Global Dual Sourcing: Tailored Base-Surge Allocation to ......the China allocation. Intuitively, this reduces the base demand while increasing the surge demand. Our formula quantifies

MANAGEMENT SCIENCEVol. 56, No. 1, January 2010, pp. 110–124issn 0025-1909 �eissn 1526-5501 �10 �5601 �0110

informs ®

doi 10.1287/mnsc.1090.1099©2010 INFORMS

Global Dual Sourcing: Tailored Base-SurgeAllocation to Near- and Offshore Production

Gad Allon, Jan A. Van MieghemKellogg School of Management, Northwestern University, Evanston, Illinois 60208

{[email protected], [email protected]}

When designing a sourcing strategy in practice, a key task is to determine the average order rates placed toeach source because that affects cost and supplier management. We consider a firm that has access to a

responsive nearshore source (e.g., Mexico) and a low-cost offshore source (e.g., China). The firm must determinean inventory sourcing policy to satisfy random demand over time. Unfortunately, the optimal policy is toocomplex to allow a direct answer to our key question. Therefore, we analyze a tailored base-surge (TBS) sourcingpolicy that is simple, used in practice, and captures the classic trade-off between cost and responsiveness. TheTBS policy combines push and pull controls by replenishing at a constant rate from the offshore source andproducing at the nearshore plant only when inventory is below a target. The constant base allocation allowsthe offshore facility to focus on cost efficiency, whereas the nearshore facility’s quick response capability isutilized only dynamically to guarantee high service. The research goals are to (i) determine the allocation ofrandom demand into base and surge capacity, (ii) estimate corresponding working capital requirements, and(iii) identify and value the key drivers of dual sourcing. We present performance bounds on the optimal costand prove that economic optimization brings the system into heavy traffic. We analyze the sourcing policy thatis asymptotically optimal for high-volume systems and present a simple “square-root” formula that is insightfulto answer our questions and sufficiently accurate for practice, as is demonstrated with a validation study.

Key words : inventory production; stochastic models; applications; probability; stochastic model applicationsHistory : Received September 25, 2008; accepted August 21, 2009, by Paul H. Zipkin, operations and supply

chain management. Published online in Articles in Advance November 6, 2009.

1. Introduction and SummaryA $10 billion high-tech U.S. manufacturer of wire-less transmission components was at a crossroadsregarding its global network.1 The company had twoassembly plants, one in China and another in Mexico.Although the Chinese facility enjoyed lower costs,ocean transportation made its order lead times 5 to10 times as long as those from Mexico. With highlyuncertain product demand—coefficients of variationsof monthly demand for some products were as highas 1.25—sole sourcing was unattractive: Mexico wastoo expensive and China too unresponsive. The firmhad to decide how it could best utilize these twosources by properly allocating product demand tothem. In practice, specifying supply allocations isa key task of any sourcing strategy—be it globalor domestic—because it affects costs and suppliermanagement. Although also relevant to domesticsourcing, the policy studied in this paper is most nat-urally applied and interpreted in a global setting dur-ing a single-season planning horizon when supplyand demand volatility dominate currency exchange

1 The sourcing strategy that motivated this paper is furtherdescribed in Mini-Case 6 in Van Mieghem (2008).

risk considerations. In this paper, we will refer to theaverage order rates as strategic allocation.The manufacturer retained a management consul-

tant company for advice. Their analysis focused oncomputing the total landed cost as a function of theallocation to China. The total landed cost representsthe end-to-end cost to transform inputs at the sourceto outputs at the destination (Van Mieghem 2008,p. 208). It captures not only the traditional cost ofgoods sold (material, labor, and overhead, shown inFigure 1), but also accounts for supply chain costssuch as transportation, customs, duties, and taxes, aswell as required working capital carrying costs. Wewill refer to all but working capital cost componentsas the “sourcing cost.” Computing the sourcing costis tedious yet straightforward. In contrast, workingcapital greatly depends on lead times (which deter-mine pipeline inventory), volatility and service levels(which determine safety stock). Whereas workingcapital is easily estimated for single sourcing usingreadily available standard inventory formulae, thereare no such formulae for dual sourcing because therequired inventory not only depends on the allocationto both sources but also on the replenishment policy.Therefore, as part of their analysis, the management

110

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Figure 1 Total Landed Cost Is the Cost to Transform Inputs at the Source to Outputs at the Destination

Material

68%9%

8%7%

Illustrative

Supply chain+ ServiceTypical COGS

3% 3%100%2%

Labor/service Overhead Outboundfreight

Customs,duties, and taxes

Inbound freight TLCWorkingcapital

consultants resorted to an extensive simulation studyof the total landed cost.The simulation study captured a variety of prod-

uct parameters as well as five distinct China alloca-tions (0%�25%�50%�75%� and 100%) using the firm’sreplenishment policy, which we will refer to as a tai-lored base-surge (TBS) sourcing policy. This policycombines push and pull controls by replenishing ata constant rate from China (push), yet ordering fromMexico only when inventory is below a target (pull).Policies that assume a constant order rate are alsoknown as “standing order policies” and have beenused for decades (Rosenshine and Obee 1976, Janssenand De Kok 1999). The presumption is that the low-cost source cannot rapidly change volumes becauseof frictions such as long lead times or an inflexi-ble level production process, which is essential toachieve this cost advantage. The benefits of this pol-icy are that it is simple to administer and it elim-inates the need to explicitly account for the longlead time. In addition, the policy aligns the orderdynamics with each source’s competencies: The slowsource replenishes “base” demand, and the fast sourcereacts to the remaining “surge” demand. As such,the TBS policy captures the classic trade-off between

Figure 2 Consultants’ Simulated Total Landed Cost Was Minimal When Allocating More Than 50% But Less Than 100% to China

SKU 1: Mexico cost = $910, China cost = $606 SKU 2: Mexico cost = $445, China cost = $344

750

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cost and responsiveness: The constant base alloca-tion allows China to operate under level production,and thereby focus on cost efficiency, whereas Mexico’squick response is utilized only dynamically to guaran-tee high service. The consultants’ simulation (Figure 2shows two representative results) indicated that thetotal cost was convex and, for the majority of parame-ter values, minimal when around 75% (i.e., more than50% but less than 100% given that only five alloca-tions were investigated) was sourced from China. Theobjective of this paper is to present an analytic modeland formulae to predict the optimal allocation, under-stand its drivers, and tailor the sourcing strategy tothe demand and supply characteristics. In our numer-ical study, we validate the robustness of the “three-quarter” allocation rule of thumb as a good startingpoint during strategic planning. The studied modelapplies to the dual-sourcing setting where (1) thelower-cost supplier has a sufficiently long lead time(making a standing order a reasonable alternative todynamically changing orders), (2) the more expen-sive supplier has a short transportation lead time(relative to his order fulfillment/production lags),

