every i ¼ 1;2; . . . ;n� 1 we have

Ietþ1;ui

¼ Iet;NðuiÞ

� Dt .

Theorem 3. If the shipping cost functions St for t ¼ 1; . . . ; Tare linear and St � ht , theneVtðI

et Þ ¼ Vtðrt;1Þ þ GtðI

et;u1; . . . ; Ie

t;un�1; IOe

t Þ. (9)

In (9), the downstream sub-system follows:

Vtðrt;1Þ � Minrt;d0�rt;1

Stðrt;d0� rt;1Þ � ht � rt;d0

nþ ELd

ðptþLd�1 � E

DðDLd

�1t

h� rt;d0

Þþ�Þ þ ED Vtþ1ðrtþ1;1Þ

� �o, (10)

where rtþ1;1 ¼ rt;d0� Dt and action rt;d0

is the downstream

echelon inventory level after shipping in time t. Function Gt is

a function of only the upstream echelon inventories. More-

over, the optimal decision rule in the downstream sub-system

corresponding to shipping quantities follows the base-stock

policy with the base stock levels rt;d0derived appropriately

from (10).

Proof. The proof is provided in Appendix B.

In Theorem 3, we show how to simplify the valuefunction of the comprehensive model. We now discuss theunderlying replenishment inventory control policy.

Corollary 1. If the procurement cost functions Ct for t ¼

1; . . . ; T are all linear in addition to St being linear and St � ht

for t ¼ 1; . . . ; T; then the replenishment decision also follows

the base stock policy.

Proof. Since the value function can be decomposed as in(9) and the sub-systems Vt and Gt are separable in termsof the state variables, we borrow concepts from the multi-echelon inventory control techniques (Scarf, 1960). Wefirst obtain the optimal downstream echelon stock levelsrt;d0

as discussed in Theorem 3. We next move upstream tofocus on the replenishment process. It is easy to see byapplying convexity that all Gt ’s are convex. Based on (19)in Appendix B, we obtain the upstream base stock levelrt;u0

in time t as in Song and Zipkin (1996). If Iet;u1

ort;u0,

the optimal order quantity is rt;u0� Ie

t;u1; otherwise, the

optimal order quantity is zero. This completes the proof.

Under the conditions stated in Corollary 1, the optimal

control policy for the comprehensive inventory model

proposed in Section 2.1 follows the echelon base stock

policy and has two threshold numbers rt;d0and rt;u0

in any

given time period t. &

3. Value of RFID in distribution

In order to identify the value of RFID in inventorycontrol systems, we study partial RFID deploymentscenarios, in which only some installations of the chainare covered by RFID. As a result, there is a possibility that

ARTICLE IN PRESS


the supplier does not know the location of shipments oncethey leave the RFID-enabled installations. We differentiatetwo types of shipments in transit: those with bothlocation and age information and those with only ageinformation. Note that the age is the number of periodssince the order was shipped out from the supplier atinstallation d0. By comparing the systems with differentscope of RFID deployments, we are able to analyticallyidentify the value of RFID in the system. For simplicity, wefocus on the downstream distribution side, i.e., thereplenishment lead time is zero.

We study the finite time horizon inventory problemwith T time periods. In this section, two partial RFIDdeployment scenarios are proposed, as shown in Figs. 5and 7. Scenario 1 represents the case where the firsts downstream installations are RFID enabled, while inscenario 2 an additional downstream installation dsþ1 isdeployed with RFID. We assume that the remainingconfiguration in the two scenarios is identical with theaim to isolate the contribution of RFID. Furthermore, bothtime-labeled (age of outstanding shipments) and location-labeled inventory information is included in the two partialscenarios, while only location inventory information iscaptured in the complete RFID deployment model discussedin Section 2. RFID system deployed in the first s installationsprovides information of both types, however, only the time-labeled inventory information within the non-RFID zone isused since there the location cannot be captured.

3.1. Two partial RFID deployment scenarios

In order to capture ages of outstanding shipments, thefollowing new notation is required. In the first s installa-tions the system dynamics are as in Section 2. Installa-tions dsþ1; dsþ2; . . . ; dn do not have RFID and thus only ageis captured. Once a shipment leaves installation ds, thelocation and its progress is no longer known, only age isavailable. Shipments then arrive randomly based on age asin Kaplan (1970).

qj
probability that outstanding shipments, which are j periods old or
more, arrive during the current time period

J
random variable with the distribution based on qj
Rt;k
amount of the outstanding shipments in time t placed k periods
ago; we set Rt;k ¼ 0 if such shipments have arrived before time t

At;j
age of the ‘‘oldest’’ shipment at installation dj in time t
at;j
age of the ‘‘youngest’’ shipment at installation dj in time t
m(t)
maxfAt;1;At;2; . . . ;At;sg
m0(t)
maxfAt;1;At;2; . . . ;At;s;At;sþ1g; clearly, we have m0(t)Zm(t)
Vst

the value function with RFID deployed in the first s installations

Ist

state vector in time t with RFID deployed in the first s installations

Ist ¼ ðIOt ; It;dn

; at;1; . . . ; at;s;At;1 ; . . . ;At;s ;Rt;1 ; . . . ;Rt;T�1Þ

RFID-enabled Network

• • •Yt

IOtIt, d1

d0 d1

Xt

d2

It, d2

ds

It, d

Fig. 5. Partial RFID impleme

If no outstanding shipments appear at installation dk,we set at;k ¼ At;k ¼ 0. We require Rt;0 ¼ 0. The state spacezs

t under s RFID-enabled installations in time t is definedas follows. Vector Is

t 2 zst if and only if:

�

s

nta

IOt � 0,
� for every i ¼ 1;2; . . . ; s we have 0 � at;i � At;i, � if for any i ¼ 1;2; . . . ; s it holds that at;i ¼ 0; then
At;i ¼ 0,
� if for any i ¼ 1;2; . . . ; s such that at;j40, then Rt;at;i
40,
� if for any i ¼ 1;2; . . . ; s such that At;i40; then Rt;At;i
40,
� if for every pair (i, j) with 1 � ioj � s, it holds: (1)
At;i40; at;j40 and (2) for every k with iokoj

has at;k ¼ At;k ¼ 0, then we haveAt;i � at;j � 1 and Rt;at;k

¼ Rt;At;k¼ 0.

We observe that the inventory at installation di for every1 � i � s is given by

PAt;i

k¼at;iRt;k. Suppose 1 � ioj � s. Given

any two consecutive installations di and dj with positiveinventory, as shown in Fig. 6, we have 0oat;i � At;ioat;j �

At;j and for every l;At;ioloat;j we have Rt;l ¼ 0.The main goal in this section is to show that Vsþ1

t � Vst .

This clearly shows that by installing RFID at installationdsþ1 the system cost is reduced. We call the system with s,s+1 RFID enabled installations scenarios 1 and 2, respec-tively. In addition, we call installations ds, dsþ1 the borderin scenarios 1 and 2, respectively. We make the followingassumptions with respect to scenario 1.

�
Transitions moving from non-RFID zone beyond in-stallation ds across the border back into the RFID-enabled network are not allowed. � Random vector W describing transitions in the RFID-
enabled portion is conditioned on random variable J

describing the non-RFID area. In other words, we firsthave the observation of random variable J and then therealization of W must be compatible with J in terms ofthe shipment transitions. For example, if a realizationof J requires all shipments of age p to arrive at dn andfor installation kos we have at;k � p � At;k, then clearlyW cannot lead shipments at locations di for k � i � s toa different installation but dn.

The shipment transitions originating from the RFID-enabled installations depend on the exogenous randomvector W. The random vector W assures location-labeledinstallation inventory information in each time period.The difference from Section 2 is in the fact that the

dn

Dt

Rt, m+1

It, dn

• • • Rt, T−1

non-RFID Zone

tion in scenario 1.

ARTICLE IN PRESS


shipments governed by W have to be compatible with J. Asseen in Fig. 8, we first observe a realization j of J. It isobvious that all the outstanding shipments with age atleast j must accordingly arrive at installation dn. Therealization of W provides additional information about theshipment transitions among installations d0;d1; d2. Thetop portion of Fig. 8 represents a possible and compatibleshipment transition case, while the bottom portionrepresents a case in which the shipment transitions basedon W and J are incompatible.

