Disruption Risk and Optimal Sourcing in Multitier …...manufacturers of automotive air conditioning systems, and the tier 2 suppliers might be semiconductor manu-facturers that provide

MANAGEMENT SCIENCEArticles in Advance, pp. 1–24ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2016.2471

© 2016 INFORMS

Disruption Risk and Optimal Sourcing inMultitier Supply Networks

Erjie AngFacebook, Menlo Park, California 94025, [email protected]

Dan A. IancuGraduate School of Business, Stanford University, Stanford, California 94305, [email protected]

Robert SwinneyFuqua School of Business, Duke University, Durham, North Carolina 27708, [email protected]

We study sourcing in a supply chain with three levels: a manufacturer, tier 1 suppliers, and tier 2 suppliersprone to disruption from, e.g., natural disasters such as earthquakes or floods. The manufacturer may not

directly dictate which tier 2 suppliers are used but may influence the sourcing decisions of tier 1 suppliersvia contract parameters. The manufacturer’s optimal strategy depends critically on the degree of overlap inthe supply chain: if tier 1 suppliers share tier 2 suppliers, resulting in a “diamond-shaped” supply chain, themanufacturer relies less on direct mitigation (procuring excess inventory and multisourcing in tier 1) andmore on indirect mitigation (inducing tier 1 suppliers to mitigate disruption risk). We also show that whilethe manufacturer always prefers less overlap, tier 1 suppliers may prefer a more overlapped supply chainand hence may strategically choose to form a diamond-shaped supply chain. This preference conflict worsensas the manufacturer’s profit margin increases, as disruptions become more severe, and as unreliable tier 2suppliers become more heterogeneous in their probability of disruption; however, penalty contracts—in which themanufacturer penalizes tier 1 suppliers for a failure to deliver ordered units—alleviate this coordination problem.

Keywords : disruption risk; multisourcing; supply chains; multiple tiersHistory : Received January 15, 2014; accepted January 17, 2016, by Yossi Aviv, operations management. Published

online in Articles in Advance July 14, 2016.

1. IntroductionIn the wake of the March 11, 2011, Tohoku earthquakeoff the eastern coast of Japan, many companies facedmassive disruptions in their supply chains. Automotivemanufacturers—particularly Toyota—were especiallyaffected, as a number of key suppliers were located inthe impacted region and were disabled as a result of acombination of earthquake damage, tsunami damage,and radiation exposure resulting from the meltdownof the Fukushima Daiichi nuclear power plant. Inaddition, some automotive suppliers that were notdirectly damaged were forced to cease productionbecause of power shortages that followed in the weeksafter the disaster. The consequences for manufacturerswere severe: Toyota faced immediate shortages on over400 parts, and production capacity was reduced for sixmonths following the disaster (Tabuchi 2011).

Prior to the disaster, Toyota had primarily concerneditself with its tier 1 (immediate) suppliers and had leftthe management of higher tiers in the supply chain toits direct partners. In most cases, Toyota did not evenknow who its tier 2 suppliers were, much less theirrisk profiles or how tier 1 suppliers managed that risk

via, for instance, sourcing decisions or holding safetystocks. As part of the recovery process, Toyota createda systematic map of the upper levels of its supply chainand the disruption management strategies of its tier 1suppliers for the first time. During this investigation, afundamental problem emerged (Greimel 2012, Masuiand Nishi 2012): much to the company’s surprise,Toyota discovered that while it might have attemptedto mitigate disruption risk by multisourcing with tier 1suppliers, many tier 1 suppliers shared some or all oftheir tier 2 suppliers, leading to a significant degree ofcorrelation in risk for supply coming from tier 1. Thisproblem manifested itself with a wide variety of tier 2suppliers, ranging from large firms with specific exper-tise, such as Renesas, an automotive semiconductormanufacturer, to smaller firms with generic capabilities,such as Fujikura, a rubber manufacturer (Masui andNishi 2012). The substantial overlap in the upper levelsof the network, which Toyota internally referred toas a “diamond-shaped” supply chain, clearly calledfor the firm to rethink its disruption management andsourcing decisions.

The findings of Toyota and other major manu-facturers quickly led firms to realize that a more

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Ang, Iancu, and Swinney: Disruption Risk and Optimal Sourcing in Multitier Supply Networks2 Management Science, Articles in Advance, pp. 1–24, © 2016 INFORMS

Figure 1 Supply Chain Overview

Tier 2 suppliersProne todisruption

Tier 1 suppliersSelect Tier 2

suppliers

ManufacturerContracts withTier 1 suppliers

comprehensive disruption management strategy,extending beyond a firm’s immediate suppliers andinto the entire supply network, is necessary to builda robust enterprise (Greimel 2012, Masui and Nishi2012, Tucker 2012). Motivated by this example andnumerous recent instances of disruption caused by nat-ural disasters (e.g., flooding in Thailand, as describedin Soble 2011), financial distress of suppliers (e.g., inthe automotive supply chain, as described in Hoffman2012), or industrial accidents (e.g., at apparel suppliersin Bangladesh, as described in Greenhouse 2012), inthis paper we consider the issue of optimal sourcingwhen the disruption risk originates in the upper levelsof a multitier supply chain. Specifically, we examine astylized model in which a downstream manufacturer(such as Toyota) sources identical critical componentsfrom tier 1 suppliers. Those tier 1 suppliers, in turn,source subassemblies or raw materials from tier 2suppliers. We assume that disruption may occur intier 2, which in turn will cause a shortage impactingtier 1 and ultimately the manufacturer; see Figure 1.

Following the classic supply chain disruptionapproach (Tomlin 2006, Aydin et al. 2010), we assumethat tier 2 suppliers have heterogeneous cost and dis-ruption risk profiles, and hence the choice of tier 2suppliers is critical in determining the risk profile ofthe supply chain. However, unlike the existing dis-ruption risk literature, in our model the manufacturercannot directly dictate that risk profile by choosingtier 2 suppliers. Instead, the manufacturer acts as asequential leader, first deciding contract terms withits tier 1 suppliers; those tier 1 suppliers then choosesourcing quantities from one or more tier 2 suppliers tomaximize their own expected profits, given the contractterms offered by the manufacturer. Consequently, themanufacturer’s contract decisions with its tier 1 suppli-ers will induce some sourcing arrangement from tier 1to tier 2, ultimately determining both the probability ofdisruption and the correlation of disruption risk for thetier 1 suppliers. Prompted by Toyota’s findings after theTohoku earthquake, we pay particular attention to theimpact of the supply chain configuration—the numberof tier 2 suppliers and the manner in which they areconnected to tier 1 suppliers—on the manufacturer’soptimal risk management strategy, focusing on threekey questions. First (in Sections 3–6), how should amanufacturer manage disruption risk when both theprobabilities and the correlations in the disruptionprofile of its immediate suppliers are endogenously

determined by the contracts that the manufactureroffers? How does the manufacturer’s optimal sourcingstrategy depend on the configuration of the supplynetwork? Second (in Section 7), if tier 1 suppliers werecapable of choosing the supply chain configuration,would they choose the optimal configuration from themanufacturer’s point of view? And third (in Section 8),if not, does a simple mechanism exist to align thosepreferences and coordinate the supply chain?

2. Literature ReviewThe existing disruption management literature hasextensively explored mitigation strategies employ-able by a firm—primarily safety stocks and supplierdiversification—to manage the risk of random supplyat its own facilities or from its immediate tier 1 sup-pliers. Examples include Anupindi and Akella (1993),Yano and Lee (1995), Elmaghraby (2000), Minner (2003),Tomlin (2006), Chopra et al. (2007), Dada et al. (2007),Federgruen and Yang (2008), Tomlin (2009), Yang et al.(2009), Aydin et al. (2010), Tomlin and Wang (2011),Hu and Kostamis (2014), Wadecki et al. (2012), andmany others. As we consider the issue of endogenouscorrelation in the supply network, our paper is par-ticularly related to models with correlated risk andendogenous risk.

From the former category (correlated risk), Babichet al. (2007) find that a monopolistic firm might preferpositively correlated supplier defaults, as this increasesprice competition among its suppliers, while Tangand Kouvelis (2011) show that firms that competein a common consumer market and share a supplierbase will earn lower profits when supplier defaults arehighly correlated. Our model differs from these in that,instead of focusing on the tension between competitionand diversification in a firm’s tier 1 supplier base, weintroduce supply correlation between tier 1 suppliersthrough their common suppliers in tier 2. In thiscase, the firm can use diversification as a strategy toinduce its tier 1 suppliers to source from different tier 2suppliers.

From the latter category (endogenous risk), Swinneyand Netessine (2009) consider how to contract witha supplier when that supplier’s (financial) defaultrisk depends on the contract price. Wang et al. (2010)explore how a firm can invest to improve the reliabilityof its suppliers and how this can be used alongsidedual sourcing as a two-pronged strategy against yield

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Ang, Iancu, and Swinney: Disruption Risk and Optimal Sourcing in Multitier Supply NetworksManagement Science, Articles in Advance, pp. 1–24, © 2016 INFORMS 3

uncertainty. Babich (2010) considers how financial sub-sidies can be used to mitigate the risk of supplierdefault. Hwang et al. (2014) explore how contracts anddelegation can be employed to induce an immediatesupplier to improve its reliability. In contrast to thesemodels, in which risk is determined by firm actionsor financial subsidies, we consider how the choiceof tier 1 and tier 2 suppliers impacts the shape, andsubsequently the risk, of the supply network. Bimpikiset al. (2013) also consider how the endogenous choiceof risky tier 2 suppliers impacts the supply chain;however, they focus on how nonconvexities of theproduction function affect supply chain risk, whereaswe focus on how supplier asymmetries and accessto different sets of tier 2 suppliers can influence theoptimal sourcing strategy of the downstream manu-facturer. To summarize, our model is, to the best ofour knowledge, among the first to explore disruptionrisk management in supply chains with more than twotiers and the first to consider how supplier selectionand contracting impacts the shape and disruption risklevel of the supply chain.

3. ModelWe analyze a supply chain consisting of three tiers:tier 0, tier 1, and tier 2. Tier 0 contains a single firm, the“manufacturer,” which assembles finished goods andsells to a consumer market. The manufacturer sourcesa critical component from tier 1, which consists oftwo identical suppliers that provide fully substitutableproducts. Each tier 1 supplier, in turn, sources a criticalintermediate component from tier 2. Using an examplefrom our introduction, the manufacturer in this supplychain might be Toyota, the tier 1 suppliers might bemanufacturers of automotive air conditioning systems,and the tier 2 suppliers might be semiconductor manu-facturers that provide the electronics used in the airconditioning units.

3.1. Tier 2Disruption in the supply chain originates in tier 2.Following Tomlin (2006) and others, we assume thereare two “classes” of tier 2 suppliers: reliable sup-pliers (denoted by the subscript r), which alwaysdeliver the requested quantity, and unreliable suppliers(denoted by u) that are prone to disruption but areless costly than reliable suppliers. This may be thecase if, for example, some suppliers are located in anearthquake-prone region (e.g., Japan) that is close tothe downstream manufacturing facilities, while othersuppliers are located in more geologically stable butdistant regions, requiring greater transportation costs(e.g., North America, Europe). We assume that reliablesuppliers have infinite capacity, while unreliable sup-pliers have random capacity that follows a two-pointdisruption distribution: with probability 1 −�j , there is

no disruption and effective capacity is infinite, whereaswith probability �j , a disruption occurs, and productioncapacity is reduced to some finite K ≥ 0 (Yano and Lee1995, Aydin et al. 2010). There are at most two unreli-able suppliers, labeled j ∈ 81129, and these suppliersmay be heterogeneous in their disruption probability.Without loss of generality, we assume that �1 ≤ �2,i.e., unreliable supplier 1 is less likely to experiencea disruption than is supplier 2. Disruptions at theunreliable suppliers are independent events.

The tier 2 sourcing cost is exogenous (e.g., becausesuppliers at this level tend to offer more commoditizedgoods, with prices determined by the market), and theper-unit sourcing cost from a reliable supplier is cr ,while the sourcing cost from an unreliable supplieris cu, where cr > cu. Note that we assume that unreliabletier 2 suppliers, while potentially heterogeneous intheir disruption probability, have identical sourcingcosts. We make this assumption for two reasons. First,it is reflective of our motivating example of Toyota’ssupply chain. In this case, the “unreliable” suppliersare, for the most part, domestic Japanese suppliers whomay face slightly different levels of risk based on theirlocation within Japan (e.g., suppliers located in thecoastal area of the Tohoku region may face a differentchance of disruption than suppliers in the interior ofthe Kansai region) but, because of intense competitionin the domestic marketplace and similar transportationcosts, sell at similar prices to tier 1. It also includes,as a special case, identical (but independent) unreli-able tier 2 suppliers. Second, this assumption enablestractable analysis of our problem, as simultaneousheterogeneity in cost and disruption risk increases thecomplexity of the model substantially; in the supple-mental appendix (available as supplemental material athttp://dx.doi.org/10.1287/mnsc.2016.2471), we numeri-cally analyze the general case of heterogeneous risk andcost, and we find that our key insights continue to hold.Aside from their reliability in delivering productionorders and their sourcing cost, the tier 2 suppliersare otherwise identical, i.e., they offer products ofidentical quality that are fully substitutable, and alltier 2 suppliers have sufficient capacity to fulfill anydownstream order if they are not disrupted.

3.2. Tier 1The two suppliers in tier 1, labeled i ∈ 8A1B9, afterreceiving a contract from the manufacturer (describedbelow), select which tier 2 suppliers to source from withthe goal of maximizing their expected profit. If a tier 1supplier selects a tier 2 supplier that disrupts, thenall parts from that supplier in excess of the disruptedcapacity K are lost, and the tier 1 supplier potentiallyfaces a shortage of components, which may in turnlead to a failure to deliver product to the manufacturer.We assume that both tier 1 suppliers have access to

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Figure 2 The Two Possible Supply Chain Configurations

U U UTier 2

Tier 1

Tier 0

shaped Diamond shaped

Notes. Only unreliable (U) tier 2 suppliers are pictured. Both tier 1 suppliersalso have access to a perfectly reliable tier 2 supplier.

one reliable and one unreliable tier 2 supplier. Tier 1suppliers may choose to single source from eithersupplier type or to dual source, with an arbitraryquantity split between the suppliers.