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and (3) there is a single-season planning period dur-ing which demand is reasonably stationary. A globaloffshore/nearshore dual-sourcing setting is a naturalexample, provided demand and supply risks domi-nate currency exchange risks.We consider a model of a single-stage inventory

system that replenishes from two supply sourcesusing a TBS policy. The demand and supply processescan be general, correlated stationary stochastic pro-cesses. Even our simple TBS policy is not amenableto exact analysis. There are two options to proceed:(1) solve the exact problem numerically or via simula-tion, or (2) solve an approximate problem analytically.Given that we seek simple formulae to determine theallocation and its key determinants, we develop aBrownian analytic model that is asymptotically opti-mal for high sourcing volumes. Analytic optimizationof the Brownian model provides us with an analyticprescription for the sourcing allocations, the base-stock level, and its corresponding cost.Our main results can be summarized as follows.(1) We present performance bounds on the opti-

mal cost and prove that economic optimization bringsthe system into the so-called heavy-traffic regime. Weprovide an analytic characterization of the asymp-totically optimal TBS dual-sourcing policy, includingits strategic allocation, base-stock level, and expectedcost, as well as an analytic expression for the cor-responding “overshoot” process. In addition, wepresent a simple square-root formula to predict thenear-optimal allocation and cost.(2) The analytic characterizations, including the

simple square-root formula, capture and quantify theclassic trade-off between cost and responsiveness.They highlight the key drivers of the dual sourc-ing allocations: (i) the monetary ratio of the Chinacost advantage to the unit holding cost; (ii) averagedemand rate; (iii) the volatility of demand and Chinasupply; and (iv) demand–supply correlations as wellas serial time correlations. Our results not only con-firm intuition but also provide new insight and permiteasy quantification of the allocation and correspond-ing cost. For example, an increase in the monetaryratio (either because of a larger China cost advantageor a smaller holding cost) results, as expected, in alarger China allocation. Our formulae predict that thisrelationship is nonlinear and follows a square root.Similarly, an increase in demand volatility decreasesthe China allocation. Intuitively, this reduces the basedemand while increasing the surge demand. Ourformula quantifies what constitutes “base demand,”thereby providing the scientific underpinnings of theprinciple of strategic alignment when applied to dualsourcing. We also quantify and investigate the valueof dual sourcing over single sourcing.

(3) A numerical study shows that our analyticcharacterization and the simple square-root formulaprovide sufficiently accurate prescriptions relative tosimulation-based optimization of the TBS policy aswell as more complex policies. This study initiallyassumes parameter values traditionally used in theliterature but then continues with applying the modelto real data from the motivating example. During thelatter, we discuss how to calibrate model parametersin practice and validate the robustness of the “three-quarter” allocation rule of thumb. This suggests thatour results are readily applicable.The remainder of this paper is structured as fol-

lows. The next section provides a review of the rele-vant literature and is followed by a discussion of themodel. Section 4 specifies inventory dynamics underthe TBS policy, presents bounds on the optimal cost,and proves that optimization brings the system intoheavy traffic. Section 5 analyzes the Brownian modeland the asymptotic performance of the TBS policy.Section 6 identifies key drivers and the value of dualsourcing. Section 7 discusses the impact of demand–supply correlations as well as serial time correlations.Section 8 reports the numerical validation study. Sec-tion 9 provides a conclusion and discussion of limita-tions. All proofs are relegated to the online appendix(provided in the e-companion).2

2. Literature ReviewThe dual-sourcing literature dates back to Barankin(1961), who studied a single-period model with emer-gency orders. The literature distinguishes betweensingle- and dual-index policies, depending onwhether one or two inventory positions are tracked,and, somewhat independently, between single- anddual-base-stock policies, depending on the number oforder-up-to levels used by the policy.3 Our TBS policyis a single-index, single-base-stock policy. The dual-sourcing literature can also be divided into discreteand continuous review models.Discrete review models include Fukuda (1964) who

studies a dynamic inventory model with stochasticdemand in which the deterministic lead times ofboth sources differ by exactly one period. He showsthat single-index, dual-base-stock policies are optimal

2 An electronic companion to this paper is available as part of the on-line version that can be found at http://mansci.journal.informs.org/.3 A single-index policy uses one state variable, usually the totalinventory position I . A single-base-stock policy uses one parameterand usually brings the inventory position after ordering as close tothe base stock level s as possible (Porteus 2002, p. 67). A single-index, dual-base-stock policy has two parameters s < s0. As witha TBS policy, the fast source shuts off when I > s. In addition, theslow source also shuts off when I > s0. A dual-index policy trackstwo state variables, e.g., inventory position of the emergency sup-plier and the total inventory position.

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under mild conditions. Whittemore and Saunders(1977) extend Fukuda’s (1964) model to allow for arbi-trary (yet still deterministic) lead times. They showthat when lead times differ by more than one period,the optimal policy is no longer a simple function ofone or two inventory positions, but depends on theentire ordering history. The model in Rosenshine andObee (1976) assumes a regular lead time, but imme-diate emergency replenishment. Their standing orderpolicy, which was evaluated numerically, assumes aconstant order rate from the regular source, a fea-ture shared by our TBS policy. Tagaras and Vlachos(2001) allow emergency replenishment within theregular review period. Veeraraghavan and Scheller-Wolf (2008) introduce a dual-index policy for capaci-tated dual-source models that can be computed usinga simple simulation-based optimization procedure.They show that such a dual-index policy is nearlyoptimal when compared to state-dependent policiesfound via multidimensional dynamic programming.Scheller-Wolf et al. (2006) establish computationallythat a single-index policy can be highly effective, andeven outperform a dual-index policy. Sheopuri et al.(2007) generalize the dual-index policy by consider-ing two classes of policies that have an order-up-to structure for the emergency supplier. The authorsshow that the “lost sales inventory problem” is aspecial case of the dual-sourcing problem. They usethis property to suggest near-optimal policies withinthis class that often improve on the already excel-lent performance of dual-index policies.4 One of theirpolicy classes uses a single order-up-to level through-out and then allocates, in each period, demand toeach supplier. The idea of determining the alloca-tion is similar in spirit to our approach. However, toaddress our research question of strategic allocation,we first determine the average allocation through-out. This average allocation then determines a singleorder-up-to level that specifies dynamically when tosource from the fast supplier. In addition, the papersabove consider deterministic lead times; in contrast,one of the goals of our model is to explore the rela-tionship between the optimal strategic allocation toeach source and the volatility of the supply sources.Continuous review models include Moinzadeh and

Nahmias (1988), who consider two sources withdeterministic lead times and fixed order costs. Theyextend the �Q�R� policy to two different lot sizes andtwo different reorder levels, and optimize over thesefour parameters. Assuming negligible fixed order

4 Dual-index policies and their generalizations require the station-ary distribution of an “overshoot” process, which typically isobtained through simulation. We provide an analytic expression ofan overshoot distribution that may be useful in the computation ofthe former policies.