The system dynamics for scenario 1 are divided intotwo groups. Part I shows the transitions of installationinventories and outstanding shipments, while part IIdemonstrates the age transitions at installations. If j is arealization of J, they read:

IOstþ1 ¼ IOt þ Xt � Yt ;

Istþ1;dnðjÞ ¼

It;dnþPT�1

k¼at;i

Rt;k � Dt ; 1 � at;i � j � At;i � mðtÞ

and 1 � i � s;

It;dnþPT�1

k¼at;i

Rt;k � Dt ; 1 � At;iojoat;i � mðtÞ

and for every k with 1 � iok

oi0 � s we have at;k ¼ At;k ¼ 0;

It;dnþPT�1

k¼j

Rt;k � Dt ; j � mðtÞ þ 1;

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

tþ1;1ðjÞ ¼ Yt ;

Rstþ1;kðjÞ ¼

Rt;k�1; j4mðtÞ and 2 � k � j;

0; mðtÞojok and k � 2;

Rt;k�1; 2 � k � at;g � j � mðtÞ;

0; otherwise when k � 2;

8>>><>>>:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:(I)

Rt, At, i > 0, At, i > 0

At, i ≤ at, j −1

0, 0

0, 0

di di+1

Fig. 6. An illustration of the action

RFID-Enabled Network

• • •Yt

IOtIt, d1

d0 d1

Xt

d2

It, d2

ds

It, dsIt

Fig. 7. Partial RFID impleme

astþ1;iðjÞ ¼

minfat;k þ 1 : Mðdk;wÞ ¼ di;d0

� dk � dg�1ðjÞ;Rt;atk40g; 1 � i � s and

1 � j � mðtÞ;

minfat;k þ 1 : Mðdk;wÞ ¼ di;d0

� dk � ds;Rt;atk40g; 1 � i � s and

j4mðtÞ;

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

Astþ1;iðjÞ ¼

maxfAt;k þ 1 : Mðdk;wÞ ¼ di;d0

� dk � dg�1ðjÞ;Rt;Atk40g; 1 � i � s and

1 � j � mðtÞ;

maxfAt;k þ 1 : Mðdk;wÞ ¼ di;d0

� dk � ds;Rt;Atk40g; 1 � i � s and

j4mðtÞ:

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

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:(II)

Here, we denote g(j) ¼ arg max{at,i|jZat,i} In (II), if theminimum or maximum is over an empty set, we definethe corresponding quantity as

tþ1;iðjÞ or Astþ1;iðjÞ to be 0.

The interface is defined as the position of the youngestoutstanding shipment which arrives at installation dn atthe end of time t. In Is

tþ1;dnðjÞ, as

tþ1;iðjÞ, and Astþ1;iðjÞ we

distinguish two cases. The first case 1 � j � mðtÞ (forIstþ1;dnðjÞ it actually consists of two sub-cases) corresponds

to the case where the interface is in the RFID-enabledarea. Note that mðtÞ is the oldest age in the RFID-enabledarea and j is a realization of J, which implies that allshipments of age equal to or more than j arrive at dn. Itfollows that the shipments arriving at dn at the end oftime period t include shipments in the RFID-enabled areaand all outstanding shipments in the non-RFID zonebecause of the non-crossover property. In the other case ofj4mðtÞ, the interface is beyond the RFID-enabled area. The

Rt, At, j > 0, At, j > 0

Rt, at, j > 0, at, j > 0

0, 0

0, 0

0, 0

0, 0

dk dj−1 dj

space of scenario 1 in time t.

dn

Dt

Rt, m'+1

It, dn

• • • Rt, T−1

Non-RFID Zone

ds+1

, ds+1

ntation in scenario 2.

ARTICLE IN PRESS

A realization j of J with at, 2 ≤ j ≤ At, 2

A realization j of J with at, 2 ≤ j ≤ At, 2

Enter Arrival

M (w) for a realization w of W

M (w) for a realization w of W

Enter Arrival

dnd2 d3 d0...

...

d1 ds

Dt

...

dnd1 d2 d3 ds d0 ...Dt

It, d1Yt

Yt It, d1

Rt, at, 2Rt, j Rt, At, 2

It, d2

It, d3 It, ds

It, dn

It, dn

Rt, at, 2Rt, j Rt, At, 2

It, d2

It, d3 It, ds

Fig. 8. Shipment transition compatibility between W and J.


arriving shipments at installation dn in time period t areonly from the non-RFID zone. We similarly distinguishtwo cases in the remaining equations. In Rs

tþ1;kðjÞ forj4mðtÞ, all outstanding shipments which are j periods oldarrive in time t. However, in the case of j � mðtÞ,outstanding shipments which are at;gðjÞ old or more arrivebecause we have more detailed information from RFIDand shipments are not broken. By definition, gðjÞ describesthe installation beyond which all outstanding shipmentsarrive during current time period t.

The optimality equation for partial RFID deploymentscenario 1 reads

VstðI

stÞ ¼ Min

Xt�0Xt�0

Yt�IOt

CtðXtÞ þ StðYtÞ þ htðIOt þ Xt � YtÞ þ pt �XmðtÞj¼1

qj

8<:� ED;W=J¼j Dt � It;dn

�XT�1

l¼at;kðjÞ:Mðdk ;wÞ¼dn

Rt;l

0@ 1Aþ24 35þ pt �

XT�1

j¼mðtÞþ1

qj � ED Dt � It;dn

�XT�1

k¼j

Rt;k

0@ 1Aþ24 35þXT�1

j¼1

qj � ED;W=J¼j

½Vstþ1ðI

stþ1Þ�

9=;. (11)

Notation W/J ¼ j denotes the random variable W condi-tional on a realization j of J.

In order for the optimality equation to be well defined,we need to show the following lemma, which is proved inAppendix C.

Lemma 1. If Ist 2 z

st , then Is

tþ1 2 zstþ1.

In the partial RFID deployment scenario 2, the RFID-enabled network is extended to capture the information ofone more downstream installation dsþ1. It is clear thatstate vector Isþ1

t in scenario 2 has two more coordinatesasþ1

tþ1;sþ1 and Asþ1tþ1;sþ1. The state transitions in scenario 2 are

also divided into two parts. Part I captures the sametransitions of inventories as in scenario 1, except that wereplace s with s+1 and mðtÞ with m0ðtÞ in the stateequations for scenario 2. Part II equations of systemdynamics regarding asþ1

tþ1;i and Asþ1tþ1;i for i ¼ 1;2; . . . ; s have

exactly the same transitions as shown in scenario 1 due tothe assumption of restricted backward movements fromnon-RFID area beyond installation dsþ1 to RFID-enabledarea. The additional transitions special to scenario 2regarding installation dsþ1 read

asþ1tþ1;sþ1ðjÞ ¼

minfat;k þ 1 : Mðdk;wÞ ¼ dsþ1;

d0 � dk � dg�1ðjÞ;Rt;atk40g; 1 � j � m0ðtÞ;

minfat;k þ 1 : Mðdk;wÞ ¼ dsþ1;

d0 � dk � dsþ1;Rt;atk40g; j4m0ðtÞ;

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

Asþ1tþ1;sþ1ðjÞ ¼

maxfAt;k þ 1 : Mðdk;wÞ ¼ dsþ1;

d0 � dk � dg�1ðjÞ;Rt;Atk40g; 1 � j � m0ðtÞ;

maxfAt;k þ 1 : Mðdk;wÞ ¼ dsþ1;

d0 � dk � dsþ1;Rt;Atk40g; j4m0ðtÞ:

8>>>>><>>>>>:We accordingly replace s with s+1 and mðtÞ with m0ðtÞ inthe optimality equation for partial scenario 2.

Shipments moving to installation dsþ1 at the end oftime period t are only from the RFID-enabled area. Theycan come from installation dsþ1 itself if the outstandingshipments at installation dsþ1 in time period t stay put.