Motivated by Toyota’s discovery of substantial over-lap in the upper levels of its supply chain, we assumethat there are two possible supply chain configurations,as shown in Figure 2: either unreliable tier 2 suppliersare independent, which we refer to as a V -shapedconfiguration, or the unreliable tier 2 supplier (eithersupplier 1 or 2) is shared by the tier 1 suppliers, whichwe refer to as a diamond-shaped configuration.1 In theV -shaped configuration, the risk of disruption from theunreliable tier 2 suppliers is assumed to be statisticallyindependent. In the diamond-shaped configuration, ifthe common unreliable tier 2 supplier disrupts, bothtier 1 suppliers are impacted by reduced capacity at thedisrupted supplier. There is no possibility of disruptionoriginating in tiers 0 and 1.

For the initial part of our analysis, the supply chainconfiguration is assumed to be exogenously speci-fied. In practice, this is frequently the case: given thehigh cost and long-term nature of establishing a listof approved suppliers, tier 1 suppliers operate overthe short term with an effectively fixed set of tier 2suppliers, around which the manufacturer optimizes itsday-to-day procurement decisions. In addition, tier 2suppliers are often chosen for reasons outside of ourmodel, such as capability or quality, implying a keyproblem of how to optimally manage disruption riskgiven a fixed supply chain configuration. The degree ofconcentration of the tier 2 supplier base may also deter-mine whether the supply chain is likely to end up withshared or independent tier 2 suppliers; if there are onlytwo tier 2 suppliers in the world that manufacture the

1 Since the reliable tier 2 suppliers are perfectly reliable, it is irrelevantwhether they are shared between the tier 1 suppliers, and the keydifferentiating feature is whether the unreliable tier 2 suppliers areshared; hence, in the figure, we suppress the reliable tier 2 supplierof each tier 1 supplier.

component, one reliable and one unreliable, then thetier 2 supply base is very “concentrated,” and the tier 1suppliers necessarily share those two tier 2 suppliers,resulting in a diamond-shaped configuration. This maybe the case for highly specialized tier 2 suppliers, suchas semiconductor manufacturers like Renesas. Con-versely, this may be driven by industry trends outsideof our model: for example, Sasaki (2013) describes howconsolidation in the Japanese automotive supply chainstemming from cost pressure throughout the 1990s and2000s has generally lead to a much more concentratedsupply base than in the past. By contrast, if there aremany tier 2 suppliers, and the likelihood that the tier 1suppliers select identical tier 2 suppliers is negligible,then the tier 2 supply base is very “diffuse,” resultingin a V -shaped configuration; this can occur particularlywith tier 2 suppliers that offer commoditized productswith generic capabilities, such as rubber componentmanufacturers like Fujikura. While understanding themanufacturer’s optimal strategy under a fixed supplychain configuration is important, an equally interestingquestion is how the supply chain configuration isformed in the first place, i.e., as tier 1 suppliers selecttier 2 suppliers. Hence, once we have determined theoptimal actions of the manufacturer and tier 1 suppliersgiven a fixed supply chain configuration, we considerthe longer-term strategic issue of how the supply chainconfiguration is formed in Section 7.

3.3. The ManufacturerThe manufacturer faces deterministic market demand D(Zimmer 2002, Babich et al. 2012) and makes fixed per-unit revenue of � for every unit it sells up to D.2 Weassume that � ≥ cu, so that it is possible for the supplychain to profitably sell the product; all manufacturercosts outside of procurement from tier 1 suppliers arenormalized to zero. The sequence of events is as follows.In the first period, the manufacturer offers its tier 1suppliers A and B a price and quantity contract of theform 4pi1Qi5, i ∈ 8A1B9. Such contracts are commonlyobserved in practice and in the literature, and theystipulate that the manufacturer pays the supplier pi forevery unit delivered up to the specified amount Qi.The manufacturer makes no payment for ordered unitsthat are not delivered. We assume that the choiceof tier 2 suppliers is not directly contractible. Evenmanufacturers as powerful as Toyota are typicallyunable to unilaterally dictate supplier choice; afterthe 2011 earthquake, approximately 50% of Toyota’stier 1 suppliers refused to even share the identitiesof their tier 2 suppliers, let alone allow Toyota tounilaterally dictate those suppliers, citing competitive

2 Assuming deterministic demand allows us to focus solely onthe element of primary interest: the role of the supply networkstructure in determining sourcing and contracting decisions. Inaddition, incorporating stochastic demands significantly increasesmodel complexity (see, e.g., Tomlin 2006).

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reasons (Masui and Nishi 2012). In addition, whileother tools may be available to influence the choice oftier 2 suppliers by tier 1 suppliers, such as manufacturerinvolvement in the development of the tier 2 supplybase, these tools are longer-term mechanisms, leavingprocurement contracts as the most obvious immediateshort- and medium-term tool available to influence thesourcing decisions of tier 1 suppliers.

After receiving the manufacturer’s contract, eachtier 1 supplier then places orders qiu and qir with its tier 2unreliable and reliable suppliers, to maximize expectedprofit. We call the pair 4qiu1 q

ir5 the sourcing strategy

of tier 1 supplier i. In the second period, the tier 2suppliers either disrupt or fully fulfill their deliveriesto the tier 1 suppliers. After collecting all deliveriesfrom tier 2 suppliers, tier 1 suppliers then attempt tofulfill the order submitted by the manufacturer. Anyexcess inventory delivered by tier 2 suppliers to tier 1suppliers has zero value. In turn, after collecting deliv-eries from tier 1 suppliers, the manufacturer attemptsto fulfill consumer demand D, and if supply is lessthan demand, lost sales occur. Any excess compo-nents delivered by tier 1 suppliers to the manufacturerhave zero value. The manufacturer seeks to maximizeits expected profit from the consumer market at theend of the second period, and its decision variablesare the price and quantity in the contracts it givesto its tier 1 suppliers, 4pA1QA5 and 4pB1QB5; hence,we define the manufacturer’s sourcing strategy to be84pA1QA51 4pB1QB59, the pair of contract offers to itstwo tier 1 suppliers. We assume that the disruptionprobabilities, sourcing costs, and supply chain config-uration are known to all parties in the supply chain:there is no private information in the model.

4. Optimal Sourcing in a V-ShapedSupply Chain

4.1. Tier 1’s Sourcing DecisionWe first examine the manufacturer’s optimal sourcingstrategy when facing a supply chain with independenttier 2 suppliers. This resembles the type of V -shapedsupply network that Toyota (mistakenly) believed itpossessed prior to the Tohoku earthquake and, assuch, will serve as a benchmark for understandinghow increased overlap in tier 2 impacts the manu-facturer’s sourcing strategy. We begin our analysisby characterizing the optimal sourcing strategy foreach tier 1 supplier i ∈ 8A1B9 with access to unreliabletier 2 supplier j ∈ 81129, in response to a contractualoffer 4pi1Qi5 from the manufacturer. Given a sourcingstrategy 4qiu1 q

ir 5, the expected profit of supplier i is

çi4qiu1qir 5

=�j 6pimin8qir +min8K1qiu91Qi9−crqir −cumin8K1qiu97

+41−�j56pimin8qiu+qir1Qi9−cuqiu−crq

ir 70 (1)

Let the superscript asterisk denote the optimal sourcingstrategy. Optimizing the tier 1 supplier’s expected profitover the quantity sourced from each tier 2 supplierleads to the following result.

Theorem 1. In a V -shaped supply chain, the optimalsourcing strategy of tier 1 supplier i ∈ 8A1B9 with access tounreliable tier 2 supplier j ∈ 81129 is

4qiu1 qir 5

∗=

4Qi105 if cu ≤ pi < cu +cr − cu�j

1

4K1Qi −K5 if pi ≥ cu +cr − cu�j

0

Proof. All proofs appear in the appendix. �Observe that it is never optimal for the supplier to

single source from the reliable tier 2 supplier. Thisis because the supplier may completely eliminatedisruption risk by sourcing only the disrupted output Kfrom the unreliable supplier and sourcing the remainder,Qi −K, from the reliable supplier, achieving the sameoutcome (at a lower cost) as sourcing everything fromthe reliable supplier. Moreover, sourcing inventory inexcess of the manufacturer’s contracted quantity Qi isnever optimal, because disruptions are manifested asrandom supplier capacity (as opposed to random yield;see Hwang et al. 2014), combined with the presence of aperfectly reliable tier 2 supplier. This implies that a tier 1supplier effectively chooses from two possible sourcingstrategies: sourcing from the unreliable supplier orsourcing from both suppliers. The former strategy isalso known as passive acceptance (i.e., paying a lowcost and merely accepting the risk of disruption), andthe latter strategy is known as dual sourcing (Tomlin2006). Intuitively, for low contract prices pi, the supplierfollows a passive acceptance strategy and single sourcesfrom the unreliable supplier, while for high prices, thesupplier employs dual sourcing to completely eliminatethe risk of disruption.3 The price necessary to inducethe supplier to dual source is greater than the marginalcost of the reliable supplier, an agency cost arising fromthe moral hazard feature inherent in a decentralizedsupply chain.

4.2. The Manufacturer’s Optimal Sourcing StrategyHaving derived the optimal sourcing strategy of atier 1 supplier in response to a contract 4pi1Qi5 fromthe manufacturer, we may now determine the optimal

3 In fact, when pi = cu + 4cr − cu5/�j , the tier 1 supplier is indifferentbetween single sourcing from the unreliable supplier and dualsourcing from both suppliers; in this boundary case, any convexcombination of the two strategies is equivalent. Throughout thepaper, for expositional convenience, we will assume that when inany such boundary case, the sourcing strategy with the maximumamount of inventory from the reliable supplier—the “safest” sourcingstrategy—is chosen. The manufacturer clearly prefers this outcometo a passive acceptance strategy, and indeed, this outcome could beachieved if the manufacturer merely raises the price an infinitesimalamount above cu + 4cr − cu5/�j .

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contracts offered by the manufacturer to the two tier 1suppliers. Without loss of generality, we assume thattier 1 supplier A has access to unreliable tier 2 sup-plier 1 (the lower-risk supplier), while tier 1 supplier Bhas access to unreliable tier 2 supplier 2 (the higher-risksupplier). In what follows, we use the term “supplychain structure” to refer to an induced set of sourcingquantities between tiers 0, 1, and 2, given a partic-ular supply chain configuration (either V -shaped ordiamond-shaped). In theory, the manufacturer couldinduce four sourcing strategies at each tier 1 supplier(no participation, single sourcing from either tier 2supplier, or dual sourcing from both tier 2 suppliers),implying a total of 16 possible induced structures.However, from Theorem 1, the manufacturer can onlyfeasibly induce three strategies for each of its tier 1suppliers: no participation, single sourcing from theunreliable supplier (by offering a low price), or dualsourcing (by offering a high price). This implies that,via its contractual offers to tier 1 suppliers, the manu-facturer is capable of generating a total of five uniquesupply chain structures (excluding the case of no pro-duction and all symmetric cases). We denote these asstructures 1–5, and we summarize them in Figure 3.

To determine the manufacturer’s optimal sourcingstrategy, we must consider the performance of eachof these possible induced structures. In any structure,the manufacturer’s expected profit as a function of itssourcing strategy 84pA1QA51 4pB1QB59 is

çM(

84pA1QA51 4pB1QB59)

= f14pA1 pB5[

� min8QA +QB1D9− pAQA − pBQB

]

+ f24pA1 pB5[

� min8min8K1QA9+QB1D9

− pA min8K1QA9− pBQB

]

+ f34pA1 pB5[

� min8QA + min8K1QB91D9

− pAQA − pB min8K1QB9]

+ f44pA1 pB5[

� min8min8K1QA9+ min8K1QB91D9

− pA min8K1QA9− pB min8K1QB9]

0 (2)

In the above equation, f1, f2, f3, and f4 denote theprobability that the full orders are fulfilled by both

Figure 3 The Five Distinct Structures That the Manufacturer Can Induce in a V-Shaped Supply Chain

U U U U U U UR R R U RTier 2

Tier 1

Tier 0

1 2 3 4 5

Note. R, reliable; U, unreliable.

tier 1 suppliers, only supplier B, only supplier A, orneither tier 1 supplier, respectively. Note that this issimilar in form to the tier 1 supplier’s profit function inEquation (1), except that (2) allows for the disruption ofboth immediate suppliers, and the probability terms in(2) are endogenously determined by the manufacturer’soffered prices.

Each structure in Figure 3 corresponds to a partic-ular set of prices offered to tier 1 suppliers, and thisin turn determines the disruption probabilities f1–f4.For instance, structure 4 may be induced by offeringsupplier A a high price and supplier B a low price,and the resultant probabilities are f1 = 1 −�2, f2 = 0,f3 = �2, and f4 = 0, because only supplier B may fail todeliver the manufacturer’s order, as supplier A dualsources and achieves perfect reliability. To determinethe manufacturer’s optimal strategy, we must find theoptimal sourcing quantities in each structure giventhe prices and probabilities associated with the struc-tures and then compare the optimal expected profitacross the five structures. For the sake of brevity, werelegate the derivation of these profit expressions toLemma 1 in the appendix, and we focus our subse-quent discussion on the outcome of this analysis: thedetermination of the manufacturer’s optimal sourcingstrategy and induced supply chain structure. We beginby considering the case of minor disruptions.

Theorem 2. In a V -shaped supply chain, if disruptionsare minor (K ≥D/2), the manufacturer’s optimal sourcingstrategy is 84pA1QA51 4pB1QB59

∗ = 84cu1 �D51 4cu1 41−�5D59for any � ∈ 64D−K5/D1K/D7.

The theorem shows if that the disrupted capacityis high (i.e., if sufficient disrupted capacity existsacross the entire unreliable tier 2 supply base forthe manufacturer to cover all of its demand), themanufacturer should induce structure 3 in Figure 3:single sourcing from the unreliable tier 2 suppliersby both tier 1 suppliers. As a result, tier 1 suppliersdo nothing to mitigate disruption risk in the supplychain, and all risk mitigation is pursued directly bythe manufacturer, who is able to completely eliminate

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risk via its own sourcing actions. We thus refer tothis approach as a manufacturer dual sourcing (DS)strategy—the manufacturer mitigates risk by dualsourcing, while the tier 1 suppliers employ passiveacceptance, by single sourcing from a risky tier 2supplier. The only restriction on the manufacturer’ssourcing strategy is that no more than K units shouldbe sourced from each tier 1 supplier, to ensure thattier 2 suppliers are always able to deliver in the eventof a disruption; this determines the feasible range of �,the division of quantities between tier 1 suppliers,which the manufacturer can employ.