costs, Moinzadeh and Schmidt (1991) consider a moresophisticated dual-base-stock policy in which real-time supply information on the age of all outstand-ing orders and the inventory level is used. Songand Zipkin (2009) extend Moinzadeh and Schmidt(1991) by considering a system with multiple sup-ply sources under stochastic demand and lead times.The authors develop performance evaluation toolsfor a family of policies that utilize real-time sup-ply information and under which the supply sys-tem becomes a network of queues with a routingmechanism called an overflow bypass. Bradley (2004),which is the closest to our model and inspired ouranalysis, considers a production–inventory problemwhere the inventory can be replenished from in-houseproduction or through a subcontractor. The authorconstructs a Brownian approximation of the optimalcontrol problem, assuming that the manufacturer usesa single-index, dual-base-stock policy. By using only asingle base stock, our replenishment policy is simplerand provides greater tractability. This allows us tospecify and investigate the optimal allocations explic-itly. Zipkin (2000) highlights the connection betweeninventory and queuing theory, and argues that “queu-ing theory remains our richest source of models forsupply processes” (p. 13).The aforementioned papers focus on determining

or optimizing the control parameters, or on eval-uating the performance of dual-sourcing policies.Although we also derive the optimal base-stock levelof the TBS policy, our focus is on determining theoptimal allocation of demand to either source. Thelatter is also the focus of the literature on “order split-ting,” which studies inventory models with determin-istic demand. Lau and Zhao’s (1994) paper belongsto this stream, and studies a system with stochasticlead times and explores the impact of splitting ruleson inventory costs and stockout risks.In addition to dual-source inventory models, our

work is also related to the literature on inventorymodels with returns. Under a TBS policy, the netdemand experienced by the firm after subtracting thebase-demand replenishment can be negative. Inven-tory models with returns, as studied by Fleischmannet al. (2002) and DeCroix et al. (2005), are characterizedby the same feature. Fleischmann et al. (2002) studieda Markovian model with fixed cost. The behavior ofthe inventory cost as a function of the return ratio isclosely related to the behavior of the total cost in ourmodel as a function of the allocation to China. DeCroixet al. (2005) studied a more general serial system withreturns and show that an echelon base-stock policy isoptimal.Our model explores the cost-responsiveness trade-

off when allocating supply to a responsive yet expen-sive source, and a low-cost but remote source within

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an existing network. It does not explore financialhedging of currency exchange rate risk or configuringglobal networks. For such models, we refer the readerto Ding et al. (2007) and Lu and Van Mieghem (2009)and references therein.

3. ModelConsider a continuous-time model of a single-stageinventory system with two supply sources. The cumu-lative demand up to time t is a stationary stochasticprocess D�t�; demand in excess of available inventoryis backlogged. Initially D, is modeled as a counting(renewal) process whose independent and identicallydistributed (i.i.d.) interarrival times have mean 1/�and coefficient of variation vD; later we will general-ize to allow for correlated interarrival times. Similarly,to model production variability as well as congestionand disruption, the actual supply from either source isstochastic around its mean rate. To be precise, let Si�t�denote the cumulative quantity received from sourcei if it were continuously supplying during �0� t�. Toalign with our motivating example, we will use i ∈�M�C for Mexico and China as concrete placeholdersfor the nearshore and offshore sources, respectively.Initially, we assume that Si�t� is a renewal processwhose associated i.i.d. service times have mean 1/i

and coefficient of variation vi; later we will gener-alize to allow for correlated intersupply as well asfor cross-correlations between intersupply and inter-demand times. We shall refer to i as the capacityof source i, which is a decision variable and incurs acapacity cost of cK

i i per unit of time.The variable order cost from source i is cV

i perunit ordered. As discussed in the Introduction, thissourcing cost includes all components of the totallanded cost with the exception of working capital (i.e.,inventory) cost. Source M is responsive but expen-sive, whereas source C is cheap but slow. M is moreexpensive both on a variable-cost basis (cV

M > cVC ) as

well on a full-cost basis: cM = cKM + cV

M > cC = cKC + cV

C .Let the control Ti�t� denote the actual cumulative

amount of time that source i is supplying during �0� t�so that Si�Ti�t�� is the actual supply from source iduring �0� t�. Let I�t� denote the net-inventory pro-cess, i.e., the amount of inventory on hand minus theamount on backorder at time t. We then have the fol-lowing dynamics:

I�t� = I�0� + SC�TC�t�� + SM�TM�t�� − D�t��

Let I��� denote the steady-state net-inventory pro-cess for a given control policy T . On-hand inventoryI+ incurs the familiar per-unit holding cost h per unitof time.5 Stockouts are backlogged, and backorders

5 Here, h is the average unit holding cost rate. Under the policy ana-lyzed in this paper, its opportunity cost component is a weighted

I− incur a per-unit backlogging penalty cost b perunit of time. In the usual way, the average inventory(or demand–supply mismatch) cost rate under thispolicy is G = Ɛg�I����, where g�x� = hx+ + bx− = hx +�b + h�x−. Let � denote the critical fractile b/�b + h�and � = 1− � .

The research question is to determine the capac-ity vector and the allocation policy T that mini-mizes total cost C, the sum of capacity, inventory, andsourcing costs. We seek simple characterizations ofhow the sourcing volume � should be allocated to thetwo sources. In other words, we want to characterizethe “base demand” that should be allocated to China,and when tailored dual sourcing outperforms singlesourcing.Addressing these questions involves determining

the optimal dynamic order policy, which is generallycomplex and not amenable to exact analysis. There-fore, in what follows, we first restrict attention to aparticular allocation policy (the TBS policy) for whichwe provide some general results. To further quantifyits performance, we then provide an analytic charac-terization using a Brownian model of the TBS pol-icy that is asymptotically correct for high volume(� → �). In a third step, we present a simple square-root formula that is a lower bound of the predictedoptimal allocation in the Brownian model. Finally,our numerical study validates the accuracy of ourapproximation.

4. The Tailored Base-Surge PolicyThe simplest tailored allocation policy orders a con-stant rate from the offshore source and orders onlyoccasionally when needed from the nearshore source.Specifically, China supplies at a constant rate C ;clearly, 0 ≤ C < � to prevent unlimited inventorybuildup. In contrast, the policy orders from Mexicoonly when the net inventory falls below a targetlevel s. During that time, supply from Mexico isreceived at rate M . Obviously, C + M > � to keepup with demand.As stated in the Introduction, a TBS policy is used

in practice because it is simple to administer and itallows the efficient source to operate under level pro-duction. It also is amenable to analysis and, hence,simple to tailor to particular demand–supply char-acteristics. The underlying assumption of the TBSpolicy is that the offshore source is not capable ofimplementing feedback control because of various

average r��1− C�cM + CcC�, where C denotes the fraction sourcedfrom China and r is the cost of capital. We shall see that ∗

C is closeto 1, so that the opportunity holding cost � rcC , and we assume h isconstant. Incorporating the dependence of h on the allocation doesnot impact our main asymptotic results, but it does significantlycomplicate exposition.