ARTICLE IN PRESS


Similarly, as before, we distinguish two cases: j � m0ðtÞ

and j4m0ðtÞ. Note that starting with the same outstandingshipments in time period t in scenarios 1 and 2, the twosystems may possibly end up with different states in timeperiod t+1. From here on, we denote the state differencebetween scenarios 1 and 2 by using the superscripts s ands+1. For example, as

tþ1;i denotes the youngest age ofoutstanding shipments at installation di in time periodt+1 corresponding to scenario 1. Likewise, Asþ1

tþ1;i denotesthe oldest age of outstanding shipments at installation di

in time period t+1 corresponding to scenario 2.

3.2. Value of RFID in the serial distribution process

Intuitively, the broader the RFID deployment is in thedistribution network, the greater should the benefits be. Ifthe RFID deployment and maintenance costs are ne-glected, we are able to show that additional benefits areobtained by a broader RFID deployment because ofavailability of additional information. We compare twopartial RFID deployment scenarios, based on identicalmodel configurations and assumptions, e.g., the same leadtime distributions and non-crossover of the outstandingshipments in time.

The following is our main result.

Theorem 4. For any Ist 2 z

st and ðat;sþ1;At;sþ1Þ, we have

VstðI

st Þ � Vsþ1

t ðIOt ; It;dn; at;1; . . . ; at;s; at;sþ1;At;1; . . . ;At;s,

At;sþ1;Rt;1; . . . ;Rt;T�1Þ. (12)

The proof of this result is very technical and is providedin Appendix D. It clearly states that under optimal policies,the total system cost of s+1 RFID deployments is equal toor lower than the cost of having only s RFID deployments.In other words, RFID reduces the total inventory cost. Aweakness of our result is the fact that the deployment costin not taken into consideration. The significance of thecost reduction depends on the RFID deployments.

4. Concluding remarks

After the initial lab experiments and case studies, thereal breakthrough of RFID applications came with theRFID mandates imposed by Wal-Mart and the US Depart-ment of Defense. Most of their suppliers are required toapply at least pallet level RFID transponders to theproducts being shipped to selected distribution centers.The well-known slap-and-ship strategy provides valuableinformation and benefits to the retailers. However, it ismuch harder to identify a return on investment forsuppliers. We first present a comprehensive inventorymodel and show that the underlying multivariate cost-to-go decomposes into two lower dimensional functions.Under certain assumptions, the optimal control policy forboth replenishments and shipments is obtained. Further-more, we analytically show that larger RFID deploymentsyield reduced overall expected cost. This clearly estab-lishes that there is potential benefit of using RFID if novelprocesses are used. There are several important contribu-tions of this work. Prior research mainly studies applica-

tions of RFID internally within a company or inventorycontrol problems in a multi-firm setting with RFID supplyinformation. This research is the first one to analyticallystudy the inventory control policies with explicit RFIDinformation at the distribution side. This is a non-trivialtask since we have to simultaneously deal with twoactions (replenishment and shipping quantities). Anothercontribution is, first, modeling a system with only a partialRFID deployment, and, second, comparing the expectedcost of the resulting models with respect to the extent ofthe RFID deployment. Perhaps the most important con-tribution concerns the next generation business applica-tions built on top of RFID data. We show that even byemploying slap-and-ship, it is possible to generate addi-tional value. This research is closing the gap between theever increasing supply chain data and challenging analy-tical tools to process such information. The main con-tribution of our work is to establish models and policiesfor slap-and-ship that benefit suppliers.

It is clear that in short-term slap-and-ship providesa quick and relatively low cost solution compared with afull-scale RFID deployment to achieve compliance with aretailer’s mandate. On the other hand, experiences fromslap-and-ship can shape suppliers’ future internal deploy-ments and even further upstream, where suppliers canreap all the benefits from RFID. In this work, we show suchpossibility by an analytical approach exploiting the vastrichness of RFID data. We first propose a comprehensiveinventory model and then evaluate a serial distributionprocess with RFID deployments, which mimics the slap-and-ship processes of suppliers facing RFID mandates. Theinventory control problem in the serial distribution chainis modeled as a dynamic program. Furthermore, theoriginal problem is reduced in size to a simpler modelwithout losing optimality. Based on this state reductionresult, the optimal inventory control policy is the echelonbase stock policy. It establishes that by knowing thedownstream flow of goods through the distributionchannel the supplier can improve decision making withrespect to inventory control. The proposed comprehensiveinventory model contains both supply and distributioninformation in serial systems. It asserts that armed withentire supply chain status information from using RFID,suppliers could better manage their internal and externalprocesses and make integrated decisions that wouldimprove their bottom line.

The value of RFID in inventory control systems is alsodiscussed in detail within the studied context. It is usuallyclaimed that slap-and-ship is a cost-bearing solution. Weargue that beyond the cost associated with slap-and-ship,suppliers could indeed find the benefits of RFID deploymentsdownstream. We show analytically how RFID can improvethe system performance in terms of the expected overall costover a finite time horizon. The partial RFID deploymentscenario with s+1 installations covered by RFID yields lowerprocurement and shipping costs than the costs based on s

installations covered by RFID. The cost difference can beregarded as the true value of RFID in the system, settingaside the RFID system deployment and maintenance costs.

The most important revelation and message of ourpaper is the fact that even by employing slap-and-ship,

ARTICLE IN PRESS


which is a basic deployment, benefits in inventory controlare possible. Such benefits could offset the underlyingcosts and thus establish the elusive ROI for an RFIDdeployment. Clearly a positive ROI is not possible bysimply slapping the tag and then shipping. An informationsystem with analytical capabilities demonstrated heremust be put together. RFID leaves an enormous trail ofdata that can be creatively analyzed and used to improvedecision making. The results in Section 3 clearly demon-strate that wider RFID deployments bring additionalbenefits. Firms must carefully leverage deployment costsvs. benefits when deciding the scope of an RFID deploy-ment. We demonstrated that additional benefits aredefinitely possible with broader deployments.

A drawback of our study is that the actual deploymentcosts are not explicitly captured. We do not claim of showinga positive ROI, but without the analytical aspects presentedherein, a positive ROI is definitely not possible under the slap-and-ship strategy.

Many suppliers to Wal-Mart are struggling with theunderlying costs of slap-and-ship. Through its power, Wal-Mart is able to dictate the RFID terms, while riskinginsolvency of smaller suppliers. Instead of these relentlesspressures, Wal-Mart could more tightly collaborate withsuppliers by using its tremendous IT resources and knowl-edge. To this end, it should show the suppliers how to gainbenefits from its RFID mandate, and not only incur cost. Byusing the presented findings, the big-box retailer could showthe suppliers how to improve inventory management simplyby using the feedback data from Wal-Mart; thus not goingdeeper into the processes of the suppliers and more complexdeployments of this pervasive technology.

With the real-time information generated from RFID,many new research directions are possible. This researchaddresses a partial RFID deployment issue in order to studythe value of RFID in inventory control. We assume RFID isdeployed from the supplier downstream. An alternativesetting not addressed here is to start deploying RFID at thepoint-of-sale and expanding it upward. It would be interest-ing to investigate the system differences between a forwardand backward deployment. All partial RFID deploymentscenarios in this work are based on perfect RFID read rates.This immediately raises the question of the impact ofimperfect RFID read rates on system performance.

Acknowledgments

We acknowledge two anonymous reviewers for pro-viding valuable feedbacks. Their comments and sugges-tions refine many subtle points of the research framework.In addition, we are obliged to the Illinois Department ofTransportation for their financial support. Without it, thiswork would not see the light.

Appendix A

Proof of Theorem 1. We use induction. We defineaTþ1 ¼ 0. Since all the terminal costs are zero, we clearlyhave VTþ1 ¼ VTþ1 þ aTþ1 ¼ 0.