Next, we consider the case when disruptions aresevere, meaning disrupted capacity is small. We firstdefine the following cost threshold for the reliablesupplier:

cr ≡ cu + cu41 −�15�2D− 2KD−K

0 (3)

Observe that this cost threshold varies dependingon the severity and likelihood of disruptions, and itlies between cu and 2cu if K <D/2. As the followingtheorem shows, the manufacturer’s optimal sourcingstrategy depends critically on this threshold.

Theorem 3. Define �L ≡ 44cr − cu5/�1�2544D−K5/4D− 2K55 + cu and �H ≡ 44cr − cu5/�1�

22544D−K5/

4D− 2K55− cu441 −�1 −�25/�1�25. In a V -shaped supplychain, if disruptions are severe (K < D/2), we have thefollowing:

(i) If cr ≤ cr , the manufacturer’s optimal sourcing strat-egy is

84pA1QA514pB1QB59∗

=

84cu1D−K514cu1K59 if cu≤�<�L1

{

4cu1K51

(

cu+cr −cu�2

1D−K

)}

if �≥ �L0

(ii) If cr > cr , the manufacturer’s optimal sourcing strat-egy is

84pA1QA514pB1QB59∗

=

84cu1D−K514cu1K59 if cu≤�<cu�1

1

84cu1D−K514cu1D−K59 ifcu�1

≤�<�H1

{

4cu1K51

(

cu+cr −cu�2

1D−K

)}

if �≥ �H 0

The theorem shows that for small unit revenues (�),regardless of the cost of the reliable tier 2 suppliers, themanufacturer’s approach is similar to the case when themagnitude of disruptions is small: the optimal strategyis to induce both tier 1 suppliers to single source from

their unreliable tier 2 suppliers. The manufacturerprefers to source the majority of its order from tier 1supplier A, who has access to the less risky tier 2supplier, and places a smaller order with supplier B,who has access to the riskier tier 2 supplier. Hence,manufacturer dual sourcing (DS) is optimal for lowunit revenues. As the unit revenue increases, the costof reliable tier 2 suppliers begins to play a role. Ifthey are inexpensive (cr ≤ cr ), the manufacturer movesfrom DS to inducing one tier 1 supplier to dual sourceby paying a high price, while paying the other tier 1supplier a low price to induce single sourcing. We callthis a manufacturer dual sourcing + supplier mitigation(DS + SM) strategy, since it combines dual sourcingin tier 1 with induced supplier mitigation via dualsourcing in tier 2. Note that this strategy completelyeliminates disruption risk for the manufacturer, but themanufacturer must pay a higher price to achieve thisoutcome.

If, on the other hand, reliable tier 2 suppliers areexpensive (cr > cr ), at intermediate unit revenues, themanufacturer continues to pay tier 1 suppliers a lowprice but increases the quantity sourced from eachtier 1 supplier to D−K, meaning the total amount ofinventory procured exceeds demand. Tomlin (2006)refers to this strategy of excess procurement—in essence,holding safety stock—as “inventory mitigation.” Hence,when reliable tier 2 suppliers are costly, the optimalsourcing strategy switches from DS to manufacturerdual sourcing + inventory mitigation (DS + IM), while stillinducing tier 1 suppliers to follow a passive acceptancestrategy by paying a low unit price. This strategyresults in the manufacturer having more inventorythan is necessary to satisfy demand if no disruptionoccurs; however, because the reliable tier 2 suppliersare expensive, the excess inventory cost is preferred topaying a high marginal procurement cost for reliablesupply. Manufacturer efforts alone, however, cannotcompletely eliminate the risk of disruption, and evenwith the DS+ IM strategy, there is still a chance that themanufacturer will have insufficient inventory to satisfydemand (i.e., if both tier 2 unreliable suppliers disrupt).Hence, as the unit revenue continues to increase, themanufacturer eventually switches to a DS+SM strategy,which eliminates disruption risk by engaging tier 1suppliers in mitigation efforts. The optimal strategieswhen disruptions are severe are summarized in Table 1.

Theorem 3 illustrates that when disruptions aresevere, the manufacturer’s optimal strategy builds riskmitigation from the bottom of the supply chain up. Putdifferently, it is optimal to begin with manufacturerdual sourcing and, as the unit revenue increases, addmanufacturer inventory mitigation (if reliable tier 2suppliers are costly) and finally supplier mitigation.Consequently, for small and moderate unit revenues,tier 1 is not involved with disruption risk mitigation:

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Table 1 Optimal Sourcing Strategies in a V-Shaped Supply Chain forSevere Disruptions

Sourcing Use Quantity Tier 2strategy when 0 0 0 sourced suppliers used

Manufacturer � low D 2 Udual sourcing (DS)

DS + Inventory � medium + cr high 24D −K5 2 Umitigation (DS + IM)

DS + Supplier � high D 2 U and 1 Rmitigation (DS + SM)


the manufacturer bears all such responsibility. For largeunit revenues, however, it is optimal to engage tier 1 indisruption risk mitigation efforts, by paying a higherprice and inducing supplier mitigation. ComparingTheorems 2 and 3, we see that the manufacturer’soptimal sourcing strategy is qualitatively different forminor and severe disruptions. For minor disruptions,the manufacturer bears all mitigation responsibility,following a strategy of pure manufacturer mitigationimplemented via dual sourcing. For severe disruptions,the manufacturer may also employ inventory mitiga-tion (at moderate unit revenues if reliable suppliersare costly) or induce supplier mitigation (at high unitrevenues), sharing the burden of disruption risk mit-igation with its suppliers. In addition, note that themanufacturer always selects tier 1 supplier B, whohas access to the riskier tier 2 supplier, to engage insupplier mitigation. This is because the price necessaryto induce supplier B to dual source is lower than theprice needed to induce supplier A to dual source, asindicated in Theorem 1. In other words, because sup-plier B faces a “riskier” unreliable supplier option, theagency costs incurred by the manufacturer are lowerwhen inducing sourcing from the reliable supplier, andhence the manufacturer always prefers to engage insupplier mitigation with supplier B. This fact will havesignificant consequences in the supply chain gameanalyzed in Section 7.

After the 2011 Tohoku earthquake, a key determina-tion of Toyota executives was that they had placedtoo much emphasis on direct mitigation efforts andtoo little on supplier mitigation. Given that our modelshows inducing supplier mitigation is indeed optimalfor high unit revenues and severe disruptions, a naturalquestion is, precisely how much does the manufacturersuffer when using the wrong strategy? Our final resultin this section answers this question, by comparingmanufacturer profit under direct dual sourcing (withor without inventory mitigation) to the profit under theoptimal sourcing strategy (using supplier mitigation)for large unit revenues.

Corollary 1. Let çM 4X5 be the manufacturer’s ex-pected profit under sourcing strategy X ∈ 8DS1DS + IM1

DS + SM9, and define L4X5≡ 1 −çM 4X5/çM 4DS + SM5to be the percentage loss in expected profit of strategy Xrelative to a DS + SM strategy. In a V -shaped supply chainand under severe disruptions, we have the following:

(i) The percentage profit loss from employing manufacturerdual sourcing is L4DS5≤ �141 − 2K/D5.

(ii) The percentage profit loss from employing manufac-turer dual sourcing and inventory mitigation is L4DS +

IM5≤ �1�241 − 2K/D5.

For example, if the probability of disruption in tier 2is 10%, the result shows that using an “incorrect” strat-egy that ignores inducements to tier 1 suppliers andsolely uses manufacturer dual sourcing results in atmost a 10% loss in profit compared with the optimalstrategy. By adding inventory mitigation, the manufac-turer’s profit is substantially improved, achieving aloss of at most 1% compared with the optimal profit.Note that the loss under either strategy depends simi-larly (i.e., linearly) on the severity of the disruption, asmeasured by the demand fill rate achievable when allunreliable tier 2 suppliers disrupt (i.e., 2K/D). As onewould expect, the performance of an incorrect strategybecomes increasingly poor as disruptions become verylikely or very severe (i.e., high � or low K), and allstrategies perform equally well when disruptions arehighly unlikely or not impactful (i.e., � close to zeroor K close to D/2). Importantly, however, the resultshows that in a V -shaped supply chain, manufacturerperformance is substantially worse under a stand-alonedual sourcing strategy than under a dual sourcingand inventory mitigation strategy. Thus, focusing ondual sourcing alone in tier 1, as Toyota and manyother lean manufacturers did prior to the Tohokuearthquake, is potentially quite costly. This impliesthat, if the manufacturer cannot engage suppliers indisruption mitigation efforts when such efforts areoptimal, a strategy that performs well is to employmanufacturer dual sourcing and inventory mitigation.While this cannot achieve the optimal profit earnedunder supplier mitigation, it results in an order ofmagnitude smaller loss than using dual sourcing alone.Indeed, Toyota has begun to implement inventorymitigation in instances where suppliers either refuseto engage in mitigation or find mitigation impractical(Greimel 2012, Masui and Nishi 2012).

5. Optimal Sourcing in aDiamond-Shaped Supply Chain

5.1. Tier 1’s Sourcing DecisionWe now consider the case of a shared unreliable tier 2supplier and the resulting “diamond”-shaped supplychain. Observe that whenever the tier 1 suppliers Aand B are induced to source from their common unre-liable tier 2 supplier, they potentially compete for

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limited capacity, i.e., when they respectively order qAu

and qBu units, and the disrupted output is K < qA

u + qBu .4

Hence, we must discuss how scarce capacity in tier 2 isallocated in the event of a disruption. While there aremany possible allocations, we select the uniform rule,which has the desirable property of simultaneouslyminimizing shortage gaming and maximizing totalsupply chain profit in a number of settings (Sprumont1991, Cachon and Lariviere 1999). A uniform allocationsplits the disrupted capacity of the unreliable tier 2supplier equally between tier 1 suppliers, unless oneof the tier 1 suppliers requested less than half thedisrupted capacity, in which case the other tier 1 sup-plier is allotted all unused capacity. In other words,supplier i’s allocation in the event of a disruption ismin8qiu1 Ki9, where Ki ≡K/2 + 4K/2 − q−i

u 5+ is the effec-tive disrupted capacity. Under this allocation rule, theunique Nash equilibrium of the shortage game betweentier 1 suppliers is, essentially, for both suppliers tooperate as if they have a dedicated tier 2 unreliablesupplier with exogenously fixed capacity equal to Ki

(see the appendix for details), leading to the followingresult.

Theorem 4. In a diamond-shaped supply chain, theoptimal sourcing strategy of tier 1 supplier i ∈ 8A1B9 withaccess to unreliable tier 2 supplier j ∈ 81129 is

4qiu1 qir 5

∗=

4Qi105 if cu ≤ pi < cu +cr − cu�j

1

4Ki1Qi − Ki5 if pi ≥ cu +cr − cu�j

0

Note that this is nearly identical to the suppliers’optimal strategy in a V -shaped supply chain derivedin Theorem 1, except for the disrupted capacity Ki;in particular, the manufacturer may still only inducesingle sourcing from the unreliable tier 2 supplier ordual sourcing from both tier 2 suppliers, and the pricesnecessary to achieve each outcome are identical to thecase with independent tier 2 suppliers.

5.2. The Manufacturer’s Optimal Sourcing StrategyAs a result of Theorem 4, there are once again fivepossible unique structures that the manufacturer caninduce, graphically depicted in Figure 4. We derive the

4 Note that we assume the unreliable tier 2 supplier, when disrupted,has capacity K, which is the same as the disrupted capacity of eachunreliable supplier in the V -shaped supply chain. In practice, thismay or may not be the case. For example, if the diamond-shapedsupply chain arises from consolidation of unreliable tier 2 suppliers,we might expect that the total capacity (and hence the disruptedcapacity) of the shared unreliable tier 2 supplier in a diamond-shapedsupply chain is larger than that of each independent unreliable tier 2supplier in a V -shaped supply chain. Our key results extend to thecase of arbitrary disrupted capacity in the diamond-shaped supplychain, provided that capacity is no more than 2K.

manufacturer’s optimal sourcing strategy by followingthe same procedure as in Section 4, beginning with thecase of minor disruptions.

Theorem 5. If tier 2 suppliers are shared and disruptionsare minor (K ≥D), the manufacturer’s optimal sourcingstrategy is 84pA1QA51 4pB1QB59

∗ = 84cu1 �D51 4cu1 41−�5D59for any � ∈ 60117.

This is similar to the optimal sourcing strategy forminor disruptions with independent tier 2 suppliers(Theorem 2), with two key differences. First, withshared tier 2 suppliers, what qualifies as a “minor”disruption is a greater capacity than with independenttier 2 suppliers; in fact, the disrupted capacity must begreater than the manufacturer’s demand to achieve thiscase, implying disruptions are effectively meaningless.Second, with shared tier 2 suppliers, any division ofsourcing quantities between tier 1 suppliers will resultin the same manufacturer profit, including single sourc-ing. Thus, while minor disruptions can be completelyovercome by the manufacturer with either independentor shared tier 2 suppliers, dual sourcing with a specificdivision of sourced quantities is necessary in the formercase, while single sourcing suffices in the latter. As aresult, we say that a manufacturer single sourcing (SS)strategy is optimal in this case. Moving next to thecase of severe disruptions, we have the following.

Theorem 6. Define �j ≡ cu+ 44cr −cu5/�2j 544D−K/25/

4D−K55. In a diamond-shaped supply chain with sharedunreliable supplier j ∈ 81129, if disruptions are severe(K <D), the manufacturer’s optimal sourcing strategy is

84pA1QA51 4pB1QB59∗

=

84cu1 �D51 4cu1 41 − �5D59 for any � ∈ 60117

if cu ≤� < �j1

{(

cu +cr − cu�j

1D−K/2)

1 4cu1K/25}

or{

4cu1K/251(

cu +cr − cu�j

1D−K/2)}

if � ≥ �j 0

Note that, once again, while dual sourcing is anoptimal strategy at low unit revenues, single sourcingperforms just as well. In other words, in a diamond-shaped supply chain, manufacturer dual sourcing hasno explicit value at low margins—regardless of theseverity of disruptions—because tier 1 suppliers haveaccess to identical tier 2 suppliers. There is no tier 2supplier that supplier A can access that supplier Bcannot; hence, there is no compelling reason to dual

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Figure 4 The Five Distinct Structures That the Manufacturer Can Induce in a Diamond-Shaped Supply Chain

Tier 2

Tier 1

Tier 0

U U U UR RU R

1 2 3 4 5


source in tier 1. In particular, if there are economies ofscale with tier 1 suppliers or other reasons to valuesingle sourcing, the manufacturer may wish to adopt asourcing strategy employing just one tier 1 supplier.Moreover, when the unreliable tier 2 supplier is shared,inventory mitigation—sourcing more from tier 1 thantotal demand D—is also not valuable, since this strat-egy is only beneficial if it is possible that one tier 1supplier delivers while the other does not. Hence, themanufacturer never needs to adopt direct disruptionmitigation efforts in a diamond-shaped supply chainat low unit revenues.5 At high unit revenues, how-ever, the manufacturer does find dual sourcing to beoptimal, as it provides a way to lower the averageunit procurement cost when supplier mitigation is alsoemployed.