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Allon and Van Mieghem: Global Dual SourcingManagement Science 56(1), pp. 110–124, © 2010 INFORMS 115

frictions such as long transportation times or inflexi-ble production.Flow balance dictates that the long-run average

supply from Mexico is �−C , and the long-run aver-age sourcing cost rate is cV

M� − C�cV , where �cV =cV

M − cVC > 0. The difference M − ��−C� > 0 between

the Mexican capacity and its long-run average sup-ply rate is called the Mexican safety capacity, whichis positive. In contrast, the constant order obviates theneed for China safety capacity, and its average supplyrate equals its capacity. Observe that, by design, thereplenishment lead time from the slow source doesnot impact the TBS policy. However, we can easilyaccount for any pipeline inventory holding costs.6

Under the TBS policy, continuous supply fromChina implies that the control TC�t� = t so that themodel simplifies to a single-source inventory modelwith remaining demand D�t� − SC�t�, which can benegative. This is mathematically equivalent to aninventory system with returns, and it is well estab-lished that a base-stock policy is optimal. In the typicalbase-stock dynamics, once inventory falls below s(after a potential transient initial regime), the inven-tory position stays at or below s and is a demand-replacement policy. This is not the case under a TBSpolicy because the slow source supply may occasion-ally exceed the actual demand, resulting in excessinventory excursions above s. A similar “overshoot”phenomenon is observed in the dual-index policies ofVeeraraghavan and Scheller-Wolf (2008) and the gen-eralizations by Sheopuri et al. (2007). This overshootis a key disadvantage that is not present in Bradley’s(2004) dual-base-stock policy.We will adopt a continuous-review base-stock pol-

icy that requests supply at rate M from the fastsource whenever the net inventory falls below s. LetZ = I −s denote the “excess inventory process,” whichis the inventory above the base stock. Under a TBSpolicy, the excess inventory dynamics simplify to

Z�t� = Z�0� + SM�TM�t�� + SC�t� − D�t��

whereTM�t� =

∫ t

01�Z�u� < 0 du�

Essentially, Z is a random walk stemming fromthe conventional order-up inventory dynamics with asuperimposed GI/G/1 queue capturing the occasionalexcess inventory excursions. For a given capacity vec-tor , let F denote the stationary distribution of Z (wewill show that such limiting distribution does exist).

6 Let LM and LC denote the average transportation times fromMexico and China, respectively, so that the associated in-transitholding costs is LM �� − C�h + LCCh. The allocation only affectsthe terms in C and that effect can be captured by inflating cV

M byh�L, where �L = LC − LM > 0.

The benefit of analyzing the excess inventory processZ is that it is independent of the actual value of thebase stock s.The average steady-state total cost rate under a TBS

policy with base stock s and capacity vector is thesum of inventory, sourcing, and capacity costs:

C�C�M�s� = Ɛg�Z���+s�+cVC C +cV

MM��Z���<0�

+cKC C +cK

MM

= G��s�+cVM�+�cK

C −�cV �C +cKMM�

given that a stationary solution requires stability sothat M��Z��� < 0� = � − C . The inventory cost G =Ɛg�Z���+ s� = hƐ�Z���+ s�+ �b +h�Ɛ�Z���+ s�− andintegration by parts of the last term yields

G�� s� = hs + h∫ +�

−�x dF�x� + �b + h�

∫ −s

−�F�x�dx�

Proposition 1. The inventory cost G�� s� is convexin s for any , and the optimal base stock s∗ is a fractileof the steady-state excess inventory distribution: If F iscontinuous, then F�−s∗� = � .

This type of newsvendor solution has appearedin previous analyses of inventory shortfall as dis-cussed by Bradley and Glynn (2002). To optimize thetotal cost, it “only” remains to specify the station-ary distribution F. Given that our system involvesGI/G/1 queue dynamics, its stationary distributioncannot be solved analytically in general. We can, how-ever, obtain a useful upper bound on the optimal costas follows. Observe that the optimal dual-sourcingcost dominates the minimal cost under single sourc-ing from Mexico with s = 0� minC�C�M� s� ≤ minC�0�M�0�. Under such single sourcing, Z��� ≤ 0,and the backlog −Z is a GI/G/1 queue so that

C�0�M�0� = −bƐZ��� + McKM + cV

M�

≤ b�

M − �

v2M + v2

D

2+ McK

M + cVM�,

using Kingman’s bound. The right-hand side is con-vex in M and reaches a minimum at M = � +√

�b/cKM���v2

M + v2D�/2��, which yields an exact upper

bound: minC�0�M�0� ≤ C�, where

C� = �cVM + cK

M�� +√2bcK

M�v2M + v2

D���

The upper bound also bounds the inventory and thecapacity cost. This directly shows how the optimalinventory and capacities depend on the volume �,which is key to our analysis. To emphasize this depen-dence, we will add a superscript � to the notation. Forexample, C� denotes the total cost given volume �,and ��∗

C ��∗M � s�∗� denotes an optimal solution.

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Proposition 2. The optimal cost C���∗C ��∗

M � s�∗� isbounded,

�cVC + cK

C �� +√2cK

C h�v2C + v2

D� ln �−1√

� + o�√

��

≤ C���∗C ��∗

M � s�∗� ≤ C��

and there exist nonnegative scalars M , C , and s suchthat the optimal solution satisfy

�∗C = � − C

√� + o�

√��� (1)

�∗M = M

√� + o�

√��� (2)

s�∗ = s√

� + o�√

��� (3)

Proposition 2 has two important implications. First,the optimal allocation sources the majority from thecheap source, but a small amount of Mexican capac-ity is necessary. Second, economic optimization natu-rally brings the system into a parameter regime called“heavy traffic.” Loosely speaking, this means that theChina resource is heavily utilized. Indeed, expres-sion (1) implies that the optimal China utilization�∗

C /� � 1− C/√

� tends to 100% as � → �. In addi-tion, the optimal Mexican capacity, although small,is just sufficient to stabilize the inventory process.The theoretical significance of the proposition is thatheavy traffic is not assumed, but the proved result ofcapacity optimization. From a practical perspective,the proposition guarantees that the system convergesto a tractable Brownian limiting system as � → �.

Proposition 3. The scaled excess inventory Z�/√

�converges almost surely uniformly in compact sets7 to adual-drift Brownian motion �Z�

Z��t�√�

→ �Z�t� = �Z�0� + M�T �t� − Ct + �B�t� a�s��

where �2 = v2C + v2

D, B is a standard Brownian motion,and �T �t� = ∫ t

0 1� �Z�u� < 0 du. Furthermore, the steady-state limit �Z��� exists if M > C .

The limiting scaled inventory �Z is a diffusion pro-cess with negative drift −C if �Z ≥ 0, and positivedrift M − C elsewhere. As we will show, the limit-ing system is amenable to analytic optimization andallows us to prescribe a solution for a system withvolume � as follows. Denote the total cost of theBrownian limiting system by

�C�C�M�s� = Ɛg� �Z���+ s�+cKMM

+�cVM −cK

C −cVC �C� (4)

and let �∗C� ∗

M� s∗� denote a minimizer of �C. We cannow state our prescribed solution.