We assume that (6) holds for t+1. In other words,

Vtþ1ðItþ1;dÞ ¼ Vtþ1ðIOtþ1; rtþ1;1Þ þ atþ1,

where

atþ1 ¼ ELd XLd�2

j¼0

ptþjþ1 � ED;W Dj

tþ1 � Itþ1;dn

0BB@2664

0BB@1CCA:

�Xjþ1

p¼1

Xdk:M

pðdk ;WÞ¼dn

1�k�n�1

Itþ1;dk

1CCAþ37751CCA

Based on the system dynamics of the full model, we haveXdk :M

pðdk ;WÞ¼dn

1�k�n�1

Itþ1;dk¼

Xdk :M

pðdk ;WÞ¼dn

1�k�n�1

Xdg :Mðdg ;WÞ¼dk

0�k�n�1

It;dg

¼X

dg :Mpþ1ðdg ;WÞ¼dn

0�k�n�1

It;dg. (13)

Next, we manipulate atþ1. We substitute

Itþ1;dn¼ It;dn

þX


It;dk� Dt

and use (13) in atþ1 to obtain

atþ1 ¼ ELd XLd�2

j¼0

ptþjþ1 � ED;W Djþ1

t � It;dn�

Xdk :Mðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@2664

8>><>>:�Xjþ1

p¼1

Xdg :M

pþ1ðdg ;WÞ¼dn

0�k�n�1

It;dg

1CCAþ37759>>=>>;

¼ ELd XLd�3

j¼0

ptþjþ1 � ED;W Djþ1

t � It;dn�

Xdk :Mðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@2664

8>><>>:�Xjþ1

p¼1

Xdg :M

pþ1ðdg ;WÞ¼dn

1�k�n�1

It;dg

1CCAþ3775þ ptþLd

�1 � ED;W

DLd�1

t � It;dn�

Xdk:Mðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@2664

�XLd�1

p¼1

Xdg :M

pþ1ðdg ;WÞ¼dn

1�k�n�1

It;dg� It;d0

1CCAþ37759>>=>>;

¼ ELd XLd�2

i¼1

ptþi � ED;W Di

t � It;dn�

Xdk:Mðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@2664

8>><>>:�Xiþ1

q¼2

Xdg :M

qðdg ;WÞ¼dn

1�k�n�1

It;dg

1CCAþ3775þ ptþLd

�1 � ED;W

DLd�1

t � It;dn�

Xdk:Mðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@2664

�XLd

q¼2

Xdg :M

qðdg ;WÞ¼dn

1�k�n�1

It;dg� It;d0

1CCAþ37759>>=>>;

ARTICLE IN PRESS


¼ ELd XLd�2

i¼1

ptþi � ED;W Di

t � It;dn�Xiþ1

q¼1

Xdg :M

qðdg ;WÞ¼dn

1�k�n�1

It;dg

0BB@1CCAþ26643775

8>><>>:

þptþLd�1 � E

D;W DLd�1

t � It;dn�XLd

q¼1

Xdg :M

qðdg ;WÞ¼dn

1�k�n�1

It;dg� It;d0

0BB@1CCAþ266437759>>=>>;.

In atþ1 above, the summation over j is decomposed intotwo groups: from 0 to Ld

�3 and j ¼ Ld�2. Moreover, the

non-crossover property is applied for j ¼ Ld�2 as follows.

Term It;d0can be taken out and termX

dg :MLdðdg ;WÞ¼dn

1�g�n�1

It;dg

can be folded into the summation term

XLd

q¼1

Xdg :M

qðdg ;WÞ¼dn

1�g�n�1

It;dg

because of MLd

ðdk;WÞ � MLd

ðd0;WÞ ¼ dn for every k � 1.

In order to establish the result, we analyze atþ1 � at : We

have

atþ1 � at

¼ ELd XLd�2

i¼1

ptþi � ED;W Di

t � It;dn�Xiþ1

q¼1

Xdg :M

qðdg ;WÞ¼dn

1�k�n�1

It;dg

0BB@1CCAþ26643775

8>><>>:9>>=>>;

þ ELd

ptþLd�1 � E

D;W DLd�1

t � It;dn�XLd

q¼1

Xdg :M

qðdg ;WÞ¼dn

1�k�n�1

It;dg� It;d0

0BB@1CCAþ26643775

8>><>>:9>>=>>;

� ELd XLd�2

j¼0

ptþj � ED;W Dj

t � It;dn�Xjþ1

p¼1

Xdk :M

pðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@1CCAþ26643775

8>><>>:9>>=>>;

¼ ELd

ptþLd�1 � E

D;W DLd�1

t � It;dn�XLd

q¼1

Xdg :M

qðdg ;WÞ¼dn

1�k�n�1

It;dg� It;d0

0BB@1CCAþ26643775

8>><>>:9>>=>>;

� pt � ED;W Dt � It;dn

�X

dk:Mðdk ;WÞ¼dn1�k�n�1

It;dk

0BB@1CCAþ26643775. (14)

In (14), the penalty cost terms from t+1 to t+Ld�2 cancel

out. We end up with two terms for time period t and

t þ Ld� 1.

It is clear that

It;dnþXld

q¼1

Xdg :M

qðdg ;wÞ¼dn

1�g�n�1

It;dg¼ It;dn

þXn�1

j¼1

It;dj¼ rt;1

for every realization ld of Ld and w of W. We are now ready

to show the induction step starting from (2). For ease of

notation, let bt ¼ CtðIt;u0Þ þ StðIt;d0


Þ.

We have

VtðIt;dÞ ¼ MinIt;u0�0

It;d0�0

It;d0�IOt

bt þ pt � ED;W Dt � It;dn

�X


It;dk

0BB@1CCAþ26643775

8>><>>:

þ Vtþ1ðItþ1;dÞ

9>>=>>; (15)

¼ MinIt;u0�0

It;d0�0

It;d0�IOt

bt þ pt � ED;W Dt � It;dn

�X


It;dkÞ

0BB@1CCAþ26643775

8>><>>:

þ Vtþ1ðIOtþ1; rtþ1;1Þ þ atþ1

9>>=>>; (16)

¼ MinIt;u0�0

It;d0�0

It;d0�IOt

bt þ ELd

ðptþLd�1 � E

D½ðDLd

�1t � It;d0

� rt1Þþ�Þ

n

þ Vtþ1ðIOtþ1; rtþ1;1Þ þ atþ1 � ELd

ðptþLd�1 � E

D½ðDLd

�1t � It;d0

� rt1Þþ�Þ þ pt � E

D;W Dt � It;dn�

Xdk :Mðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@1CCAþ266437759>>=>>;

¼ MinIt;u0�0

It;d0�0

It;d0�IOt

VtðIOt ; rt1Þ þ atþ1 � ELd

ðptþLd�1ED½ðDLd

�1t � It;d0

n

� rt1Þþ�ÞptE

D;W Dt � It;dn�

Xdk :Mðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@1CCAþ266437759>>=>>;

(17)

¼ MinIt;u0�0

It;d0�0

It;d0�IOt

VtðIOt ; rt1Þ þ at þ atþ1 � at � ELd


�1t

n

� It;d0� rt1Þ

þ�Þ þ ptE

D;W Dt � It;dn�

Xdk :Mðdk ;WÞ¼dn

1�k�n�1

It;dk

0BB@1CCAþ266437759>>=>>;

¼ VtðIOt ; rt1Þ þ at . (18)

Eq. (15) is the optimality equation of the distribution

process. In (16) we use the induction hypothesis. The

optimality Eq. (3) of the reduced model is applied in (17).

In (18) we use the derived result for atþ1 � at and the

observation

It;dnþXld

q¼1

Xdg :M

qðdg ;wÞ¼dn

1�g�n�1

It;dg¼ rt;1

for each realization ld of Ld and w of W. After simplifying

the remaining terms and pulling term at out of minimiza-

tion because it is independent of the decision variables

It;u0and It;d0

; we conclude that VtðIt;dÞ ¼ VtðIOt ; rt1Þ þ at :

This completes the proof. &

ARTICLE IN PRESS


Appendix B

Proof of Theorem 3. We prove the statement by induc-tion. It is straightforward to show the theorem for T+1. Weassume that (9) holds for t+1, i.e.,

eVtþ1ðIetþ1Þ ¼ Vtþ1ðrtþ1;1Þ þ Gtþ1ðI

etþ1;u1

; . . . ; Ietþ1;un�1

; IOetþ1Þ.