Our final result in this section considers the loss inprofit that the manufacturer suffers from failing to usesupplier mitigation.

Corollary 2. Let L4X5 be as defined in Corollary 1. Ina diamond-shaped supply chain with shared unreliable tier 2supplier j and under severe disruptions, the percentage profitloss from employing single sourcing is L4SS5≤ �j

(

1−K/D)

.

As with the case of a V -shaped supply chain, theprofit loss depends on the likelihood as well as theseverity of the disruption. Continuing with the sameexample as before, if there is a 10% chance of disruption,the worst-case profit loss from ignoring supplier mitiga-tion is 10% of the optimal profit. However, unlike theV -shaped supply chain case, with a diamond-shapedsupply chain, there is no intermediate strategy thatperforms well, i.e., achieving the �1�2 profit loss thatdual sourcing and inventory mitigation would yieldwith independent suppliers. This once again illustratesthe critical importance of inducing tier 1 suppliers tomitigate disruption when there is overlap in tier 2.

5 If disruption can occur in tier 1, direct manufacturer mitigation—including dual sourcing and inventory mitigation—will likely havesome value. Nevertheless, the qualitative impact of increased overlapin tier 2 reducing the value of manufacturer mitigation shouldremain even if disruption can occur in tier 1.

6. Comparing Supply ChainConfigurations

We now compare the V -shaped and diamond-shapedconfigurations to see how the manufacturer is impactedby overlap in the upper tiers of the network.6 First,we note that by comparing Theorems 3 and 6, it canbe seen that increased overlap in tier 2 decreases themanufacturer’s reliance on direct mitigation efforts(dual sourcing and inventory mitigation). In addition,we have the following result.

Corollary 3. If �1 = �2, the manufacturer finds sup-plier mitigation optimal at lower unit revenues in a diamond-shaped supply chain than in a V -shaped supply chain.

In other words, if �1 = �2, in a diamond-shapedsupply chain, the manufacturer should rely more onsupplier mitigation and less on manufacturer mitigationthan in a V -shaped supply chain. Returning to ourmotivating example of an unexpectedly high degree ofoverlap in the automotive supply chain discoveredfollowing the 2011 Tohoku earthquake, this suggeststhat an appropriate reaction from the downstream man-ufacturer is to reduce the emphasis placed on its owndirect mitigation efforts and instead offer higher pricesto its tier 1 suppliers; these suppliers, facing highermargins and potentially greater losses in the event ofa disruption, will in turn invest more in their ownrisk mitigation efforts, thereby lowering the overallrisk to the downstream manufacturer. The presence ofdisruption risk and overlap in tier 2 leads the manu-facturer to offer higher prices to tier 1 suppliers thanwould otherwise be optimal; hence, it is crucial for themanufacturer to consider risk originating throughoutthe supply chain, even if it is not optimal to directlymitigate that risk.

6 A key determinant of this is the relationship between the variousthresholds that determine the manufacturer’s optimal sourcingstrategy in the two configurations. Lemmas 3 and 4 in the appendixderive several properties of these thresholds; in what follows, wediscuss the implications of these results in a series of corollaries andexamples.

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Figure 5 The Manufacturer’s Optimal Sourcing Strategy as a Function of Unit Revenues (�) and Disrupted Capacity (K )

51 2 3 4

100

0

20

40

60

80

Disrupted capacity

51 2 3 4

Disrupted capacity

Man

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DS + SM

DS

DS + IM

(a) -shaped supply chain (b) Diamond-shaped supply chain

DS + SM

SS

Note. In the figure, cu = 5, cr = 505, �1 = �2 = 002, and D = 10.

The manufacturer’s optimal strategy as a function ofboth unit revenue and disrupted capacity is depictedgraphically in Figure 5 for the case of severe disruptions.In the figure, darker shading corresponds to greaterdisruption risk mitigation efforts. As the figure shows,an increase in either unit revenues or the severity of dis-ruptions leads the manufacturer to increase mitigationefforts, either directly (via inventory mitigation) or indi-rectly (via induced supplier mitigation). The result ofCorollary 3 can be seen in panel (b) of Figure 5, whichdepicts the optimal strategy in a diamond-shapedsupply chain: compared with panel (a), which depictsthe optimal strategy in a V -shaped supply chain, theregion of DS + SM optimality is larger in panel (b), andthere are no regions of DS+ IM or DS optimality (in thestrict sense)—SS performs equally well. Importantly,Corollary 3 assumes that the two unreliable tier 2suppliers have identical disruption risk; i.e., �1 = �2. Ingeneral, the result continues to hold if differences inrisk are small, but if tier 2 suppliers are sufficientlyheterogeneous, a different picture may emerge.

Corollary 4. Consider the case K = 0, and defineå ≡ 1

2 6√

4f − 152 + 4f�22 − f + 17, where f ≡ 4cr − cu5/

4cu�241 −�2550 If �1 < max4�221 å5, the manufacturer finds

supplier mitigation optimal (i) at the lowest unit revenuesin a diamond-shaped supply chain with tier 2 supplier 2shared; (ii) at moderate unit revenues in a V -shaped supplychain; and (iii) at the highest unit revenues in a diamond-shaped supply chain with tier 2 supplier 1 shared. If �1 >max4�2

21 å5, the order of (ii) and (iii) is reversed. Further-more, these results continue to hold provided disruptions aresufficiently severe (i.e., K is sufficiently small).

The key intuition for this result is that the manufac-turer always pays a lower price for supplier mitigationwhen the more risky tier 2 supplier is in the sup-ply chain. Interestingly, however, a diamond-shaped

supply chain with supplier 2 shared and a V -shapedsupply chain are not necessarily equivalent, becausethe manufacturer has at its disposal an alternative strat-egy in a V -shaped supply chain—namely, inventorymitigation. Hence, while overlap in tier 2 makes themanufacturer value supplier mitigation more, ceterisparibus, which tier 2 supplier is shared also makes adifference. If the tier 1 suppliers share the most riskysupplier, the manufacturer will engage in suppliermitigation over the widest range of unit revenues;if they share the least risky unreliable supplier, themanufacturer will engage in supplier mitigation only atvery high unit revenues. The results of this corollary aredepicted in Figure 6, which shows the manufacturer’soptimal strategy using the same parameter valuesas Figure 5, except for �1 = 001 and �2 = 004.7 Theseparameter values are chosen such that the probabilityof two simultaneous disruptions (�1�2) is identical tothe case in Figure 5, but in Figure 6, the individualtier 2 suppliers have different disruption probabilities;hence, the suppliers are “more heterogeneous” in thelatter example.8

In the next result, we say that the manufacturer is“more likely to use supplier mitigation” if the parameter

7 Note that �1 <�22, so for sufficiently severe disruptions (K small

enough), the ordering described in Corollary 4 holds.8 Throughout the paper, we define “more heterogeneous tier 2suppliers” to mean the case when �1 decreases and �2 increaseswhile their product, �1�2, remains constant. This has an appealinginterpretation in our model because it means the probability of asimultaneous disruption at both tier 2 suppliers is unchanged, butthe individual suppliers have increasingly different probabilities ofdisruption as �1 and �2 move away from one another. We numericallyinvestigated other definitions of “more heterogeneous,” such asincreasing �2 while holding �1 constant and decreasing �1 andincreasing �2 while holding their sum constant, and we found similarqualitative insights.

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Figure 6 The Manufacturer’s Optimal Sourcing Strategy as a Function of Unit Revenues (�) and Disrupted Capacity (K )

51 2 3 4

100

0

20

40

60

80

Disrupted capacity

51 2 3 4

Disrupted capacity

51 2 3 4

Disrupted capacity

DS + SM

DS

DS+ IM

(a) -shaped supply chain

DS + SM

SS

(b) Diamond-shaped supply chainwith supplier 1

DS + SM

SS

(c) Diamond-shaped supply chainwith supplier 2

Man

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Man

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Note. In the figure, cu = 5, cr = 505, �1 = 001, �2 = 004, and D = 10.

region under which supplier mitigation is optimalgrows as a parameter changes.

Corollary 5. If K <D/2, in any supply chain config-uration, the manufacturer is more likely to use suppliermitigation as unit revenues increase (� increases) anddisruptions become more severe (K decreases), and in aV -shaped supply chain, as tier 2 suppliers become moreheterogeneous (�1 decreases and �2 increases, holding theprobability of simultaneous disruptions, �1�2, constant).

The first part of the corollary can be seen from themanufacturer’s optimal strategy in Theorems 3 and 6.The second part can be seen in Figure 6. More severedisruptions simultaneously make supplier mitigationmore attractive (as the loss during a disruption isgreater) and inventory mitigation less attractive (as theamount of excess inventory the manufacturer mustpurchase increases), and as a result, supplier mitigationis adopted at lower unit revenues as disruptions becomemore severe in all configurations. The impact of tier 2heterogeneity on the manufacturer’s incentive to usesupplier mitigation is more nuanced. In Figure 6(a),note that compared to Figure 5(a), the manufacturer’sregion of DS + SM optimality is weakly larger whenunreliable tier 2 suppliers are heterogeneous thanwhen they are homogeneous. In fact, it can be shownthat if cr < cr , the manufacturer’s optimal strategyis insensitive to changes in �1 and �2, holding theirproduct constant (see Lemma 4 in the appendix). Thereason for this is that increased tier 2 heterogeneitysimultaneously increases profit when using suppliermitigation (because it results in a lower price necessaryto induce supplier mitigation) and increases profitwhen using manufacturer dual sourcing (because itreduces the risk of a disruption at the less risky tier 2supplier). These two effects precisely counteract oneanother, leading to no impact of tier 2 heterogeneity onthe manufacturer’s strategy if reliable suppliers arecheap. However, if cr > cr , i.e., reliable suppliers are

costly, greater heterogeneity favors supplier mitigationover inventory mitigation because of the increasedrisk of disruption at the riskier tier 2 supplier in thelatter strategy; for a similar reason, heterogeneity alsofavors dual sourcing over inventory mitigation, andthe sum of these effects implies that as suppliersbecome more heterogeneous, the manufacturer reliesless on inventory mitigation and more on suppliermitigation and dual sourcing. Thus, as �1 decreasesand �2 increases, holding �1�2 constant, the region ofDS + IM optimality in Figure 6(a) will shrink, and theregions of DS + SM and DS optimality will grow.

Finally, we compare the manufacturer’s profits inthe possible supply chain configurations.

Corollary 6. (i) If 0 <K <D, manufacturer profit isstrictly greatest in a V -shaped supply chain.

(ii) Among the two diamond-shaped supply chains, themanufacturer’s profit is larger when the less risky tier 2supplier is shared if and only if unit revenues are small; i.e.,� ≤ cu + 44cr − cu5/�1�2544D−K/25/4D−K55.

Part (i) shows that, while there may be other fac-tors that influence the manufacturer’s preferencesbetween the supply chain configurations in practice—for instance, if a single tier 2 supplier can achievegreater economies of scale than two independent tier 2suppliers, the diamond shape may be more profitable;conversely, if product quality risk is a significant con-cern, the V shape may be preferable—the corollaryconfirms that, ceteris paribus, the manufacturer doesindeed prefer a V -shaped supply chain purely from adisruption risk mitigation perspective.9 Part (ii) con-firms that, when faced with a choice between diamond-shaped supply chains, the manufacturer would prefer

9 If disrupted capacity is nonzero, this preference is strict; in thespecial case where disrupted capacity is equal to zero, the manufac-turer may achieve equal profit in the V -shaped and diamond-shapedsupply chains at some unit revenues but would still always weaklyprefer the V -shaped supply chain.

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having the less risky supplier shared at low unit rev-enues and the riskier supplier shared at high revenues.The underlying reason is that having a less risky sup-plier is advantageous for strategies that rely moreextensively on sourcing from unreliable suppliers,which are prevalent at low unit revenues. By con-trast, at larger unit revenues, the presence of a riskierunreliable supplier is preferable, since this makes itcheaper for the manufacturer to induce sourcing fromthe reliable supplier. While intuitive, both of theseresults will play a critical role in our analysis in thenext section.

7. The Supply Chain GameAs the results in the previous sections show, the optimalsourcing strategy may vary substantially dependingon the degree of overlap in the tier 2 supplier base.Aside from the observations regarding the behaviorof the manufacturer’s optimal sourcing strategy, thisfact has two additional implications: first, because themanufacturer’s optimal sourcing strategy influenceswhich tier 1 suppliers are selected and at what pricethey are compensated, the tier 1 suppliers may preferone supply chain configuration over the other; second,these preferences may not align with the manufacturer’spreference, which is always (per Corollary 6) for aV -shaped supply chain. As a result, if tier 2 suppliersare chosen strategically by tier 1 suppliers, there is apotential for the supply chain to be uncoordinated—for the tier 1 suppliers to select tier 2 suppliers thatdo not maximize the manufacturer’s profit—an issuethat we investigate in this section. Specifically, weanalyze a model in which, prior to contractual offersfrom the manufacturer, the tier 1 suppliers engagein a simultaneous move supplier selection game inwhich they choose which unreliable tier 2 supplier toestablish a relationship with, determining the supplychain configuration (either V shaped or diamondshaped).10 After the preferred tier 2 suppliers havebeen selected, the remainder of the game proceeds as inour base model; i.e., the manufacturer offers contractsto tier 1 suppliers and induces a particular supplychain structure. This sequence of events is reflectiveof our motivating example of the automotive supplychain, in which tier 2 supplier selection and approval

10 We restrict tier 1 suppliers to selecting exactly one unreliablesupplier to keep the model parsimonious and consistent with ourpreceding analysis. Note that, to make positive profit, a tier 1 suppliershould always select at least one unreliable supplier. While the costof qualification and vetting makes it reasonable to assume that atmost one unreliable supplier is selected in most cases, our numericalexperiments suggest that allowing two or more does not substantiallyimpact our qualitative results (from the manufacturer’s perspective,it would still generate a fixed—but considerably more complex—disruption profile in tier 2, hence only significantly complicating theanalysis of the optimal sourcing decisions in the various tiers).

is a long-term decision, while the manufacturer’s day-to-day procurement contracts are short- or medium-term decisions. In other words, having analyzed themanufacturer’s optimal sourcing strategy, given a fixedsupply chain configuration, in this section we considerthe critical question of which supply chain—V -shapedor diamond-shaped—the tier 1 suppliers would formgiven the manufacturer’s optimal sourcing strategy.