7 This means that, with probability 1, for every compact set A ⊂�+,lim�→� supt∈A �Z��t�/

√� − �Z�t�� = 0�

Proposition 4. The prescription �� − ∗C

√�� ∗

M

√��

s∗√�� is asymptotically optimal:

lim�→�

C��� − ∗C

√�� ∗

M

√�� s∗√�� − �cV

C + cKC ��√

�

=min �C�C� M� s��

The solution that we prescribe for a system withvolume � is based on the optimal solution of theBrownian limiting model, and thus is guaranteed toperform well for large volumes. In the remainder, weanalyze the asymptotic cost �C, characterize its optimalsolution �∗

C� ∗M� s∗�, and validate the prescription for

various volume levels.

5. Asymptotic Analysis of theTBS Policy

5.1. Steady-State Distribution �F

The steady-state distribution of �Z��� follows directlyfrom Browne and Whitt (1995):

Proposition 5. The steady-state limit �Z��� has dis-tribution function

�F�x� =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

C

M

exp(2 �M − C�

�2x

)x < 0�

1− �M − C�

M

exp(

−2C

�2x

)x ≥ 0�

(5)

which decreases as −C or M increases.

For Markovian systems (demand follows a Pois-son process with rate �, service times from source iare i.i.d. exponentially distributed with rate i, andall service and demand times are independent), wecan calculate the exact steady-state distribution of Z�

using the detailed balance equations

limt→���Z��t� ≤ x� =

⎧⎪⎨⎪⎩

�0

1− ��−�x� for x < 0�

1− �0

1− ���x�+1 for x ≥ 0�

where

� = �

M + C

� � = C

��

�0 = �� − C��M + C − ��

�M

.

Using the optimal capacities (1) and (2) and the fact thatlim�→��1+ x/��� = ex, it is easy to verify that indeed

lim�→�

limt→��

(Z��t�√

�≤x

)= �F�x�= lim

t→� lim�→�

�

(Z��t�√

�≤x

)�

The scaled excess inventory has a biexponentialdensity, as shown in Figure 3. As expected, higher

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Figure 3 Stationary Scaled Inventory I = �Z��� + s Has aBiexponential Distribution

1 –2�C

2(�M –�C)

Base stock s Inventory I

Pr(I ≥ s) = 1 –�C

�C

�C

�M

�M

�M

�2

�2( )

supply rates (hence, smaller C or larger M )stochastically increase inventory, and thus the averagestationary inventory:

Ɛ �Z��� = − � 2

2 �M − C�

C

M

+ �2

2C

M − C

M

�

Notice that the proposition provides an analyticcharacterization of the overshoot process that may beuseful in the computation of dual-index policies andtheir generalizations. It certainly allows us to computeand optimize the holding cost of the TBS policy.

5.2. Optimal Scaled Base Stock s∗ andAssociated Inventory Cost G�� s∗�

The explicit characterization (5) allows the specifica-tion of the optimal base stock s∗ = − �F −1

��� and of theassociated inventory cost for fixed capacities. Giventhe specific biexponential structure of �F , we distin-guish two operating regimes: s∗ ≥ 0 versus s∗ < 0. Fol-lowing Bradley (2004), we say that the control policyis preventive if the fast Mexican source is engagedwhile inventory is on hand (s∗ ≥ 0), and reactive whenMexico supplies only backorders (s∗ < 0).

Proposition 6. Consider fixed capacities M and C .If � = h/�h + b� ≤ C/M , then the optimal base stock s∗

is positive so that the fast source supplies to stock (“pre-ventive mode”) with

s∗ = − �2

2 �M − C�ln

M

C

� ≥ 0 and

G�� s∗� = hs∗ + h�2

2C

≥ 0�

Otherwise, s∗ < 0, and the fast source engages only to coverbacklog (“reactive mode”) with

s∗ = �2

2C

lnM

M − C

� < 0 and

G�� s∗� = −bs∗ + b�2

2�M − C�≥ 0�

As expected, the optimal base stock s∗ is decreasingin M and −C . In addition, for a constant alloca-tion, the absolute value of the optimal base stock andthe total cost are increasing in volatility � . Operatingin preventive (reactive) mode is similar to operatingthe Mexico source in a make-to-stock (make-to-order)fashion. The optimal regime is preventive when rela-tive holding costs h/b and the contingent supply M

are small; otherwise, it is better to move to a make-to-order model in which we operate in reactive modeand use the ample capacity of the fast source to coverbacklogs.

5.3. Optimal Scaled Capacities ∗

and Cost �C∗

We can now characterize the optimal capacities thatminimize the total asymptotic cost �C:

Proposition 7. The inventory cost G�� s∗�, and thusthe total cost, is strictly convex over the set 0≤ C ≤ M .The optimal cost is

�C∗ = �C�∗C� ∗

M� s∗� = 2cKM∗

M + 2�cVM − cC�∗

C (6)

where the optimal capacities �∗C� ∗

M� are the unique solu-tions to

∗C =

√h�2

2�c+ �2 − �� where � = 1

∗M

h�2

4�c� (7)

2h�2

cKM = 1

∗M�∗

M − ∗C�

− 1�∗

M − ∗C�2

ln∗

M

∗C

� (8)

if s∗ > 0, and otherwise,

∗M −∗

C =√

b�2

2cKM

+�2−�� where �= 1∗

M

b�2

4cKM

� (9)

2b�2

�cC −cVM�= 1

∗2C

ln∗

M

∗M −∗

C

� + 2∗C −∗

M

∗C�∗

M −∗C�2

� (10)

At the optimal control variables, the inventory costG�∗� s∗� equals the sourcing and capacity costs sothat the optimal cost �C∗ equals twice the latter. Thisproperty is similar to the familiar economic orderquantity model and useful to verify whether the con-trols are close to optimal in numerical work. Indeed,the transcendental first-order equations (7) and (9) areeasily solved numerically, yet also suggest the follow-ing simple square-root expressions to be used as astarting point for allocations in the preventive andreactive modes:

p = �

√h

2�cand r = �

√b

2cKM

� (11)

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Proposition 8. The simple square-root formulae pro-vide upper bounds on allocation,

p − 1∗

M

h�2

4�c≤ ∗

C ≤ p if s∗ >0 �preventive�� (12)

r −1

∗M

b�2

4c1K≤ ∗

M −∗C ≤ r

otherwise �reactive�� (13)

and on the optimal cost,

�C∗ ≥ 2�c∗C ≥ �

√2h�c� (14)

Our analysis thus culminates in these simplesquare-root formulae which provide a useful esti-mate or starting point for the optimal China allocationand cost in the more likely preventive mode. Indeed,when dual sourcing a given volume �, our analysisgives the following prescriptions:

China allocation �∗C ��−∗

C

√�≥�−�

√h�

2�c� (15)

Total cost C�∗ �cC�+ �C∗√�≥cC�+�√2h��c� (16)

which are accurate up to o�√

�� and directly identifykey drivers that we discuss next.

6. Drivers and Value of Dual Sourcing6.1. Key Drivers of Strategic AllocationThe square-root formula (15) directly provides the fol-lowing key drivers, insights, and quantification onstrategic allocation. First, the key monetary trade-offin determining the China allocation is �c/h, whichcan be expressed as follows. Recall that the unitholding cost h = �cost of capital r+physical holdingcost p�cC , so that the key trade-off simplifies to

�c

h= �csource − h�L

h= �c/cC

r + p− �L

= relative cost advantagecost of capital+ physical holding cost

− transportation time difference.