By examining the downstream sub-system, standardarguments are applicable showing that Vt ’s are all convex.The base stock levels

rt;d0¼ arg min

rt;d0�0fðSt � htÞrt;d0

þ ELd


�1t � rt;d0

Þþ�Þ

þ ED½Vtþ1ðrt;d0

� DtÞ�g

are used to determine the optimal shipping decisions tothe distribution process. In other words, if rt;1ort;d0

, theoptimal shipping quantity in time t is rt;d0

� rt;1; other-wise, the optimal shipping quantity is zero. In order toshow (9), we need to distinguish two cases.

Case I: Let first rt;d0oIOe

t : We have

eVtðIet Þ ¼ Min

It;u0�0

rt;d0�rt;1

rt;d0�IOe

t

fCtðIt;u0Þ þ St � ðrt;d0

� rt;1Þ þ ht � ERðIe

t;NðunÞ� rt;d0

Þ

þ ELd

ðptþLd�1 � E

D½ðDLd

�1t � rt;d0

Þþ�Þ þ ED;W ;R

½eVtþ1ðIetþ1Þ�g

¼ MinIt;u0�0fCtðIt;u0

Þ þ ht � ERðIe

t;NðunÞÞ þ ER

½Gtþ1ðIetþ1;u1

; . . . ; Ietþ1;un�1

; IOetþ1Þ�g þ St � ðr

t;d0� rt;1Þ � ht

� rt;d0þ ELd

ðptþLd�1 � E

D½ðDLd

�1t � rt;d0

Þþ�Þ þ Vtþ1ðrtþ1;1Þ.

It implies that

eVtðIet Þ � Vtðrt;1Þ ¼ Min

It;u0�0fCtðIt;u0

Þ þ ht � ERðIe

t;NðunÞÞ þ ER

½Gtþ1ðIetþ1;u1

; . . . ; Ietþ1;un�1

; IOetþ1Þ�g.

Case II : Let now rt;d0� IOe

t : It is clear that the optimal

shipping decision satisfies rt;d0¼ IOe

t because of the

convexity of function St � ðrt;d0� rt;1Þ � ht � rt;d0

þ

ELd

ðptþLd�1 � E

D½ðDLd

�1t � rt;d0

Þþ�Þ and Vtþ1 in rt;d0

. Finally,

we derive that

eVtðIet Þ ¼ Min

It;u0�0

rt;d0�rt;1

rt;d0�IOe

t

CtðIt;u0Þ þ St � ðrt;d0

� rt;1Þ þ ht � ERðIe

t;NðunÞ� rt;d0

Þ

n

þELd

ðptþLd�1 � E

D½ðDLd

�1t � rt;d0

Þþ�Þ þ ED;W ;R

½eVtþ1ðIetþ1Þ�

o¼ Min

It;u0�0

CtðIt;u0Þ þ ht � E

RðIe

t;NðunÞÞ þ ER

½Gtþ1

n Ie

tþ1;u1; . . . ; Ie

tþ1;un�1; IOe

tþ1

� �ioþ St � ðIO

et � rt;1Þ � ht � IO

et

þ ELd

ðptþLd�1 � E

D½ðDLd

�1t � IOe

t Þþ�Þ þ Vtþ1ðrtþ1;1Þ.

It implies that

eVtðIet Þ � Vtðrt;1Þ


Þ þ ht � ERðIe

t;NðunÞÞ þ ER

½Gtþ1

ðIetþ1;u1

; . . . ; Ietþ1;un�1

; IOetþ1Þ�g þ St � ðIO

et � rt;1Þ � ht � IO

et

þ ELd

ðptþLd�1 � E

D½ðDLd

�1t � IOe

t Þþ�Þ þ ED

½Vtþ1ðrtþ1;1Þ

� St � ðrt;d0� rt;1Þ þ ht � r

t;d0� ELd

ðptþLd�1 � E

D½ðDLd

�1t

� rt;d0Þþ�Þ � ED

½Vtþ1ðrtþ1;1Þ


Þ þ ht � ERðIe

t;NðunÞÞ þ ytðIO

et Þ þ ER

½Gtþ1ðIetþ1;u1

; . . . ; Ietþ1;un�1

; IOetþ1Þ�g,

where ytðIOet Þ ¼ ðht � StÞðrt;d0

� IOet Þ þ ELd

ðptþLd�1 � E

D½ðDLd

�1t

�IOet Þþ� ðDLd

�1t � rt;d0

Þþ�Þ is the penalty term correspond-

ing to the shortage of on-hand inventory for an optimalshipment.

As a result, we have

eVtðIet Þ � Vtðrt;1Þ ¼ GtðI

et;u1; . . . ; Ie

t;un�1; IOe

t Þ


Þ þ ht � ERðIe

t;NðunÞÞ þ ytðIO

et Þ

þ ER½Gtþ1ðI

etþ1;u1

; . . . ; Ietþ1;un�1

; IOetþ1Þ�g, (19)

where the penalty term reads

ytðIOet Þ ¼

0; rt;d0oIOe

t ;

ðht � StÞðrt;d0� IOe

t Þ þ ELd

ðptþLd�1 � E

D

½ðDLd�1

t � IOet Þþ� ðDLd

�1t � rt;d0

Þþ�Þ; rt;d0

� IOet ;

8>>><>>>:This completes the proof of Theorem 3. &

Appendix C

Proof of Lemma 1. We need to show that if Ist 2 z

st , then

Istþ1 2 z

stþ1. First, we consider non-negativity require-

ments. It is clear from system dynamics that IOstþ1 ¼

IOst þ Xt � Yt � 0 because Xt � 0, Yt � 0 and Yt � IOs

t . Non-negativity of other components follows by definition andfrom system dynamics.

If Rt;at;k40 for a k, then at;k40 (otherwise at;k ¼ At;k ¼ 0,

which implies Rt;at;k¼ 0). In turn we have At;k40 and thus

Rt;At;k40. System dynamics imply that for every j

0 � astþ1;iðjÞ � As

tþ1;iðjÞ.

If astþ1;iðjÞ ¼ 0, then the underlying set defining the

quantity is |. The underlying set defining Astþ1;iðjÞ is also

empty, which in turn implies Astþ1;iðjÞ ¼ 0.

If astþ1;iðjÞ40, then we have Rt;at;k

40 for some 1 � k � s

and astþ1;iðjÞ ¼ at;k þ 1: Since as

tþ1;iðjÞ40, we obtain that j in

time period t must be 4at;k. Then either j4mðtÞ or

at;koj � mðtÞ. System dynamics imply in both cases that

Rstþ1;as

tþ1;iðjÞ ¼ Rt;at;k

40. The same argument applies to the

case Astþ1;iðjÞ40 implying Rs

tþ1;Astþ1;iðjÞ

40.

It remains to show that if for 1 � i0oj0 � s such that

Astþ1;i040; as

tþ1;j040 and for every i0ok0oj0 we have

astþ1;k0

¼ Astþ1;k0 ¼ 0, then we have As

tþ1;i0 � astþ1;j0� 1: A

possible scenario of the order transitions from time period

t to t+1 is shown in Fig. 9. Outstanding orders in the

installation range [u,v] in time period t move to installa-

tion i0 in time period t+1. Similarly, outstanding orders

in [u0,v0] move to installation j0 in time period t+1.

ARTICLE IN PRESS


The light-shaded installations correspond to installations

with no inventory in time period t. Installationsei and i are

the two farthest installations with positive inventory

within [u,v], i.e., inventory in installations within ½ei; i� is 0,

while j and ej are two such installations within [u0,v0]. As

can be easily verified by the non-crossover assumption,

the dark-shaded area represents installations with no

outstanding orders in time period t. Formally, we obtain

Astþ1;i0 ¼ At;i þ 1 and as

tþ1;j0¼ at;j þ 1: From At;i � at;j � 1, we

then obtain Astþ1;i0 � as

tþ1;j0� 1.

In conclusion, the presented dynamic program is well-

defined. This completes the proof of Lemma 1. &

Appendix D

In order to show Theorem 4, we need the followinglemma.