If the manufacturer treats tier 1 suppliers asym-metrically—i.e., by offering one a high price to inducesupplier mitigation and the other a low price to inducepassive acceptance—and is indifferent between whichsupplier receives which contract (as in the diamond-shaped supply chain; Theorem 6) we assume themanufacturer randomly selects tier 1 suppliers to fulfilleach role; hence, during the supplier selection game,the ex ante expected profit of a tier 1 supplier issimply the average profit of the suppliers in tier 1.We also assume all information and parameters in thegame are public knowledge; i.e., tier 1 suppliers knowthe manufacturer’s unit revenues and demand whenthey choose their tier 2 suppliers. Let 8i1 j9 denote thesupply chain configuration where tier 1 supplier Asources from unreliable tier 2 supplier i, and supplier Bsources from unreliable tier 2 supplier j , i1 j = 112. Ifthe tier 1 suppliers select the same unreliable tier 2supplier, then the resultant supply chain configurationis diamond-shaped (either 81119 or 82129). If tier 2suppliers select different unreliable tier 2 suppliers, thenthe supply chain is V -shaped (either 81129 or 82119).Consequently, the tier 1 suppliers play a two-by-twonormal form game, depicted in Table 2. The pure-strategy equilibria to this game will thus determine thesupply chain configuration. Our primary result in thissection characterizes the equilibria to this game.

Theorem 7. Let n= L if cr ≤ cr and n=H if cr > cr . Ifdisruptions are severe (K <D/2), we have the following:

(i) If � < �2, all supply chain configurations are equilibriato the supply chain game.

(ii) If �2 ≤� < min4�n1 �15, then both 81119 and 82129are equilibria to the supply chain game. Here, 81119 is a weakNash equilibrium while 82129 is a strict Nash equilibrium.

(iii) If min4�n1 �15≤� < �1, 82129 is the unique Nashequilibrium to the supply chain game.

(iv) If min4�n1 �15≤� < �n, then both 81119 and 82129are equilibria to the supply chain game. Here, 81119 is boththe payoff-dominant and risk-dominant equilibrium.

Table 2 The Supply Chain Game

Tier 1 supplier B

Tier 1 supplier A Supplier 1 Supplier 2

Supplier 1 81119 81129Supplier 2 82119 82129

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(v) If � ≥ max4�n1 �15, 82129 is an equilibrium to thesupply chain game, and 81119 is also an equilibrium if thetier 2 supplier risk is sufficiently heterogeneous. If bothequilibria exist, 81119 is the payoff-dominant and, if tier 2supplier risk is sufficiently heterogeneous, risk-dominantequilibrium.

In the theorem, note that cases (iii) and (iv) aremutually exclusive and depend on the ordering ofthresholds �n and �1 (described in detail in Lemma 3in the appendix); in turn, these depend on the relativelevel of heterogeneity, cost, and disrupted capacity.Observe that unless the manufacturer’s unit revenuesare very small, the only equilibria to the supply chaingame are the diamond-shaped supply chains. Com-bined with Corollary 6, this implies that for all � > �2,the tier 1 suppliers select a supply chain configurationthat is strictly suboptimal for the manufacturer.11 Thereason for this is that, in any V -shaped supply chain,there is at most one tier 1 supplier that is employed toengage in supplier mitigation, earning a high pricefrom the manufacturer and positive profit; this is, perTheorem 3, the tier 1 supplier with access to the riskiesttier 2 supplier. The tier 1 supplier that is not selectedfor supplier mitigation—which is the supplier withaccess to the lower risk tier 2 supplier—is paid a lowprice by the manufacturer, earning zero profit. Thissupplier thus has incentive to deviate from his actionand source instead from the tier 2 supplier with greaterrisk, leading to a diamond-shaped supply chain. Notethat both diamond-shaped supply chains can occurin equilibrium, and in particular, the shared supplychain with the riskier tier 2 supplier (82129) is alwaysan equilibrium for any �, because of the forces justdescribed. This tier 2 supplier, it should be empha-sized, is seemingly “dominated” by the less risky tier 2supplier: they have the same cost, but supplier 2 hashigher risk. Nevertheless, it is always an equilibriumfor the tier 1 suppliers to jointly select not only thesame tier 2 supplier but also the riskiest tier 2 supplier.

The theorem also shows that heterogeneity amongthe unreliable tier 2 suppliers worsens the coordinationproblem between the manufacturer and tier 1 sup-pliers. In particular, not only do the tier 1 suppliersselect a diamond-shaped supply chain, but also—underheterogeneity—they often select the worst diamond-shaped supply chain, from the manufacturer’s view-point. To see this, note that in case (ii), where the

11 Indeed, even when � < �2, a preference conflict can arise, asdiamond-shaped supply chains are equilibria to the supply chaingame (although the V -shaped configurations are also equilibria inthis case). In addition, while we do not explicitly model asymmetricinformation, if tier 1 suppliers have some uncertainty about themanufacturer’s unit revenues expressed, e.g., as a prior distributionon � , if this distribution has any mass placed above the threshold �2,the suppliers would strictly prefer a diamond-shaped supply chain.Thus, the selection of a diamond-shaped supply chain seems plausiblein practice.

unique strict equilibrium is 82129, the manufacturer’sprofit would actually be larger under the alternativediamond-shaped configuration 81119—merely a weakNash equilibrium—when unit revenues are close to �2

(as per Corollary 6). Weak Nash equilibria are typicallyunstable, which suggests that the tier 1 suppliers aremore likely to select 82129, which indeed is the worstoverall supply chain configuration for the manufacturer.Similarly, in case (iv) (and also in (v), under sufficientheterogeneity), 81119 is the payoff and risk-dominantequilibrium; equilibrium selection approaches suchas evolutionary game theory (Samuelson 1997) sug-gest that such equilibria tend to be reached over theirdominated counterparts, implying that 81119 is themore likely equilibrium to occur in practice. Whilethis may seem beneficial to the manufacturer, in fact,it is not; at these unit revenues, in either diamond-shaped configuration, the manufacturer will inducesupplier mitigation but will have to pay a higher priceto achieve this in 81119, yielding lower profits (seeCorollary 6). Hence, in this regime, the manufactureractually prefers 82129 to 81119, but the tier 1 suppliersare likely to select 81119.

Taken together, these results suggest that the strategicdecisions of tier 1 suppliers may have unfortunate long-term repercussions for the formation and evolutionof the industry’s supply base. Tier 1 suppliers havean incentive to select the same tier 2 suppliers, oftenthe worst such configuration from the manufacturer’sperspective, meaning the industry over time maygravitate toward a tier 2 supply base made up offew tier 2 suppliers, resulting in a less competitive,costlier, more risky, and more correlated tier 2 supplierbase, potentially to the detriment of the manufacturer’sprofitability. This is consistent with several observationsconcerning the evolution of the Japanese automotivesupply chain over the last 20 years (Sasaki 2013).

In our final result in this section, we define a preferenceconflict between the manufacturer and tier 1 suppliersto mean a scenario when the manufacturer’s strictlypreferred supply chain configuration (a V -shapedsupply chain) is not an equilibrium to the supply chaingame—in essence, cases (ii)–(v) of Theorem 7. Giventhis definition, we have the following.

Corollary 7. A preference conflict between the manu-facturer and tier 1 suppliers is more likely as unit revenuesincrease (� increases), disruptions become more severe(K decreases), and tier 2 suppliers become more heterogeneous(�1 decreases and �2 increases, holding �1�2 constant).

The first part of the corollary (the impact of �)can be seen directly from Theorem 7. The secondpart follows from the behavior of the equilibriumthresholds (in particular, �2) as K decreases, which is

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Figure 7 Supply Chain Game Equilibrium Cases from Theorem 7 as a Function of Unit Revenues (�) and Disrupted Capacity (K )

51 2 3 4

100

0

20

40

60

80

Disrupted capacity

51 2 3 4

Disrupted capacity

Case (v)

Case (i)

Case (ii)

(a) �1 = �2 = 0.2 (b) �1 = 0.1, �2 = 0.4

Case (v)

Case (i)

Case (ii)

Case (iii)

Case (iv)

Man

ufac

ture

r’s u

nit r

even

ues

100

0

20

40

60

80

Man

ufac

ture

r’s u

nit r

even

ues

Note. In the figure, cu = 5, cr = 505, and D = 10.

derived analytically in Lemma 4 of the appendix,12

and illustrated graphically in Figure 7.As disruptions become more severe, the size of equi-

librium region for case (i) shrinks, leading to a greaterlikelihood of a preference conflict. The reason for this,per Corollary 5, is that the manufacturer is more likelyto use supplier mitigation as K decreases, which inturn gives the tier 1 suppliers greater incentive to selecta diamond-shaped supply chain. Finally, the corol-lary shows that greater tier 2 heterogeneity increasesthe prevalence preference conflicts; this also followsfrom Corollary 5, since greater heterogeneity implies(in a V -shaped supply chain) that the manufactureris more likely to use supplier mitigation, which inturn strengthens the tier 1 suppliers’ incentives todeviate from a V shape and select a diamond-shapedsupply chain. In Figure 7, this can be seen by notingthat in panel (b), with heterogeneous tier 2 suppliers,the size of equilibrium region for case (i) is smallerthan in panel (a), with homogeneous tier 2 suppliers.Hence, we conclude that preference conflicts betweenthe manufacturer and tier 1 suppliers worsen as themanufacturer becomes more likely to induce suppliermitigation, either because unit revenues are higher,disruptions become more severe, or tier 2 suppliers aremore heterogeneous.

8. Penalty ContractsIn the preceding analysis, we restricted our attention toprice and quantity contracts of the form 4p1Q5. Whilethese are commonly observed in practice, they expose

12 Lemma 4 also provides a full characterization of how each threshold,and hence each equilibrium region in Theorem 7, changes as afunction of disrupted capacity and the probability of disruptions.

the manufacturer to a moral hazard problem, leadingto positive agency costs when inducing supplier miti-gation. Moreover, when tier 1 suppliers are capable ofstrategically choosing the supply chain configuration,they almost always select a diamond-shaped configura-tion, while the manufacturer always strictly prefersa V -shaped configuration. From these results, threequestions arise: First, is it possible to overcome themoral hazard problem using a simple coordinatingcontract? Second, do the results derived in Sections 3–5continue to hold qualitatively if a coordinating contractis used? And third, can this contract overcome thecoordination problem from Section 7 when the supplychain configuration is endogenous? In this section, wedemonstrate that the answer to all of these questions isyes, if the coordination mechanism used is a penaltycontract. Specifically, we consider contracts of the form4p1Q1f 5, where f is a per-unit fee for each unit orderedby the manufacturer that the contracted tier 1 supplierfails to deliver. The following theorem demonstratesthat these contracts are effective both at eliminatingmoral hazard and at solving the preference conflictproblem discussed in Section 7.

Theorem 8. When the manufacturer uses an optimalpenalty contract:

(i) In either a V -shaped supply chain or a diamond-shapedsupply chain, the manufacturer’s optimal sourcing strategyis the same as in the price–quantity contract case, except forthe fact that supplier mitigation is adopted at lower unitrevenues.

(ii) The manufacturer extracts all profit, and tier 1 sup-pliers always receive zero profit, regardless of the supplychain configuration, and hence tier 1 suppliers are indifferentbetween the V -shaped and diamond-shaped supply chains.

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Part (i) of Theorem 8 shows that if penalty contractsare employed, the manufacturer adopts supplier miti-gation at lower unit revenues than under simple priceand quantity contracts; otherwise, the manufacturer’spreferences between the various mitigation strategiesand induced supply chain structures are qualitativelyidentical to our earlier results.13 This increased relianceon supplier mitigation stems from the fact that penaltycontracts enable the manufacturer to extract all surplusfrom tier 1 suppliers even when inducing them to dualsource, making supplier mitigation relatively moreattractive than in our earlier setting. Beyond this effect,however, all the qualitative observations made in Sec-tions 3–6 continue to hold,14 confirming that our basicinsights persist even in the absence of a moral hazardproblem. There is, however, one critical difference whenpenalty contracts are used, as shown in part (ii) of thetheorem. Since the manufacturer can extract the entireprofit in all induced structures by using an appropriatepenalty, tier 1 suppliers become indifferent betweenall supply chain configurations. In the context of thesupply chain game analyzed in Section 7, this impliesthat a manufacturer who prefers a V -shaped supplychain can (profitably) make a small side payment totier 1 suppliers to induce them to make this selection,thus eliminating the perverse incentives generating themoral hazard problem and effectively coordinating thesupply chain.

We note that penalty contracts are commonly dis-cussed as mechanisms to induce increased risk mitiga-tion efforts by suppliers (Kleindorfer and Saad 2005,Yang et al. 2009, Hwang et al. 2014). However, it isimportant to recognize that such contracts may faceimplementation challenges in practice. First, whilebuyers and suppliers may be willing to accept penaltycontracts when disruptions arise endogenously (e.g.,because of supplier malfeasance or negligence, suchas from industrial accidents or poor quality control),buyers are often loathe to penalize suppliers becausetheir supply chain experienced an exogenous natu-ral disaster. This is especially true when buyers andsuppliers have long-term relationships, as many ofthe firms in our example of the Japanese automotiveindustry do. Second, our motivating example of the2011 Tohoku earthquake shows precisely the oppositebehavior of penalties occurred in practice: rather thanpunish suppliers for failing to deliver after the earth-quake, Toyota took an active role in assisting them to

13 It is also straightforward to show that with the optimal penaltycontracts, the manufacturer makes the same sourcing choices as acentralized system (i.e., a system in which the manufacturer sourcesdirectly from tier 2). For a discussion of this issue, please see thesupplemental appendix of the paper.14 For instance, the result that increased overlap in tier 2 leadsthe manufacturer to favor supplier mitigation more and directmanufacturer mitigation less.

quickly recover, providing manpower and financialassistance that subsidized the recovery efforts of thesuppliers—in effect, a payment from the manufacturerto the suppliers in the wake of a disruption. Thus,penalties may come with a credibility problem: once adisruption has occurred, it is in the manufacturer’s bestinterests to help suppliers quickly recover via directfinancial and resource assistance (or else the manufac-turer will experience even greater lost sales), meaninga punishment in the form of a financial penalty is notcredible. This dynamic is not captured in our stylizedsingle period model, but nonetheless, it suggests thatpenalty contracts may have limitations and might notbe easily implemented in practice.