Note that this equation is in time units and that it cap-tures the combined impact of monetary cost concernsas well as responsiveness. This is exactly the trade-offat the essence of this model. It shows that the Chinaallocation is high when (i) China has a high relativecost advantage (as expected), (ii) the cost of capitaland the physical holding cost are low (meaning smallopportunity costs as well as low risk of obsolescence),and (iii) there is a relatively small transportation timedifference between China and Mexico. Not only does

this confirm intuition, the equation also quantifies thefactors and their interaction.Second, the total China allocation as a fraction of

average demand is

�∗C

�� 1− ∗

C√�

≥ 1− �

√h

2��c�

It strongly depends on product volume, and thus onits stage in the product life cycle: As the volume grows,the China allocation should increase. Later, during thedecline phase, that allocation should decrease, therebyreflecting a shift in the relative importance from costto responsiveness.Third, the allocation depends mostly on the China

supply volatility and the demand volatility. Our ap-proximation depends equally on both, but is ratherinsensitive to the Mexico supply volatility.8 As ex-pected, as China becomes a less reliable source, itsallocation is reduced. Interestingly, as the demandvolatility increases, the allocation to China is reducedas well. Both effects reflect the fact that China is theless flexible source.The combined impact of these three key drivers is

summarized through the ratio

h��v2C + v2

D�/2�2��c

= holding cost of safety stocksourcing cost savings

�

which captures the natural trade-off in dual sourcingand quantifies it: as the ratio increases, the China allo-cation reduces.

6.2. Cost and Value of Dual SourcingThe square-root formula (16) provides similar insightson the cost of dual sourcing. Given that the major-ity is sourced from China, the first-order componentin the total cost is simply the China cost rate cC�.With proper sourcing management, the cost of safetycapacity and safety stock is of second order, O�

√��.

That cost increases linearly in volatility and sublin-early with the unit holding cost h and the total costdifferential �c.The square-root formula (16) can also be used to

quantify the value of dual sourcing, which is thereduction in total cost relative to single sourcing fromMexico. The latter is a special case of dual sourcing,and its cost is

C�∗1M = �cV

M + cKM�� +

√2cK

Mh�21 ln �−1

√� + o�

√��

= cM� + 2cKM1M

√� + o�

√��,

8 The Brownian analysis in the online appendix shows that vM isO��1/4�, and thus a third-order effect.

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where 1M can be shown to be√

�h�21 /2cK

M� ln �−1 (seeproof of Proposition 2 in the online appendix) so that

Vdual = C�∗1M −C�∗

= ��c+2�cKM�∗

1M −∗M�−�cV

M −cC�∗C�

√�+o�

√��

≥ ��c−2p

√��c+o�

√��

= ��c−�√2h��c+o�

√��,

where we used the fact that ∗C ≤ p and ∗

C ≤ ∗M . The

lower bound on value is tight if cKM is small (so that

∗1M � ∗

M ). The relative value of dual sourcing,

Vdual

Csingle�

�c

cM

− �

√2

h

�cM

�c

cM

�

is bounded from above by the relative sourcing costsavings (which are deterministic), but is reduced byincreased working capital requirements (which reflectthe cost of variability). The latter reflect the corrupt-ing influence of variability, which increases when(1) demand or supply volatility, (2) holding costs rel-ative to total (capacity + variable) Mexico cost, or(3) the relative China advantage increase, or when(4) average demand is low. Any of those four factorscan lead to the lower bound on the relative value ofdual sourcing becoming negative, suggesting that thevalue of dual sourcing would be small.The TBS policy assumes that feedback control on

China is not feasible; which precludes the comparisonof dual sourcing with single sourcing from China. Ifone allows feedback control, this comparison followsthe same lines as our comparison with single sourcingfrom Mexico.

7. Serial and Cross-CorrelatedDemand and Supply

So far we have confined the analysis to settings inwhich the demand and supply processes are tractable,independent renewal processes. The strength of theBrownian approximation, however, is not only ana-lytic tractability but also generality: It can handlecomplex correlated processes, provided the varianceterms are adjusted appropriately. Bradley and Glynn(2002) derived the asymptotic time-average varianceof a general stationary demand and supply process��D�t�� S�t��� t ≥ 0 with interarrival times ��Ui�Vi�� i ∈�. We use that result to discuss two applications thathighlight specific correlation structures observed inpractice.First, assume that demand has autocorrelation func-

tion corr(U1�Uk+1� = �k with ��� < 1, and the supply

processes are independent renewal processes. Then,covar�U1�Uk+1� = �kvarU1 and

�2 = �2varU1�1+ 2�∑

k=1

�k� + 2CvarV C

1

= v2D

(1+ �

1− �

)+ v2

C .

Relative to our earlier setting of independent renewalprocesses, demand that is serially correlated over timehas the effect of adjusting v2

D. Positive time correla-tions increase volatility, and thus reduce the Chinaallocation and the value of dual sourcing. In con-trast, negative time correlations are mean reversing,and increase the China allocation and value of dualsourcing.Second, assume the demand and China supply are

correlated renewal processes with correlation coeffi-cient �, and the Mexico supply process is an indepen-dent renewal process. Then

�2 = �2varU1 + 2CvarV C

1 − 2�Ccovar�U1�V C1 �

= v2D + v2

C

(1− 2�

vD

vC

)�

Relative to independent renewal processes, cross-correlated demand and the China supply have theeffect of adjusting the China supply volatility v2

C . Pos-itive cross-correlations � > 0 could represent a sit-uation where economic cycles impact both demandand China supply productivity. This would decreaseChina volatility, and thus increase the China alloca-tion and the value of dual sourcing. In contrast, nega-tive cross-correlations may arise because of congestedtransportation and import/customs processes, whichwould decrease the value of dual sourcing. To ourknowledge, there is no empirical evidence as to whicheffect dominates.Last, although the China and Mexico supply pro-

cesses could be correlated, this would not impact ourmodel given that Mexico volatility is a third-ordereffect. It is interesting that supply correlation—whichhas been advocated as an important reason to diver-sify the supply base—has little impact on sourcingallocation, and hence on the value of dual sourcing inour model.

8. Numerical Validation StudyWe conduct a numerical study to illustrate and vali-date some of the key results discussed above. The goalof this validation study is to answer four questions:(i) How well does the TBS policy perform relative tothe dual-base-stock policy, both in terms of cost mini-mization and allocation prediction? (We use the dual-base-stock policy as a proxy of the optimal policy

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given that Bradley (2005) showed that a dual-base-stock policy is optimal when all interarrival and inter-supply times are exponentially distributed.) (ii) Howwell does the Brownian prescription perform relativeto the simulation-based optimal allocation? (iii) Howwell does the square-root allocation perform relativeto the exact Brownian allocation? (iv) How well doesour Brownian prescription perform if Mexico has alead time?We shall address these questions using parame-

ters similar to those studied in Bradley (2004) andMoinzadeh and Nahmias (1988): interdemand timesare independent and identically normally distributedwith coefficient of variation vD = 1; intersupply timesfrom Mexico are independent and identically nor-mally distributed with vM = 1; and intersupply timesfrom China are independent and identically normallydistributed with vC = 0�5. (When simulating, negativesampled interarrival times were truncated to 0.9) Theholding and backlogging costs are h = $1 per periodof time per unit and b = $50 per period of time perunit. We set cK

M/cM = 0�25, and results were ratherinsensitive to changes in this fraction. Results shall bedisplayed typically as a function of the relative Chinacost 0< cC/cM < 1 and of the demand rate �.