Lemma 2. For simplicity, we denote

Vsþ1tþ1ðI

sþ1tþ1;dn

;Rsþ1tþ1Þ ¼ E½Vsþ1

tþ1ðIOsþ1tþ1 ; I

sþ1tþ1;dn

; asþ1tþ1;1; . . . ; a

sþ1tþ1;sþ1;A

sþ1tþ1;1,

. . . ;Asþ1tþ1;sþ1;R

sþ1tþ1;1; . . . ;R

sþ1tþ1;T�1Þ�,

where the state variables are derived from the recursion in

scenario 2, and

Vsþ1tþ1ðI

stþ1; a

sþ1tþ1;sþ1;A

sþ1tþ1;sþ1Þ

¼ E½Vsþ1tþ1ðIO

stþ1; I

stþ1;dn

; astþ1;1; . . . ; a

stþ1;s; a

sþ1tþ1;sþ1;A

stþ1;1; . . . ,

Astþ1;s;A

sþ1tþ1;sþ1;R

stþ1;1; . . . ;R

stþ1;T�1Þ�,

where state variable Istþ1 2 z

stþ1 is derived from the recursion

in scenario 1. If Istþ1;dn

� Isþ1tþ1;dn

, then for every k ¼

f1;2; . . . ; T � 1g we have Rstþ1;k � Rsþ1

tþ1;k,

Istþ1;dn

þXT�1

k¼1

Rstþ1;k ¼ Isþ1

tþ1;dnþXT�1

k¼1

Rsþ1tþ1;k

and

Vsþ1tþ1ðI

stþ1; a

sþ1tþ1;sþ1;A

sþ1tþ1;sþ1Þ � Vsþ1

tþ1ðIsþ1tþ1;dn

;Rsþ1tþ1Þ. (20)

Proof. Let us first show that for every k ¼ f1;2; . . . ; T � 1gwe have Rs

tþ1;k � Rsþ1tþ1;k. We distinguish three cases.

(1)

t:

t+1:

Let j4m0ðtÞ and k � 2: Following the state transitionsin partial scenarios 1 and 2, for j4m0ðtÞ and k � j, wehave Rs

tþ1;kðjÞ ¼ Rt;k�1 and Rsþ1tþ1;kðjÞ ¼ Rt;k�1; which im-

plies Rstþ1;kðjÞ ¼ Rsþ1

tþ1;kðjÞ: If m0ðtÞojok; we have

Installations

Installations

u v u' v'

i’ j’

~ji ji

~

Fig. 9. Graphical representation of order transitions name.

Rstþ1;kðjÞ ¼ 0 and Rsþ1

tþ1;kðjÞ ¼ 0; which also impliesRs

tþ1;kðjÞ ¼ Rsþ1tþ1;kðjÞ.

(2)
Let mðtÞoj � m0ðtÞ and k � 2. FormðtÞoat;g � k � j � m0ðtÞ, we have Rs
tþ1;kðjÞ ¼ Rst;k�1ðjÞ

and Rsþ1tþ1;kðjÞ ¼ 0; which implies Rs

tþ1;kðjÞ � Rsþ1tþ1;kðjÞ.

Otherwise, we clearly have Rstþ1;kðjÞ ¼ Rsþ1

tþ1;kðjÞ.
(3) Finally, let 1 � j � mðtÞ and k � 2: For k � at;g � j �
mðtÞ � m0ðtÞ; we have Rstþ1;kðjÞ ¼ Rs

t;k�1ðjÞ and Rsþ1tþ1;kðjÞ ¼

Rst;k�1ðjÞ; which implies Rs

tþ1;kðjÞ ¼ Rsþ1tþ1;kðjÞ: Otherwise,

we have Rstþ1;kðjÞ ¼ 0 and Rsþ1

tþ1;kðjÞ ¼ 0, which againimplies Rs

tþ1;kðjÞ ¼ Rsþ1tþ1;kðjÞ. &

We also observe that Rstþ1;1ðjÞ ¼ Rsþ1

tþ1;1ðjÞ ¼ Yt : Thus, we

conclude that for every k ¼ f1;2; . . . ; T � 1g we have

Rstþ1;k � Rsþ1

tþ1;k.

Next, let us show Istþ1;dn

� Isþ1tþ1;dn

. We similarly distin-

guish cases as follows.

(1)
Let first 1 � at;i � j � At;i � mðtÞ and 1 � i � s: By usingthe state transitions in scenarios 1 and 2, we have
Istþ1;dnðjÞ ¼ Isþ1

tþ1;dnðjÞ ¼ It;dn

þXT�1

k¼at;i

Rt;k � Dt .

(2)
Let 1 � At;iojoat;i0 � mðtÞ. For every g such that 1 �iogoi0 � s and at;dg
¼ At;dg¼ 0; we have



þXT�1

k¼at;i0

Rt;k � Dt .

(3)
Let now mðtÞoj � m0ðtÞ. We have
Istþ1;dnðjÞ ¼ It;dn

þXT�1

k¼j

Rt;k � Dt ,

Isþ1tþ1;dnðjÞ ¼ It;dn

þXT�1

k¼mðtÞþ1

Rt;k � Dt .

(4)
Finally, let j4m0ðtÞ. We have


þXT�1

k¼j

Rt;k � Dt .

Therefore, we obtain Istþ1;dnðjÞ � Isþ1

tþ1;dnðjÞ because

XT�1

k¼j

Rt;k �XT�1

k¼mðtÞþ1

Rt;k.

Now, we show

Istþ1;dn

þXT�1

k¼1

Rstþ1;k ¼ Isþ1

tþ1;dnþXT�1

k¼1

Rsþ1tþ1;k.

We first investigate

Istþ1;dn

þXT�1

k¼1

Rstþ1;k

ARTICLE IN PRESS


and distinguish two cases.

(1) Let 1 � j � mðtÞ and g ¼ arg maxifat;ijj � at;i;

d1 � di � dsg. By state transitions in scenario 1 we have

XT�1

k¼2

Rstþ1;kðjÞ ¼

Xat;g

k¼2

Rstþ1;kðjÞ þ

XT�1

k¼at;gþ1

Rstþ1;kðjÞ ¼

Xat;g�1

p¼1

Rt;p.

Let e ¼ minfd : Mðd;WÞ ¼ dn; d1 � d � dsg and f ¼

maxfd : Mðd;WÞ ¼ dn; d1 � d � dsg: Let also B be the subset

of fe; eþ 1; . . . ; f g such that for every db 2 B we have

Rt;at;b40 and for every db 2 fe; eþ 1; . . . ; f g\B we have

Rt;at;b¼ Rt;At;b

¼ 0: We also denote db1¼ minfdb : db 2 Bg

and db2¼ maxfdb : db 2 Bg: We have at;b1

¼ at;g and mðtÞ ¼

maxb2BfAt;bg ¼ At;b2: In addition, Rs

tþ1;1ðjÞ ¼ Yt . Adding

Istþ1;dnðjÞ and

PT�1k¼1 Rs

tþ1;kðjÞ, we obtain

Istþ1;dnðjÞ þ

XT�1

k¼1

Rstþ1;kðjÞ ¼ It;dn

þXAt;b2

p¼at;b1

Rt;p þXT�1

k¼mðtÞþ1

Rt;k � Dt

0@ 1Aþ Yt þ

XT�1

k¼2

Rstþ1;kðjÞ

!

¼ It;dnþXAt;b2

p¼at;b1

Rt;p þXT�1

k¼mðtÞþ1

Rt;k

� Dt þ Yt þXat;g�1

p¼1

Rt;p

¼ It;dnþXmðtÞp¼1

Rt;p þXT�1

k¼mðtÞþ1

Rt;k � Dt þ Yt

¼ It;dnþXT�1

k¼1

Rt;k � Dt þ Yt .

(2) Let now j4mðtÞ: We have Istþ1;dnðjÞ ¼ It;dn

þPT�1k¼j Rt;k � Dt ;R

stþ1;1ðjÞ ¼ Yt ; and

XT�1

k¼2

Rstþ1;kðjÞ ¼

Xj

k¼2

Rstþ1;kðjÞ ¼

Xj�1

p¼1

Rt;p.