9. ConclusionIn this paper, we have analyzed the sourcing strategyof a manufacturer subject to disruptions in a multitiersupply chain. In contrast to the existing disruptionrisk literature, we have considered the impact of riskoriginating in tier 2 of the supply chain, rather thantier 1. We find that the degree of overlap in tier 2(whether unreliable tier 2 suppliers are shared by tier 1suppliers) has a substantial impact on the manufac-turer’s optimal sourcing strategy, as greater overlapcauses the manufacturer to rely less on direct mitigationefforts (dual sourcing and procuring excess inventoryfrom tier 1) and more on indirect methods (inducingtier 1 suppliers to mitigate risk via contract terms).Moreover, we showed that the manufacturer and tier 1suppliers may have conflicting preferences regardingthe configuration of the supply network, potentiallyleading to a coordination problem in the supply chainif tier 1 suppliers strategically choose tier 2 suppliers;generally speaking, tier 1 suppliers select a diamond-shaped supply chain while the manufacturer prefers aV -shaped supply chain. In essence, the tier 1 suppli-ers have an interest in exploiting the manufacturer’sdesire to mitigate disruption risk by creating the mostcorrelated supply chain possible; the manufacturer’sown disruption risk mitigation efforts thus condemn itto face a highly correlated supply chain configuration.The manufacturer can rectify this coordination problemby using penalty contracts to eliminate moral hazardand hence the perverse incentives that tier 1 suppliersfeel to select a diamond-shaped supply chain.

We also showed that both the manufacturer’s relianceon supplier mitigation and the potential for preferenceconflicts between the tier 1 suppliers and the man-ufacturer grow as the manufacturer’s unit revenuesincrease, as disruptions become more severe, and asunreliable tier 2 suppliers become more heterogeneous.The impact of heterogeneous risk is especially intrigu-ing, as it suggests that tier 1 suppliers are more likely

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to create a diamond-shaped supply chain if tier 2 sup-pliers are more “different.”15 This phenomenon occursbecause of a focusing effect generated by heterogene-ity: in essence, if tier 2 suppliers were homogeneous,tier 1 suppliers would in many cases be indifferentbetween them, making overlap unlikely. However, astier 2 becomes more heterogeneous, the incentives feltby tier 1 suppliers to select the same tier 2 suppliergrow stronger, leading to a diamond-shaped supplychain. This indicates that manufacturers should find itmost critical to confront and remedy the coordinationproblem precisely when unreliable tier 2 suppliers haveespecially heterogeneous levels of risk, e.g., if some arelocated in a coastal region prone to earthquakes andtsunamis while others are located in interior regionsthat, although still subject to geological events, are lessinherently “risky” than their coastal counterparts.

There are several assumptions of our model thatbear discussion. First, we focused on a single-periodmodel rather than one in which demand (and disrup-tions) occur over multiple periods (Tomlin 2006). In amultiperiod model, inventory mitigation—essentially,holding safety stocks—becomes a more valuable strat-egy, as excess inventory from one period can be held touse in subsequent periods. Nevertheless, we expect thatthe same sort of progression of the optimal sourcingstrategy persists even over multiple periods; i.e., themanufacturer moves from direct mitigation to suppliermitigation as overlap increases. Indeed, Toyota itself isfollowing the strategy of greater supplier mitigationfollowing the Tohoku earthquake, as it has workedwith several key tier 1 suppliers to have them dualsource and hold inventories, despite the fact that thelatter strategy is a significant departure from the Toy-ota Production System philosophy of zero inventory(Greimel 2012). Our model thus supports the optimalityof Toyota’s increased reliance on supplier mitigation inthe wake of the 2011 disaster.

Second, we have assumed that no emergency backupsupply is available. In practice, it may be possible fortier 1 suppliers to find secondary sources of componentsupply if tier 2 suppliers disrupt, an extension that wehave analyzed but omit for the sake of brevity; detailsmay be found in the supplemental appendix. As onemight expect, the availability of such an option meansthat the manufacturer is less likely to use suppliermitigation, but otherwise, the same progression ofoptimal mitigation strategies holds. Interestingly, ifthe cost of emergency supply is sufficiently low, themanufacturer induces single sourcing from each tier 1

15 Indeed, while we showed this result analytically for heterogeneityin tier 2 disruption risk, in the supplemental appendix we numeri-cally demonstrate that it—and our other key results—continues toqualitatively hold when unreliable tier 2 suppliers are heterogeneousin both cost and risk.

supplier for any unit revenues, regardless of the supplychain configuration; this implies that for componentswhere backup supply is readily available (e.g., com-modity products), the optimal manufacturer strategy isindependent of the supply chain configuration. Con-versely, knowing the supply chain configuration andengaging tier 1 suppliers in risk mitigation are criti-cally important if emergency supply is costly, e.g., forspecialized or custom products such as semiconductorcomponents. This result illustrates that it is importantfor manufacturers to focus their limited resources onunderstanding and managing disruption risk in theupper tiers of their supply chain primarily for spe-cialized, noncommodity components, while a moretraditional manufacturer-only mitigation approach maybe employed for more standard products with readilyavailable backup supply.

Third, we have assumed that the manufacturer knowsthe configuration of the supply chain when making itsoptimal sourcing decision. In practice, this may notbe the case—even Toyota, one of the most powerfulmanufacturers in the world, was unable to persuadealmost half of its tier 1 suppliers to share the identitiesof their tier 2 suppliers. Thus, an interesting questionremains unsolved by our analysis: How can a manu-facturer manage disruption risk if it does not knowthe configuration of its supply network? Future workmight explore this question, using our analysis of theoptimal sourcing strategy under public information asa baseline for comparison.

Finally, we have assumed that the only mechanismby which the manufacturer can influence tier 1 sourc-ing decisions are purchasing contracts, specificallyprice-and-quantity contracts or penalty contracts. Whilethis may be true in many cases—especially if themanufacturer has only vague knowledge about itstier 2 suppliers, such as in the case of Toyota andBoeing during the development of its 787 (Tang andZimmerman 2009)—some manufacturers may havebetter awareness of their supply chains and betterrelationships with tier 2 and higher suppliers, allow-ing them to directly supervise and work with uppertiers to improve resiliency (e.g., Honda, which oftencontracts directly with suppliers in tiers 2 and 3; seeChoi and Linton 2011). This is an intriguing aspect ofthe disruption risk management problem to explorein future work; in a different context, Huang et al.(2015) consider a similar trade-off between direct man-ufacturer control and delegation of upper-tier suppliermanagement.

Viewed as a whole, our results illustrate the sig-nificant impact that an extended supply chain canhave on optimal sourcing and disruption mitigationefforts. While much of the existing work on disruptionrisk management has focused on immediate suppli-ers (Aydin et al. 2010), the trend toward longer and

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longer global supply chains has exposed firms to riskat multiple levels, necessitating an understanding ofhow tier 1 sourcing decisions impact risk originating athigher levels of the supply chain. Taken in that light,our model provides a first step at understanding howto manage disruption risk and optimal sourcing inmultitier supply networks.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/mnsc.2016.2471.

AcknowledgmentsThe authors thank Keiji Masui from Toyota and Yoko Masuiand Naofumi Nishi from Stanford University for helpfuldiscussions about this problem, and the department editor,associate editor, three referees, and seminar participants atthe 2013 Manufacturing & Service Operations Management(MSOM) annual meeting in Fontainebleau, the 2014 MSOMannual meeting in Seattle, Singapore Management University,and INSEAD for many useful suggestions. E. Ang’s con-tribution to this project was completed while a student atStanford University.

Appendix A. Proofs

Proof of Theorem 1. For convenience, we suppress thedependence on i and j in the proof. From Equation (1),we may make several observations. First, the supplier willalways source enough inventory to cover demand in thenondisrupted state; i.e., qu + qr ≥Q. Second, the supplier willnever source more inventory than covers demand in thedisrupted state; i.e., qr + min8K1 qu9≤Q. Third, the supplieralways sources at least qu ≥ K units from the unreliabletier 2 supplier, as this quantity is risk free and less expen-sive than sourcing from the reliable tier 2 supplier. Hence,profit may be written çS4qu1 qr 5= �6p4qr +K5− crqr − cuK7+41 −�56pQ− cuqu − crqr 7. Because of the linear nature of thisfunction, it must be true that either qu =K and qr =Q−Kor qu =Q and qr = 0 (or some convex combination of thetwo) maximizes profit. Under the former strategy, profit isçS4K1Q−K5= 4p− cr 54Q−K5+ 4p− cu5K1; under the latterstrategy, profit is çS4Q105= 41 −�54p− cu5Q+�4p− cu5K0Comparing the profit functions yields the optimal strategy inthe theorem.

Proof of Theorem 2. By Lemma 1, when K ≥D/2, struc-ture 3 can achieve riskless supply at a unit cost of cu, whichis the lowest cost and highest revenue outcome and is henceoptimal.

Proof of Theorem 3. Observe that structure 4 dominates 2and 5, because all three structures have perfectly reliablesupply, but structure 4 achieves this at a lower average unitcost. Structure 3 dominates structure 1, since the strategy ofdual sourcing outperforms single sourcing by ensuring agreater level of minimum supply. Since all profit functionsare linearly increasing in �, determining the preferencebetween structures 3 and 4 reduces to determining thethreshold � for which the structures are equivalent. First,

by Lemma 1, structure 4 dominates structure 3 withoutinventory mitigation if

4� − cu5D−cr − cu�2

4D−K5> 4� − cu5441 −�15D+�12K50

This reduces to � > 44cr − cu5/�2�1544D−K5/4D− 2K55+ cu ≡

�L. In addition, structure 4 dominates structure 3 withinventory mitigation if

4� − cu5D−cr − cu�2

4D−K5> 41 −�1�254� − cu5D

+�1�24� − cu52K − 41 −�1541 −�25cu4D− 2K50

This reduces to � > 44cr − cu5/�1�22544D−K5/4D− 2K55− 41 −

�1 −�25cu/�1�2 ≡ �H . Finally, note that within structure 3,inventory mitigation is optimal if � > cu/�1.

Hence, there are two cases. If cr ≤ cr , then by Lemma 3(i),cu/�1 ≥ �L ≥ �H , and structure 4 becomes optimal beforeinventory mitigation becomes optimal within structure 3, lead-ing to case (i). If cr > cr , then cu/�1 < �L < �H by Lemma 3(i),so inventory mitigation within structure 3 becomes optimalbefore structure 4 becomes optimal, leading to case (ii).

Proof of Corollary 1. (i) Let çM 4X5 be the manufac-turer’s expected profit under strategy X ∈ 8DS1DS + IM,DS + SM9, such that the maximum percentage profit loss fromusing strategy X is L4X5= 1 − lim�→� çM 4X5/çM 4DS + SM5.Then, from Lemma 1,

L4DS5 = 1 − lim�→�

4� − cu5441 −�15D+�12K54� − cu5D− 44cr − cu5/�254D−K5

= �1

(

1 −2KD

)

0

(ii) When using strategy DS + IM,

L4DS+IM5 = 1− lim�→�

(

41−�1�254�−cu5D+�1�24�−cu52K

−41−�1541−�25cu4D−2K5)

·

(

4�−cu5D−cr −cu�2

4D−K5

)−1

= �1�2

(

1−2KD

)

1

which completes the proof.

Proof of Theorem 4. With a uniform allocation rule,supplier i’s allocation in the event of a disruption from theshared unreliable tier 2 supplier is min8qiu1K/2 + 4K/2 − qiu5

+9.This implies that supplier i’s profit is

�i4qiu1 q

ir 5 = �j

[

pi min(

Qi1 qir + min

{

qiu1K

2+

(

K

2− q−i

u

)+})

− cu min{

qiu1K

2+

(

K

2− q−i

u

)+}]

+ 41 −�j5[

pi min4Qi1 qir + qiu5− cuq

iu

]

− crqir 0

The following must be true: qir + qiu ≥ Qi, qiu ≥ K/2 +

4K/2 − q−iu 5+ ≡ Ki, and qir + min8qiu1K/2 + 4K/2 − q−i

u 5+9≤Qi.Using these facts, profit is

�i4qiu1 q

ir 5 = �j 6piq

ir + pKi − cuKi7+ 41 −�j56piQi − cuq

iu7− crq

ir 0

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Observing that the profit function is linear in supplier i’ssourcing quantities, we have two cases: either the supplierwill source qiu =Qi and qir = 0, yielding profit �i4q

iu1 q

ir 5=

�j 6pi − cu7Ki + 41 −�j56pi − cu7Qi, or the supplier will sourceqiu = Ki and qir =Qi − Ki, yielding profit �i4q

iu1 q

ir 5= piQi −

cuKi −cr 4Qi − Ki5. The latter is preferred if pi ≥ cu+ 4cr −cu5/�j ,which is identical to the condition derived with independenttier 2 suppliers (Theorem 1) and is independent of Ki andhence q−i

u . Consequently, the supplier has two sourcingstrategies (single source or dual source), with the conditionsgiven in the theorem.

Proof of Theorem 5. The result follows from part (iii)of Lemma 2. When disruptions are minor (K ≥ D), themanufacturer’s profit under the contract 84cu1�D51 4cu141 − �5D59 is çM = 4� − cu5D, which corresponds to theminimal possible sourcing cost.

Proof of Theorem 6. From Lemma 2, structures 1 and 3are equivalent, and structures 2 and 5 are clearly dominatedby structure 4, so the manufacturer’s choice is effectivelybetween structures 1 and 3 and structure 4. The latter ispreferred if � ≥ cu + 44cr − cu5/�

2j 544D−K/25/4D−K55= �j .

Proof of Corollary 2. Let çM 4X5 be the manufacturer’sexpected profit under strategy X ∈ 8DS1DS + SM9, such thatthe maximum percentage profit loss from using strategy DSis L4DS5 = 1 − lim�→� çM 4DS5/çM 4DS + SM5. Then, fromLemma 2,

L4DS5 = 1− lim�→�

�j4�−cu5K+41−�j54�−cu5D

4�−cu−4cr −cu5/�j54D−K/25+4�−cu5K/2

= �j

(

1−K

D

)

0

Proof of Corollary 3. By Lemma 3(i), note that �L > �H ⇔

cr < cr . This implies that the manufacturer prefers inducingsupplier mitigation under a V -shaped supply chain if andonly if � > max4�L1 �H 5. To show that this occurs at higherprofit margins than in a diamond-shaped supply chain, itthus suffices to show that max4�L1 �H 5≥ �11 �2.