8.1. Comparing the TBS With Dual-Base-StockPolicies

The first step in the validation study investigates howwell the optimized TBS policy performs relative tothe more complex dual-base-stock policy. For vari-ous values of China’s relative cost advantage �c/cM

and volumes �, we simulated the total cost under aTBS policy with various base-stock levels. A numer-ical search then found the cost-minimizing capacitiesand base-stock level, and the corresponding optimalcost under TBS. For the dual-base-stock policy, wefirst kept the TBS-cost-minimizing capacities. Then,we simulated the total cost for a grid of possiblebase-stock pairs and obtained both optimal base-stocklevels using a numerical search over this grid. Anextensive numerical study in which we also opti-mized over the capacities under a dual-base-stockpolicy found no significant improvement, suggest-ing that the dual-base-stock performance is relativelyinsensitive to the capacities. This also suggests thatthe optimal China supply rate for our single-base-stock policy remains nearly optimal for the dual-base-stock policy, echoing the finding in Scheller-Wolfet al. (2006) that compares single-index with dual-index policies.Figure 4 depicts the relative cost penalty of using

the (optimized) TBS policy compared to using the

9 We computed the coefficient of variation of the truncated sampleand found it was very close to that of the nontruncated distribution.

Figure 4 Relative Cost Increase of TBS Over Dual-Base-Stock Policy

10

12

4

6

8

Rel

ativ

e co

st in

crea

se (

%)

0

2

0 0.2 0.4 0.6 0.8 1.0

Relative China cost cC /cM

� = 10

� = 1

� = 100

(optimized) dual-base-stock policy for � = 1�10� and100. (The sample error was less than 2% of the eithercost.) As the China cost increases, the benefit of con-trolling the China supply (shutting it off and prevent-ing large excess inventory) increases. Yet this benefitdecreases as the volume increases, reflecting the factthat the inventory cost is of second order comparedto the sourcing cost. This suggests that the use of theoptimized TBS policy in practice (instead of a morecomplex policy) can be justified for larger volumes orwhen China’s cost advantage is large.

8.2. Comparing the Brownian Prescription toSimulation-Based Optimization

The second question addressed by our validationstudy is as follows: Under the TBS policy, how closeis the Brownian scaled allocation ∗

C to the optimalscaled allocation ��∗

C − ��/√

�, where �∗C is obtained

by simulation-based optimization?The left panel of Figure 5 shows how the optimal

scaled China allocations for � = 1, 10, and 100 con-verge to the allocation ∗

C that minimizes the asymp-totic Brownian cost. It is remarkable that even for� = 1, the relative error is less than 16% and below 8%as long as the China cost advantage exceeds 10%. For� = 10 and above, the relative error was not statisti-cally significant. Note, however, that the relative erroron the total allocation prescription � − ∗

C

√� depends

on the volume � and is much smaller, especially as� increases. The same comment applies to the rela-tive cost difference, which is shown in the right panelof Figure 5. That panel shows the scaled cost of theoptimal control ��∗

C ��∗M � s�∗� obtained by optimiza-

tion via simulation against the simulated scaled cost ofthe Brownian prescription �� − ∗

C

√�� ∗

M

√�� s∗√��,

both for � = 1, 10, and 100. As proved, both convergeto the asymptotic cost �C∗ (which is of the

√�c form).

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Figure 5 Comparing the Brownian Allocation to the Allocation Optimized via Simulation

0.35

0.40

Scaled cost: Brownian vs. optimal TBS

0.21

0.25

Scaled China allocation: Brownianvs. optimal TBS

0.20

0.25

0.30

0.13

0.17

Scal

edco

st

Scal

edal

loca

tion

Relative China cost cC/cM

0.150.05

0.09

0 0.2 0.4 0.6 0.8 1.0

Relative China cost cC/cM

0 0.2 0.4 0.6 0.8 1.0

Simulated cost of Brownian prescription

� = 1 � = 10

Optimization by simulation

� = 100

� = 1 � = 100� = 10

C*

Asymptotic (analytical)

� = 10

Optimization by simulation

� = 1 � = 100

Asymptotic (analytical)*�C

Yet even for � = 1, the relative error in scaled costbetween the prescription and the optimal control wasless than 7%. The main implication is that the Brown-ian prescription is a good and useful approximation ofthe optimal strategic China allocation, even for smallvolumes.

8.3. Comparing the Square-Root Allocation to theBrownian Allocation

The left panel in Figure 6 shows the optimal capaci-ties ∗

C and ∗M in the Brownian model, as well as the

square-root approximation p. To evaluate cost differ-ences, we also solved first-order condition (8) for theoptimal Mexican capacity given C = p and denoteit by M�p. The right panel shows the optimal cost �C∗

Figure 6 Comparing the Square-Root Allocation �p to the Brownian Allocation �∗C

0.6

0.7

Scaled allocation: Square root vs. Brownian

Scal

ed a

lloca

tion

Scal

ed c

ost

Scaled cost: Square root vs. Brownian

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1.0 0

*

0.2 0.4 0.6 0.8 1.00

0.1

0.6

0.3

0.4

0.5

0.2

Relative China cost cC /cM Relative China cost cC /cM

�C

�p

*�M, p

C(�p, �M, p)

*�M

C*

and the cost C�p� M�p� when using the square-rootformulae.One can observe that the scaled square-root allo-

cation p is a reasonable approximation of the exactscaled Brownian allocation ∗

C , but the error increasesas the China cost increases. Indeed, in our numeri-cal study, the allocation difference is about 10% andbelow 28% as long as the China cost advantageexceeds 10%. Keep in mind that the relative error onthe total allocation prescription �− ∗

C

√� depends on

the volume � and will be much smaller, especially as� increases. The same comment applies to the relativecost difference. The main implication is that the sim-ple square-root formulae provide a reasonable start-ing point for the strategic China allocation.