Adding Istþ1;dnðjÞ and

PT�1k¼1 Rs

tþ1;kðjÞ, we have

Istþ1;dnðjÞ þ

XT�1

k¼1

Rstþ1;kðjÞ ¼ It;dn

þXT�1

k¼j

Rt;k � Dt þ Yt þXT�1

k¼2

Rtþ1;kðjÞ

¼ It;dnþXT�1

k¼j

Rt;k � Dt þ Yt þXj�1

p¼1

Rt;p

¼ It;dnþXT�1

k¼1

Rt;k � Dt þ Yt .

We conclude that in both cases

Istþ1;dn

þXT�1

k¼1

Rstþ1;k ¼ It;dn

þXT�1

k¼1

Rt;k � Dt þ Yt .

By using almost identical steps as those shown above it

can be derived that

Isþ1tþ1;dn

þXT�1

k¼1

Rsþ1tþ1;k ¼ It;dn

þXT�1

k¼1

Rt;k � Dt þ Yt .

It follows that

Istþ1;dn

þXT�1

k¼1

Rstþ1;k ¼ Isþ1

tþ1;dnþXT�1

k¼1

Rsþ1tþ1;k ¼ It;dn

þXT�1

k¼1

Rt;k � Dt þ Yt .

Next, we argue that if Istþ1;dn

� Isþ1tþ1;dn

, Rstþ1;k � Rsþ1

tþ1;k and

Istþ1;dn

þXT�1

k¼1

Rstþ1;k ¼ Isþ1

tþ1;dnþXT�1

k¼1

Rsþ1tþ1;k,

then

Istþ2;dn

� Isþ1tþ2;dn

; Rstþ2;k � Rsþ1

tþ2;k

and

Istþ2;dn

þXT�1

k¼1

Rstþ2;k ¼ Isþ1

tþ2;dnþXT�1

k¼1

Rsþ1tþ2;k

We note that from time period t+1 to time period t+2 the

system follows the recursions in scenario 2. It is clear

that if Rstþ1;k � Rsþ1

tþ1;k for every k ¼ f1;2; . . . ; T � 1g; then

Rstþ2;k � Rsþ1

tþ2;k. We next show Istþ2;dn

� Isþ1tþ2;dn

by distin-

guishing two cases.

(1) Let j � m0ðt þ 1Þ: We note that for every k ¼

f1;2; . . . ; sg; it holds astþ1;k ¼ asþ1

tþ1;k and Astþ1;k ¼ Asþ1

tþ1;k. By

state transitions in scenario 2, we obtain

Istþ2;dnðjÞ ¼ Is

tþ1;dnþ

XT�1

l¼astþ1;kðjÞ:Mðdk ;wÞ¼dn

Rstþ1;l � Dtþ1,

Isþ1tþ2;dnðjÞ ¼ Isþ1

tþ1;dnþ

XT�1

l¼asþ1tþ1;kðjÞ:Mðdk ;wÞ¼dn

Rsþ1tþ1;l � Dtþ1.

By using Rstþ1;k � Rsþ1

tþ1;k for every k ¼ f1;2; . . . ; T � 1g and

Istþ1;dn

þXT�1

k¼1

Rstþ1;k ¼ Isþ1

tþ1;dnþXT�1

k¼1

Rsþ1tþ1;k,

it follows that Istþ2;dnðjÞ � Isþ1

tþ2;dnðjÞ.

(2) Let j4m0ðt þ 1Þ. It is clear that

Istþ1;dn

þXT�1

k¼j

Rstþ1;k � Isþ1

tþ1;dnþXT�1

k¼j

Rsþ1tþ1;k

by using Rstþ1;k � Rsþ1

tþ1;k and

Istþ1;dn

þXT�1

k¼1

Rstþ1;k ¼ Isþ1

tþ1;dnþXT�1

k¼1

Rsþ1tþ1;k.

Thus, we obtain Istþ2;dnðjÞ � Isþ1

tþ2;dnðjÞ for every j.

It is clear that

Istþ2;dn

þXT�1

k¼1

Rstþ2;k ¼ Is

tþ1;dnþXT�1

k¼1

Rstþ1;k � Dtþ1 þ Ytþ1,

Isþ1tþ2;dn

þXT�1

k¼1

Rsþ1tþ2;k ¼ Isþ1

tþ1;dnþXT�1

k¼1

Rsþ1tþ1;k � Dtþ1 þ Ytþ1.

ARTICLE IN PRESS


We obtain

Istþ2;dn

þXT�1

k¼1

Rstþ2;k ¼ Isþ1

tþ2;dnþXT�1

k¼1

Rsþ1tþ2;k

since

Istþ1;dn

þXT�1

k¼1

Rstþ1;k ¼ Isþ1

tþ1;dnþXT�1

k¼1

Rsþ1tþ1;k.

Now, we are ready to prove the last part of the lemma

by induction. If Istþ2;dn

� Isþ1tþ2;dn

, and Rstþ2;k � Rsþ1

tþ2;k for every

k ¼ f1;2; . . . ; T � 1g, we assume that the lemma holds for

t+2, i.e., Vsþ1tþ2ðI

sþ1tþ2; a

sþ1tþ2;sþ1;A

sþ1tþ2;sþ1Þ � Vsþ1

tþ2ðIsþ1tþ2;dn

;Rsþ1tþ2Þ. The

state variables in Vsþ1tþ2ðI

sþ1tþ2 ; a

sþ1tþ2;sþ1;A

sþ1tþ2;sþ1Þ evolve from

values of ðIstþ1; a

sþ1tþ1;sþ1;A

sþ1tþ1;sþ1Þ, while variables in

Vsþ1tþ2ðI

sþ1tþ2;dn

;Rsþ1tþ2Þ transition from values of ðIsþ1

tþ1;dn;Rsþ1

tþ1Þ.

For ease of notation, we define

LHS ¼ Ctþ1ðXtþ1Þ þ Stþ1ðYtþ1Þ þ htþ1ðIOtþ1 þ Xtþ1 � Ytþ1Þ þ ptþ1

�Xm0 ðtþ1Þ

j¼1

qj � ED;W=J¼j Dtþ1 � Is

tþ1;dn�

XT�1


Rstþ1;l

0@ 1Aþ24 35þ ptþ1 �

XT�1

j¼m0 ðtþ1Þþ1

qj � ED Dtþ1 � Is

tþ1;dn�XT�1

k¼j

Rstþ1;k

0@ 1Aþ24 35,

RHS ¼ Ctþ1ðXtþ1Þ þ Stþ1ðYtþ1Þ þ htþ1ðIOtþ1 þ Xtþ1 � Ytþ1Þ þ ptþ1

�Xm0 ðtþ1Þ

j¼1

qj � ED;W=J¼j Dtþ1 � Isþ1

tþ1;dn�

XT�1


Rsþ1tþ1;l

0B@1CAþ264375

þ ptþ1 �XT�1

j¼m0 ðtþ1Þþ1

qj � ED Dtþ1 � Isþ1

tþ1;dn�XT�1

k¼j

Rsþ1tþ1;k

0@ 1Aþ24 35.

Recall that for j � m0ðt þ 1Þ; we have

Istþ1;dn

�XT�1


Rstþ1;l � Isþ1

tþ1;dn�

XT�1


Rsþ1tþ1;l

and for j4m0ðt þ 1Þ it holds

Istþ1;dn

þXT�1

k¼j

Rstþ1;k � Isþ1

tþ1;dnþXT�1

k¼j

Rsþ1tþ1;k.