If �1 = �2, then Lemma 3(ii) implies that max4�L1 �H 5≥

�11 �2 holds at K = 0, and Lemma 3(v) implies that �H1 �L

will grow faster than �11 �2, so that max4�L1 �H 5≥ �11 �2will continue to hold for any K ≥ 0, proving the result.

Proof of Corollary 4. Consider the case �1 <�2, andK = 0. Recall that the manufacturer prefers to induce suppliermitigation under a V -shaped supply chain if and only if � >max4�L1 �H 5 (see the proof of Corollary 3). By Lemma 3(iii),two cases can arise, depending on whether max4�L1 �H 5is larger or smaller than �1. When �2 < max4�L1 �H 5 < �1holds, the manufacturer prefers supplier mitigation (a) at thelowest unit revenues when the tier 2 supplier 2 is shared,(b) at intermediate unit revenues in a V -shaped supply chain,and (c) at the highest unit revenues when tier 2 supplier 1is shared. When �2 < �1 < max4�L1 �H 5 holds, the order of(b) and (c) is flipped. Since these inequality hold strictly atK = 0, they will continue to hold for sufficiently small K > 0,proving the results for sufficiently severe disruptions.

We now show that �1 < max4�221 å5 implies max4�L1 �H 5

< �1. By Lemma 3(iii), we have that max4�L1 �H 5 < �1 holdsif and only if cr < ¯cr . This condition holds trivially when�1 ≤ �2

2, since ¯cr = �. Thus, let us assume �1 >�22, in which

case ¯cr ≡ cu + cu44�1�241−�1541−�255/4�1 −�2255. The equation

cr = ¯cr has two real solutions for �1, of which the positiveone is exactly

å= 12

[

√

4f − 152+ 4f�2

2 − f + 1]

1

where f ≡ 4cr − cu5/4cu�241 −�255. Therefore, since ¡ ¯cr/¡�1 =

−4cu�241−�254�21 +�2

2 −2�1�2255/4�1 −�2

252 < 0, it can be readily

seen that cr < ¯cr for any �1 < å. The same argument alsoshows that �1 > max4�2

21 å5 implies max4�L1 �H 5 > �1, whichcompletes the proof.

We note that a more general result is also possible, for anyvalue of K. In particular, note that having the order (i)–(iii), isequivalent to having �2 ≤ max4�L1 �H 5≤ �1. By Lemma 3(vi),we always have �2 ≤ �1 and �2 ≤ �1. Thus, it is necessaryand sufficient to have �L ≤ �1 and �H ≤ �1. By parts (vii)and (ix) of Lemma 3, this is equivalent to

�L ≤ �1 ⇔ �1 ≤42D−K54D− 2K5

24D−K52�21

�H ≤ �1 ⇔ �1 ≤ max(

42D−K54D− 2K524D−K52

�221

¯å

)

1

where ¯å is the positive root of the equation cr = cu +

cu442�1�241 − �1541 − �254D−K54D − 2K55/42�14D−K52 −

�2242D−K54D− 2K555 in �1. The proof for the latter equiva-

lence follows a similar logic to the case K = 0, by arguing thatthe positive root is always at most �2 and that the right-handside of the equation is always decreasing in �1. Full detailsare omitted for space considerations.

Proof of Corollary 5. The behavior as a function �follows immediately from Theorems 3 and 6. To see that thelikelihood of using SM increases as K decreases, note fromLemma 4 that ¡�H/¡K ≥ ¡�L11/¡K ≥ ¡�2/¡K ≥ 0; i.e., all thesupplier mitigation thresholds are increasing in K. Moreover,¡cr/¡K ≤ 0, which implies that as K decreases, holding allelse constant, the “costly” reliable supplier threshold crgrows. This means it is possible to transition from thecostly to the “cheap” case as K decreases, which furtherincreases the chance of employing SM . Finally, to see thatthe likelihood of using SM increases as suppliers becomemore heterogeneous, note that, from Lemma 4, holding �1�2constant, ¡cr/¡�2 ≥ 0, ¡�L/¡�2 = 0, and ¡�H/¡�2 ≤ 0 if cr ≥ cr .Hence, the SM thresholds in a V -shaped supply chain are(weakly) decreasing in tier 2 heterogeneity, and moreover,the costly reliable supplier threshold is increasing in tier 2heterogeneity, confirming the result.

Proof of Corollary 6. (i) The result clearly holds for allD/2 <K <D. Hence, we need only show it for 0 <K <D/2.First, we note that in either a V- or diamond-shaped supplychain, per Lemmas 1 and 2, the manufacturer chooses betweenstructures 3 and 4. Hence, a sufficient condition for optimalmanufacturer profit in a V -shaped supply chain to strictlyexceed optimal profit in a diamond-shaped supply chain isfor the profit within each structure to be strictly greater in a

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V -shaped supply chain. Manufacturer profit in structure 3 isstrictly greater in a V -shaped supply chain if

4� − cu5441 −�15D+�12K5> 4� − cu5441 −�j5D+�jK5

for j = 112. (Note that we assume the manufacturer doesnot use inventory mitigation in a V -shaped supply chain;allowing this option only increases the left-hand side of theabove inequality.) This inequality is strict for all K > 0. Instructure 4, manufacturer profit is greater in a V -shapedsupply chain if

4�−cu5D−cr −cu�2

4D−K5>4�−cu5D−cr −cu�j

(

D−K

2

)

1

which always holds for j = 112 and K > 0.(ii) Note first that �2 ≤ �1, by Lemma 3(vi). From Theo-

rem 6 and Lemma 2, for � ≤ �2, the manufacturer’s profitwhen the shared tier 2 supplier is j is given by 4� − cu5 ·441 −�j5D+�jK5. Since �1 ≤ �2 and D ≥K, we readily seethat the profit is larger when supplier 1 is shared. Similarly,for �2 < � ≤ �1, the condition that profit is larger whensupplier 1 is shared is equivalent to

4� − cu5441 −�15D+�1K5≥ 4� − cu5D−cr − cu�2

(

D−K

2

)

⇔ � ≤ cu +4cr − cu54D−K/25

�1�24D−K50

It can be readily checked that this threshold is always greaterthan �2 and always smaller than �1, provided that �1 ≤ �2.Finally, when � > �1, the profit when the shared tier 2supplier is j is given by 4� − cu5D− 44cr − cu5/�j54D−K/25,so that the configuration where supplier 2 is shared yieldslarger profit, as �2 ≥ �1.

Proof of Theorem 7. From Theorems 3 and 6, there arefive cases:

(i) If � < �2, under any supply chain configuration, themanufacturer will induce passive acceptance by its tier 1suppliers. This leaves all tier 1 suppliers with zero profit;hence tier 1 suppliers are indifferent between tier 2 suppliers,and any supply chain configuration is an equilibrium.

(ii) If �2 ≤� < min4�n1 �15, the manufacturer will inducesupplier mitigation only if the tier 1 suppliers share supplier 2.Let çD

2 = 44cr − cu5/�254D−K/25− cuK/2 − cr 4D−K5 be atier 1 supplier’s profit in a diamond-shaped supply chainwhen supplier mitigation is induced and the unreliablesupplier is tier 2 supplier 2, and the manufacturer ordersall product from that tier 1 supplier; the expected payoffis thus çD

2 /2. Then the payoffs to the supply chain gameare 40105 in all configurations except for 82129, where theyare 4çD

2 /21çD2 /25. Consequently, there are two equilibria:

either both tier 1 suppliers select supplier 1 or both selectsupplier 2. The former is a weak Nash equilibrium, sincetier 1 suppliers are indifferent between their actions, whilethe latter is a strict Nash equilibrium.

(iii) If min4�n1 �15≤� < �1, the manufacturer will inducesupplier mitigation if the tier 1 suppliers share unreliable thetier 2 supplier 2 or if they select different tier 1 suppliers;otherwise, the manufacturer induces passive acceptance.Let çV

2 = 44cr − cu5/�254D−K5− cuK − cr 4D− 2K5 be a tier 1

supplier’s profit in a V -shaped supply chain when suppliermitigation is induced. Thus, the payoffs to the supply chaingame are 40105 in 81119, 4çV

2 105 in 82119, 401çV2 5 in 81129,

and 4çD2 /21çD

2 /25 in 82129. Consequently, the unique Nashequilibrium to this game is for both tier 1 suppliers to selectthe dominated tier 2 supplier (supplier 2).

(iv) If min4�n1 �15≤� < �n, the manufacturer will inducesupplier mitigation in any diamond-shaped supply chainbut not in a V -shaped supply chain. Let çD

1 = 44cr − cu5/�15 ·4D−K/25− cuK/2 − cr 4D−K5 be the tier 1 supplier’s profitwhen engaged in supplier mitigation in a diamond-shapedsupply chain and sourcing from an unreliable tier 2 supplier 1.Note that çD

1 ≥çD2 . Thus the payoffs to the supply chain

game are 4çD1 /21çD

1 /25 in 81119, 40105 in 82119 and 81129,and 4çD

2 /21çD2 /25 in 82129. Consequently, either diamond-

shaped supply chain is an equilibrium. Clearly, 81119 is payoffdominant; to see that 81119 is risk dominant, note that riskdominance holds (Samuelson 1997) if 40 −çD

1 /2540 −çD1 /25≥

40 −çD2 /2540 −çD

2 /25, which holds for any �1 ≤ �2.(v) If � ≥ max4�n1 �15, the manufacturer will induce sup-

plier mitigation under all supply chain configurations. Thepayoffs to the supply chain game are 4çD

1 /21çD1 /25 in 81119,

4çV2 105 in 82119, 401çV

2 5 in 81129, and 4çD2 /21çD

2 /25 in 82129.Here, there is a distinction between the cases, depending onwhether çD

1 /2 >çV2 holds.

(v-a) If çD1 /2 >çV

2 , there are two Nash equilibria tothe game: 81119 and 82129. As in case (iv), 81119 is payoffdominant; it is risk dominant if and only if 4çV

2 −çD1 /252 ≥

4çD2 5

2/4. Both of these conditions are met if, all else beingequal, �2 is sufficiently high relative to �1.

(v-b) If çD1 /2 <çV

2 , there is a unique Nash equilibriumto the game: 82129.

Proof of Corollary 7. From Theorem 7, a preferenceconflict exists in every case except case (i). Thus, to show theresult, we show that case (i) is less likely in each instance.This immediately follows for increases in �. For decreasesin K, note that �2 increases in K per part (iii) of Lemma 4,proving the result. For increased heterogeneity, per part (iv)of Lemma 4, �2 is decreasing in �2, holding the product �1�2constant, which proves the result.

Proof of Theorem 8. (i) We show the result only for thecase of a V -shaped supply chain; the proof with a diamond-shaped supply chain is similar and is omitted. With a contract4p1Q1f 5, the profit of tier 1 supplier i sourcing from tier 2supplier j is

çi4qiu1qir 5

=�j

[

4pi+f 5min8qir +min8K1qiu91Qi9−crqir −cumin8K1qiu9

]

+41−�j5[

4pi+f 5min8qiu+qir1Qi9−cuqiu−crq

ir

]

−fQi0

The last term is a constant; the rest of the expression isidentical to the 4p1Q5 contract case with p replaced by p+ f .Hence, the tier 1 supplier’s optimal sourcing strategy is

4qiu1 qir 5

∗=

4Qi105 if cu ≤ pi + f < cu +cr − cu�j

1

4K1Qi −K5 if pi + f ≥ cu +cr − cu�j

0

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In the latter case, tier 1 supplier profit reduces to çi4qiu1 qir 5=

piQi − cr 4Qi − K5 − cuK. Consequently, the manufacturerhas two strategies: it can induce passive acceptance by atier 1 supplier, by offering a contract price pi = cu and nopenalty (f = 0), or it can induce dual sourcing, by offering acontract price plus penalty (pi + f ) equal to cu + 4cr − cu5/�j .To ensure participation of the tier 1 supplier, the man-ufacturer must also ensure nonnegative expected profit,meaning pi ≥ cu + 41 −K/Qi54cr − cu5. The optimal price isclearly the lowest price that achieves this outcome, i.e.,pi = cu+41−K/Qi54cr − cu5 < cu+4cr − cu5/�j , with a fee equalto f = 44cr − cu5/�j 541−�j 41 −K/Q55, such that the sum of theprice and fee is cu + 4cr − cu5/�j and the tier 1 supplier earnszero profit. This implies that inducing passive acceptance isidentical to the 4p1Q5 case, but inducing supplier mitigationis less expensive for the manufacturer than in the 4p1Q5 case.Hence, manufacturer profit when employing either DS orDS + IM is unchanged, but profit when employing DS + SMis increased; hence the manufacturer pursues supplier mitiga-tion at lower unit revenues than with 4p1Q5 contracts butotherwise follows an identical strategy.

(ii) Because the manufacturer leaves the tier 1 supplierswith zero expected profit in any structure, the result follows.

Appendix B. Supporting Results

Lemma 1. In a V -shaped supply chain, we have the following:(i) In structure 1, the optimal contract is 84cu1D51 4cu1059, and

the optimal expected profit is çM = 4� − cu5441 −�15D+�1K5.(ii) In structure 2, the optimal contract is 84cu1051 4cu +

4cr − cu5/�21D59, and the optimal expected profit is çM = 4� −

cu − 4cr − cu5/�25D.(iii-a) In structure 3, if K ≥D/2, the optimal contracts are

84cu1�D51 4cu1 41 − �5D59 for any � ∈ 60117 such that neitherquantity exceeds K, and the optimal expected profit is çM =

4� − cu5D.(iii-b) In structure 3, if K <D/2 and if � < cu/�1, the optimal

contracts are 84cu1D−K51 4cu1K59, and the optimal expected profitis çM = 4� − cu5441 −�15D+�12K50

(iii-c) In structure 3, if K < D/2 and if � > cu/�1, theoptimal contract is 84cu1D−K51 4cu1D−K59, and the optimalexpected profit is çM = 41 −�1�254� − cu5D+�1�24� − cu52K −

41 −�1541 −�25cu4D− 2K5.(iv) In structure 4, the optimal contract is 84cu1K51 4cu +

4cr − cu5/�21D−K59, and the optimal expected profit isçM = 4� − cu5D− 44cr − cu5/�254D−K5.

(v) In structure 5, the optimal contracts are 84cu + 4cr − cu5/�1105, 4cu + 4cr − cu5/�21D59, and the optimal expected profitis çM = 4� − cu − 4cr − cu5/�25D.