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Allon and Van Mieghem: Global Dual Sourcing122 Management Science 56(1), pp. 110–124, © 2010 INFORMS

8.3.1. Practice-Based Validation Study. The nu-merical study reported so far assumed parametervalues that have traditionally been used in the litera-ture to assess the quality of our analytic approxima-tions and of the TBS policy. Now, in an attempt to testthe robustness of these results, we next calibrate theparameters using the actual data observed in practicein our motivating example. In addition to choosingthe unit of time and unit of money, the key calibrationinvolves the coefficients of variation using real data.The monthly demand experienced by the firm

varied between 5,000 and 67,000 units. Using a timeunit of one month, we set � = 5�000. The sourcingcost cC varied between a few hundred and a couplethousand dollars. Using $1 as the monetary unit,we set cC = 1�000. The annual holding cost h equals�interest rate + physical holding cost� × cC and wasestimated at about 0�6cC = $600/year to reflect theshort product life cycle, or h = 50 (per month). We keptthe ratio b/h equal to 50 as before. Next, the monthlydemands in the actual observed data exhibited acoefficient of variation between 0�05 to 1�25. Thesevalues were converted into the coefficient of variationof interarrival times using renewal theory: vD =� × �the coefficient of variation of demand rate� =5�000 × �0�05to 1�25� = 15 to 88. The fact that theinterarrival coefficients of variations are significantlyhigher than the monthly coefficient of variation isbecause the latter exhibits strong aggregation effects.In addition, actual order patterns are staggered orbatched. For example, an order for 1,000 units resultsin one high interarrival time followed by 999 interar-rival times of 0. To simulate these arrival processes,we sampled interarrival times that were independentand identically gamma distributed.We used these parameters to obtain both the

allocation and the corresponding total cost using(1) simulation-based optimization, (2) the Brownianprescription, and (3) the square-root approximation,all assuming a TBS policy, and (4) simulation-basedoptimization assuming a dual-base-stock policy. Asshown in Figure 7, the Brownian prescription wasvery close to the optimal allocations under both TBS(the relative error was smaller than 1.4% and smallerthan 0.8% if China cost < 500� and dual base stock(the relative error was smaller than 2.5% and smallerthan 1.0% if China cost< 500). As proved, the square-root prescription � − p

√� is a lower bound on the

China allocation: its error is between 3.4% and 9.3%relative to optimal TBS, and between 4.8% and 10.4%relative to dual base stock.

8.4. How Well Does the Brownian PrescriptionPerform If Mexico Has a TransportationLead Time?

Our analysis has assumed that the Mexico supply in-curs an endogenous delay because of orders queuing

Figure 7 Comparing the Optimal Allocation Using ParametersConsistent with Practice

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Square-root prescription

up for stochastic production. To compare this with tra-ditional inventory models, we simulated the optimalallocation using the practice-based parameters whenMexico supply incurs an additional transportation orinformation lead time. Figure 8 shows that the opti-mal China allocation decreases as the Mexico lead timeincreases. An informal explanation is that the relevantmetric of volatility is the volatility of the lead timedemand, which increases, and our square-root formulathus predicts a decrease in the China allocation. Themain conclusion, however, is that the Brownian pre-scription derived in this paper is quite robust underpractical settings where the nearshore transportationlead time is about a week or less.

Figure 8 Brownian Prescription vs. the Optimal Allocation If MexicoHas a Transportation Lead Time

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Allon and Van Mieghem: Global Dual SourcingManagement Science 56(1), pp. 110–124, © 2010 INFORMS 123

Figure 9 Scaled China Allocation as a Function of the Batch Size

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8.5. How Well Does the Brownian PrescriptionPerform If Batch Orders Are Possible?

Our analysis has assumed that orders always occurone unit at a time. We next explore the robustnessof our results when this assumption is violated. Wereport a numerical validation study in which weallowed for batch arrivals, using the parameters asreported at the beginning of §8 with China cost =0�5×Mexico cost. For the same arrival point processwith arrival rate � = 100 as used in §8, we increasedthe expected number of units demanded (which werefer to as the batch size) at each arrival epoch. Weused a geometric distribution to generate the actualbatch size at each epoch. As Figure 9 shows, thescaled optimal China allocation increases in the batchsize (holding total expected demand per period fixed).However, the increase is very small and leads to anerror of less than 1% in the scaled allocation. Notethat this means that the error in the total allocationwill be much smaller, especially as � increases.

9. Summary and DiscussionThe dual-sourcing literature has traditionally focusedon determining sophisticated dynamic policies thatapproach optimal performance. The main researchobjective of this paper, however, is more strategic innature: to determine the near-optimal average sourc-ing allocation. We were able to answer this questionby assuming a simpler policy that is used in practice.This tailored base-surge policy echoes a fundamen-tal tenet in strategy: it aligns the ordering patternswith the core competencies of the suppliers. The con-stant base allocation allows China to focus on costefficiency, whereas Mexico’s quick response is uti-lized only dynamically to guarantee high service. Ourmodel is a first attempt to provide some theory andquantification of this intuition of tailoring the sourc-ing strategy.

The model provides the following insights andquantification on strategic allocation. First, we presentan analytic characterization of the TBS dual-sourcingpolicy that culminates in a simple square-root for-mula. This formula specifies the near-optimal strate-gic allocation that separates stochastic demand into“base” and “surge.” Second, we determine the targetinventory level and the corresponding cost under thisnear-optimal allocation. Our formulas allow an esti-mation of working capital requirements under dualsourcing, which have been lacking in the literature.Third, we identify and value the key drivers of dualsourcing. The square-root formula suggests a classifi-cation into first- and second-order drivers, and high-lights the key role of supply and demand volatilitiesin dual sourcing. Our mode of analysis allows us togo beyond the typical assumptions of independenceand also discuss the impact of serial time correlationsas well as intra-demand/supply correlations.A numerical study demonstrates that the results are

robust and validates practice. We demonstrate robust-ness by showing that the TBS policy is near optimalin terms of total cost minimization, that the Brownianmodel provides reasonably accurate predictions ofallocation and cost, and that the square-root formularesults in a simple and useful estimate of strate-gic China allocation. The numerical study also vali-dates the consultants’ recommendation of allocatingroughly three-quarters to the slow source as a start-ing point. With more specific data, the three-quarterallocation can and should be further tailored to thespecific demand and supply characteristics using ourresults.As with every model, ours has limitations. We do

not explicitly model scale economies (such as thosearising from fixed costs) in ordering, production, orcapacity costs. Our results, however, show the pres-ence of scale economies (our expressions are nonlin-ear in the demand rate �) due to statistical economiesof scale. Our policy assumes that feedback control onChina is not feasible, which precludes the comparisonof dual sourcing with single sourcing from China. Ifone allows feedback control, this comparison followsthe same lines as our comparison with single sourc-ing from Mexico. Finally, we have focused on a singleproduct and a single market setting under centralizedcontrol. Future work should extend to multiproduct,multimarket settings under decentralized control.

10. Electronic Companion andTeaching Game

An electronic companion to this paper is available aspart of the online version that can be found at http://mansci.journal.informs.org/. We have developed anin-class game to port our academic insights to the

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Allon and Van Mieghem: Global Dual Sourcing124 Management Science 56(1), pp. 110–124, © 2010 INFORMS

classroom and practice; see Allon and Van Mieghem(2009) for a description.

AcknowledgmentsThe authors are grateful to Cort Jacoby, Ruchir Nanda, andBrian Bodendein from Deloitte Consulting, and to AchalBassamboo, Marty Lariviere, Hyo duk Shin, Jing-ShengSong, and the anonymous reviewers for detailed sugges-tions that improved the content of this paper.

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