Thus, we have LHS � RHS. Finally, we obtain

Vsþ1tþ1ðI

stþ1; a

sþ1tþ1;sþ1;A

sþ1tþ1;sþ1Þ

¼ E½Vsþ1tþ1ðIO

stþ1; I

stþ1;dn

; astþ1;1; . . . ; a

stþ1;s; a

sþ1tþ1;sþ1;A

stþ1;1; . . . ,

Astþ1;s;A

sþ1tþ1;sþ1;R

stþ1;1; . . . ;R

stþ1;T�1Þ�

¼ minXtþ1�0Ytþ1�0

Ytþ1�IOtþ1

fLHSþ Vsþ1tþ2ðI

sþ1tþ2; a

sþ1tþ2;sþ1;A

sþ1tþ2;sþ1Þg

� minXtþ1�0Ytþ1�0

Ytþ1�IOtþ1

fLHSþ Vsþ1tþ2ðI

sþ1tþ2;dn

;Rsþ1tþ2Þg

� minXtþ1�0Ytþ1�0

Ytþ1�IOtþ1

fRHSþ Vsþ1tþ2ðI

sþ1tþ2;dn

;Rsþ1tþ2Þg

¼ Vsþ1tþ1ðI

sþ1tþ1;dn

;Rsþ1tþ1Þ. (21)

where (21) follows by the induction hypothesis. Thiscompletes the proof of Lemma 2. &

Proof of Theorem 4. We take a closer look at the currentone-period cost in time period t in scenarios 1 and 2. Forease of notation, we denote

bCsþ1

t ¼ CtðXtÞ þ StðYtÞ þ htðIOt þ Xt � YtÞ

þ pt �Xm0ðtÞj¼1

qj � ED;W=J¼j Dt � It;dn

�XT�1


Rt;l

0@ 1Aþ24 35þ pt �

XT�1

j¼m0 ðtÞþ1


�XT�1

k¼j

Rt;k

0@ 1Aþ24 35,

bCs

t ¼ CtðXtÞ þ StðYtÞ þ htðIOt þ Xt � YtÞ

þ pt �XmðtÞj¼1

qj � ED;W=J¼j Dt � It;dn

�XT�1


Rt;l

0@ 1Aþ24 35þ pt �

XT�1

j¼mðtÞþ1


�XT�1

k¼j

Rt;k

0@ 1Aþ24 35.

Quantity bCs

t denotes the one-period cost in scenario 1and bCsþ1

t denotes the one-period cost in scenario 2. Wefirst show that

bCs

t �bCsþ1

t . (22)

We start by showing

Xm0 ðtÞj¼mðtÞþ1

qj � pt � ED Dt � It;dn

�XT�1

k¼j

Rt;k

0@ 1Aþ24 35�



�XT�1

k¼mðtÞþ1

Rt;k

0@ 1Aþ24 35. (23)

To show (23), we distinguish three cases.

(1)
Let first m0ðtÞ ¼ mðtÞ. Then it follows:


�XT�1

k¼j

Rt;k

0@ 1Aþ24 35¼



�XT�1

k¼mðtÞþ1

Rt;k

0@ 1Aþ24 35 ¼ 0.

(2)
Next let m0ðtÞ ¼ mðtÞ þ 1. We have


�XT�1

k¼j

Rt;k

0@ 1Aþ24 35¼



�XT�1

k¼mðtÞþ1

Rt;k

0@ 1Aþ24 35¼ qm0 ðtÞ � pt � E

D Dt � It;dn�XT�1

k¼m0 ðtÞ

Rt;k

0@ 1Aþ24 35.

(3)
Finally, let m0ðtÞ4mðtÞ þ 1. For every mðtÞ þ 1 � j �
m0ðtÞ; we have

Xm0 ðtÞk¼j

Rt;k �Xm0 ðtÞ

k¼mðtÞþ1

Rt;k

ARTICLE IN PRESS


and



�XT�1

k¼j

Rt;k

0@ 1Aþ24 35�



�XT�1

k¼mðtÞþ1

Rt;k

0@ 1Aþ24 35.

Suppose that there exists k 2 fmðtÞ þ 1; . . . ;m0ðtÞg such

that Rt;k40. It is clear that mðtÞ ¼ maxfAt;1;At;2; . . . ;At;sg

and m0ðtÞ ¼maxfmðtÞ;At;sþ1g ¼ At;sþ1: By definition of zst

we conclude that mðtÞ þ 1 � at;sþ1 � At;sþ1 ¼ m0ðtÞ.

Now, we perform the final step towards showingbCs

t �bCsþ1

t . We have

bCs

t ¼ CtðXtÞ þ StðYtÞ þ htðIOt þ Xt � YtÞ þXmðtÞj¼1

qj � pt � ED;W=J¼j

Dt � It;dn�

XT�1


Rt;l

0@ 1Aþ24 35þ XT�1

j¼mðtÞþ1

qj � pt � ED

Dt � It;dn�XT�1

k¼j

Rt;k

0@ 1Aþ24 35¼ CtðXtÞ þ StðYtÞ þ htðIOt þ Xt � YtÞ þ

XmðtÞj¼1

qj � pt � ED;W=J¼j

Dt � It;dn�

XT�1


Rt;l

0@ 1Aþ24 35þ Xm0 ðtÞj¼mðtÞþ1

qj � pt � ED


k¼j

Rt;k

0@ 1Aþ24 35þ XT�1

j¼m0 ðtÞþ1

qj � pt � ED


k¼j

Rt;k

0@ 1Aþ24 35 (24)

� CtðXtÞ þ StðYtÞ þ htðIOt þ Xt � YtÞ þXmðtÞj¼1

qjpt

� ED;W=J¼j Dt � It;dn�

XT�1


Rt;l

0@ 1Aþ24 35þ



�XT�1

k¼mðtÞþ1

Rt;k

0@ 1Aþ24 35

þXT�1

j¼m0 ðtÞþ1


�XT�1

k¼j

Rt;k

0@ 1Aþ24 35¼ CtðXtÞ þ StðYtÞ þ htðIOt þ Xt � YtÞ þ

Xm0 ðtÞj¼1

qj � pt

� ED;W=J¼j Dt � It;dn�

XT�1


Rt;l

0@ 1Aþ24 35þ

XT�1

j¼m0 ðtÞþ1


�XT�1

k¼j

Rt;k

0@ 1Aþ24 35¼ bCsþ1

t . (25)

In (24) we break the indices and apply (23) in (25).

We are now ready to carry out the complete proof by

induction. Since we do not consider the cost incurred

beyond the time horizon T, we have VsTþ1 ¼ Vsþ1

Tþ1 ¼ 0. By

the induction hypothesis for t+1, if Istþ1 2 z

stþ1, we assume

that (12) holds, i.e.,

E½Vstþ1ðI

stþ1Þ� � E½Vsþ1

tþ1ðIOtþ1; Istþ1;dn

; astþ1;1; . . . ; a

stþ1;s; a

sþ1tþ1;sþ1,

Astþ1;1; . . . ;A

stþ1;s;A

sþ1tþ1;sþ1;R

stþ1;1; . . . ;R

stþ1;T�1Þ�.

(26)

Starting from (11), we obtain

VstðI

stÞ ¼ Min

Xt�0Xt�0

Yt�IOt

fbCs

t þ E½Vstþ1ðI

stþ1Þ�g

� MinXt�0Xt�0

Yt�IOt

fbCs

t þ E½Vsþ1tþ1ðIO

stþ1; I

stþ1;dn

; astþ1;1; . . . ; a

stþ1;s; a

sþ1tþ1;sþ1,

Astþ1;1; . . . ;A

stþ1;s;A

sþ1tþ1;sþ1;R

stþ1;1; . . . ;R

stþ1;T�1Þ�g (27)

�MinXt�0Xt�0

Yt�IOt

fbCsþ1

t þ Vsþ1tþ1ðI

stþ1; a

sþ1tþ1;sþ1;A

sþ1tþ1;sþ1Þg (28)

�MinXt�0Xt�0

Yt�IOt

fbCsþ1

t þ Vsþ1tþ1ðI

sþ1tþ1;dn

;Rsþ1tþ1Þg (29)

¼ Vsþ1t ðIOt ; I

sþ1t;dn; asþ1

t;1 ; . . . ; asþ1t;sþ1,

Asþ1t;1 ; . . . ;A

sþ1t;sþ1;R

sþ1t;1 ; . . . ;R

sþ1t;T�Þ.

In (27), (28), (29), we have used (26), (22) and (20),

respectively.

This completes the proof of Theorem 4. &

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