Proof of Lemma 1. Parts (i), (ii), (iv), and (v) followimmediately from Theorem 1 and Equation (2). For part (iii-a),structure 3 is induced by offering a price cu to both tier 1suppliers, implying that in Equation (2), f1 = 41 −�1541 −�25,f2 = �141−�25, f3 = 41−�15�2, and f4 = �1�2. Two observationscan be made to simplify Equation (2): at optimality, QA +

QB ≥D, and Qi +K ≤D for i =A1B. Furthermore, observethat if K >D/2, the manufacturer can simply source D/2from each tier 1 supplier and fully eliminate disruption risk.More generally, any sourcing strategy in which QA = �D,QB = 41 − �5D, for some � ∈ 60117, and in which neitherquantity exceeds K, results in profit equal to 4� − cu5D. If,

on the other hand, K < D/2, it is clearly optimal for themanufacturer to source at least K from each supplier (becausethis is the “risk-free” quantity that each supplier delivers forcertain), resulting in expected profit

çM 484cu1QA51 4cu1QB595

= �1�24� − cu52K +�141 −�254� − cu54K +QB5

+�241 −�154� − cu54K +QA5

+ 41 −�1541 −�254�D− cu4QA +QB550

It can be checked that ¡çM/¡Qi = 41 − �i54�−i� − cu5 fori =A1B. Hence, if �i� > cu, i = 112, the manufacturer willsource as much as possible from each supplier, satisfyingthe constraints QA +QB ≥D, K ≤QA, K ≤QB, D−K ≥QA,and D−K ≥QB. This implies Q∗

A =Q∗B =D−K, resulting in

the optimal expected profit given in part (iii-c) the lemma.Conversely, if cu/�2 <� < cu/�1, the manufacturer will sourceas much as possible from supplier A and as little as possiblefrom supplier B, resulting in Q∗

A =D−K, Q∗B =K. Finally,

if cu/�2 >�, the manufacturer wants to source as little aspossible while satisfying all the constraints. This impliesQ∗

A =D−K, Q∗B =K.

Lemma 2. In a diamond-shaped supply chain with sharedunreliable tier 2 supplier j ∈ 81129, we have the following:

(i) In structure 1, the optimal contract is 84cu1D51 4cu1059or 84cu1051 4cu1D59, and the optimal expected profit is çM =

4� − cu5441 −�j5D+�jK5.(ii) In structure 2, the optimal contract is 84cu1051 4cu +

4cr − cu5/�j1D59 or 84cu + 4cr − cu5/�j1D51 4cu1059, and theoptimal expected profit is çM = 4� − cu − 4cr − cu5/�j5D.

(iii) In structure 3, if D ≤ K, the optimal contracts are84cu1�D51 4cu1 41 − �5D59 for any � ∈ 60117, and the optimalexpected profit is çM = 4� − cu5D. If D>K, the optimal contractsare 84cu1 �D51 4cu1 41 − �5D59 for any � ∈ 60117, and the optimalexpected profit is çM = 4� − cu5441 −�j5D+�jK5.

(iv) In structure 4, the optimal contracts are 84cu + 4cr − cu5/�j ,D−K/25, 4cu1K/259 or 84cu1K/251 4cu + 4cr − cu5/�j1D−K/259,and the optimal expected profit is çM = 4� − cu5D − 44cr −

cu5/�j54D−K/25.(v) In structure 5, the optimal contract is 84cu1051 4cu +

4cr − cu5/�j1D59 or 84cu + 4cr − cu5/�j1D51 4cu1059, and theoptimal expected profit is çM = 4� − cu − 4cr − cu5/�j5D.

Proof of Lemma 2. Parts (i), (ii), and (v) follow imme-diately. For part (iii), in structure 3, there are two cases. IfD ≤K, the manufacturer can achieve risk-free sourcing bysplitting his order between the two tier 1 suppliers in anyway. Conversely, if D>K, under the uniform allocation rulethe manufacturer can divide its sourcing quantities betweentier 1 suppliers in any way and achieve expected profit

çM(

84cu1QA514cu1QB59)

=�j4�−cu5K+41−�j54�min4D1QA +QB5−cu4QA +QB550

In that case, any QA +QB =D is optimal, and manufacturerprofit is as given in the lemma. For part (iv), under theuniform allocation rule, to induce supplier A to source fromthe reliable tier 2 supplier, the manufacturer must allocate atleast K/2 units of inventory to supplier A (see Theorem 4). If

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the manufacturer allocates such that Qi ≥K/2 for i = A1B,then supplier A will source K/2 units from the unreliabletier 2 supplier and QA −K/2 from the reliable tier 2 supplier,while supplier B will source QB from the unreliable tier 2supplier (see Theorem 4). The manufacturer’s profit will be

çM= �j4� − cu5Ks + 41 −�j5

(

� min8D1QA +QB9− cuQB

)

−

(

cu +cr − cu�j

)

QA0

It is straightforward to see that ¡çM/¡QB < 0, and QA +

QB ≤D at optimality, suggesting the solution QA =D−K/2,QB =K/2. Conversely, if the manufacturer allocates QB <K/2,profit will be

çM=�D− cuQB −

(

cu +cr − cu�j

)

4D−QB50

Since cu < cu + 4cr − cu5/�j , the optimal allocation is clearlyQB =K/2, leading to the optimal contracts and profits asstated in the lemma.

Lemma 3. The following identities hold for the thresholds cr ,�L1 �H1 �1, and �2:

(i) The following are equivalent: cu/�1 > �L ⇔ cu/�1 > �H ⇔

�L > �H ⇔ cr < cr .(ii) If K = 0 and �1 = �2, then �1 = �2 = �L.(iii) If K = 0 and �1 <�2, then

(iii-a) �H < �2 < �L < �1 if cr < cu + cu�241 −�25,(iii-b) �2 < �H < �L < �1 if cu + cu�241 −�25 < cr < cr ,(iii-c) �2 < �L < �H < �1 if cr < cr < ¯cr , and(iii-d) �2 < �L < �1 < �H if ¯cr < cr ,

where

¯cr =

cu + cu�1�241 −�1541 −�25

�1 −�22

if �1 >�221

� otherwise0

(iv) In the limit as K ↗D/2, �2 ≤ �1 ≤ �L1 �H .(v) If �2 ≤ 2�1, 0 ≤ d�2/dK ≤ d�1/dK ≤ d�L/dK ≤ d�H/dK.(vi) Both �2 ≤ �1 and �2 ≤ �L.(vii) The inequality �L ≤ �1 holds if and only if �1 ≤ 442D−K5

4D− 2K5/424D−K5255�2.(viii) The inequality �H ≤ �2 holds if and only if cr ≤

cu + cu442�241 −�1541 −�254D− 2K54D−K55/424D−K52 −�14D− 2K542D−K555.

(ix) The inequality �H > �1 holds if and only if �1 >442D−K54D − 2K55/424D−K525�2

2 and cr > cu + cu442�1�241 − �1541 − �254D−K54D − 2K55/42�14D−K52 − �2

242D−K54D− 2K555.

Proof of Lemma 3. Part (i) follows by showing that eachcondition is exactly equivalent to cr < cr . To prove (ii) and (iii),note first that when K = 0, the thresholds become �L = cu +

4cr − cu5/4�1�25, �H = 4cr − cu5/4�1�225− 41 −�1 −�25cu/4�1�25,

�1 = cu + 44cr − cu5/�215, and �2 = cu + 44cr − cu5/�

225. Thus,

�1 = �2 = �L if �1 = �2, proving (ii). To prove (iii), note firstthat �1 <�2 implies �2 < �L < �1. Furthermore, �H < �2 isequivalent to

cr − cu�1�

22

−41 −�1 −�25cu

�1�2< cu +

cr − cu�2

2

⇔ cr < cu +�241 −�25cu0

The latter threshold is always smaller than cr ≡ cu + �2 ·

41−�15cu, since �1 <�2. Therefore, since �H < �L ⇔ cr < cr , bypart (i), we immediately obtain the results in (ii-a) and (ii-b).

Finally, note that �H < �1 is equivalent to

cr − cu�1�

22

−41 −�1 −�25cu

�1�2< cu +

cr − cu�2

1

⇔ 4cr − cu54�1 −�225≤ cu�1�241 −�1541 −�251

which reduces to cr < ¯cr . Since ¯cr ≥ cr always holds, thisyields the results in (ii-c) and (ii-d).

To prove (iv), observe that as K ↘ D/2, �L and �H goto infinity, whereas �1 and �2 remain finite, immediatelyleading to the result.

To prove (v), by differentiating each threshold withrespect to K, we have d�1/dK = 44cr − cu5/�

2154D/24D−K525,

d�2/dK = 44cr − cu5/�2254D/24D−K525, d�L/dK = 44cr − cu5/

�1�254D/4D− 2K525, and d�H/dK = 44cr − cu5/�1�225 ·

4D/4D− 2K525. It follows that d�2/dK ≤ d�1/dKand d�L/dK ≤ d�H/dK. Observe that D/4D− 2K52 >D/424D−K525 and 4d/dK54D/4D− 2K52 −D/424D−K5255 > 0.Hence, a sufficient condition for d�1/dK ≤ d�L/dK is if thisinequality holds when K = 0. In that case, the derivativesare d�1/dK = 44cr − cu5/�

21541/42D55 and d�L/dK = 44cr − cu5/

4�1�25541/D5. The inequality holds if �2 ≤ 2�1.To see (vi), note first that �2 ≤ �1 always holds when

�1 ≤ �2. Also, �2 ≤ �L is equivalent to

cu +cr − cu�2

2

D−K/2D−K

≤ cu +cr − cu�1�2

D−K

D− 2K

⇔ �142D−K54D− 2K5≤ 2�24D−K521

which always holds when �1 ≤ �2.Inequality (vii) follows since �L ≤ �1 is equivalent to

cu +cr − cu�1�2

D−K

D− 2K≤ cu +

cr − cu�2

1

D−K/2D−K

⇔ 24D−K52�1 ≤ 42D−K54D− 2K5�20

To prove (viii), note that �H ≤ �2 is equivalent to

cr −cu�1�

22

D−K

D−2K−cu

1−�1 −�2

�1�2≤cu+

cr −cu�2

2

D−K/2D−K

⇔ 4cr −cu5

[

D−K

D−2K−�142D−K5

24D−K5

]

≤cu�241−�1541−�250

Since the terms in parentheses multiplying cr − cu on the leftare always positive, this yields the desired inequality.

Finally, to prove (ix), note that �H > �1 is equivalent to

cr −cu�1�

22

D−K

D−2K−cu

1−�1 −�2

�1�2>cu+

cr −cu�2

1

D−K/2D−K

⇔ 4cr −cu5

[

D−K

4D−2K5�2−�242D−K5

24D−K5�1

]

>cu41−�1541−�250

For this to hold, the terms in parentheses multiplying cr − cumust be positive, yielding the desired result.

Lemma 4. The following identities hold for the thresholds cr ,�L1 �H1 �1, �2:

(i) ¡cr/¡�1 ≤ 0, ¡�L/¡�1 ≤ 0, ¡�1/¡�1 ≤ 0, ¡�2/¡�1 = 0. Also,¡�H/¡�1 ≤ 0 if cr ≥ cr .

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(ii) ¡cr/¡�2 ≥ 0, ¡�L/¡�2 ≤ 0, ¡�1/¡�1 = 0, ¡�2/¡�2 ≤ 0.Also, ¡�H/¡�2 ≤ 0 if cr ≥ cr .

(iii) ¡�H/¡K ≥ ¡�L11/¡K ≥ ¡�2/¡K ≥ 0 ≥ ¡cr/¡K. Also,¡�L/¡K ≥ ¡�1/¡K if and only if �1 ≥ 44D− 2K52/24D−K525�2.

(iv) Assume the product �1�2 is held constant. Then,¡cr/¡�2 ≥ 0, ¡�L/¡�2 = 0, ¡�1/¡�2 ≥ 0, ¡�2/¡�2 ≤ 0, and¡�H/¡�2 ≤ 0 if cr ≥ cr .

Proof of Lemma 4. (i) It can be readily checked that

¡cr¡�1

= −cu4D− 2K5�2

D−K≤ 03

¡�L

¡�1= −

4cr − cu54D−K5

4D− 2K5�21�2

≤ 03

¡�1

¡�1=

−24cr − cu54D−K/25�3

14D−K5≤ 01

and �2 does not depend on �1, so ¡�2/¡�1 = 0. Finally,¡�H/¡�1 ≤ 0 is equivalent to cr ≥ cu + cu44�241−�254D− 2K55/4D−K55. Since the latter threshold is smaller than cr ≡ cu +

cu44�241 −�154D− 2K55/4D−K55, the last result follows.(ii) Similarly, one can check that

¡cr¡�2

=cu4D−2K541−�15

D−K≥03

¡�L

¡�2=−

4cr −cu54D−K5

4D−2K5�1�22

≤03

¡�2

¡�2=

−24cr −cu54D−K/25�3

24D−K5≤01

and �1 does not depend on �2, so ¡�1/¡�2 = 0. Finally,¡�H/¡�2 ≤ 0 is equivalent to cr ≥ cu + cu44�241−�154D− 2K55/24D−K55. Since the latter threshold is smaller than cr , thelast result follows.

(iii) By taking derivatives with respect to K, we have

¡cr¡K

=−cuD41−�15�2

4D−K52≤03

¡�L

¡K=

4cr −cu5D

4D−2K52�1�2≥03

¡�H

¡K=

4cr −cu5D

4D−2K52�1�22

≥03¡�j

¡K=

4cr −cu5D

24D−K52�2j

≥01 j=1120

It can be readily checked that ¡�H/¡K ≥ ¡�L11/¡K ≥ ¡�2/¡K,and the inequality between ¡�L/¡K and ¡�1/¡K follows.

(iv) Assume �1�2 ≡A is constant. Then,

¡cr¡�2

=cu4D− 2K5

D−K≥ 03

¡�L

¡�2= 03

¡�1

¡�2=

4cr − cu542D−K5�2

A24D−K5≥ 03

¡�2

¡�2= −

4cr − cu542D−K5�2

4D−K5�32

≤ 03

¡�H

¡�2= −

1A

[

4cr − cu54D−K5

4D− 2K5�22

+Acu�2

2

− cu

]

0

Thus, ¡�H/¡�2 ≤ 0 is equivalent to cr ≥ cu + cu4D − 2K5 ·

4�22 −A5/4D−K5. Since �2

2 −A≡ �24�2 −�15≤ �241 −�15, thelatter threshold is smaller than cr , so that ¡�H/¡�2 ≤ 0 ifcr ≥ cr .

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