Mitigating Supply Risk: Dual Sourcing or ProcessImprovement?

Yimin WangW.P. Carey School of Business, Arizona State University

Wendell Gilland, Brian TomlinKenan-Flagler Business School, University of North Carolina

September 26, 2008. Last revised June 1, 2009.

Surveys suggest that supply chain risk is a growing issue for executives and that supplier reliability is of

particular concern. A common mitigation strategy observed in practice is for the buying firm to expend

effort improving the reliability of its supply base. We explore a model in which a firm can source from

multiple suppliers and/or exert effort to improve supplier reliability. For both random capacity and random

yield types of supply uncertainty, we propose a model of process improvement in which improvement efforts

(if successful) increase supplier reliability in the sense that the delivered quantity (for any given order

quantity) is stochastically larger after improvement. We characterize the optimal procurement quantities

and improvement efforts, and generate important insights. For random capacity, improvement is increasingly

favored over dual sourcing as the supplier cost heterogeneity increases but dual sourcing is favored over

improvement if the supplier reliability heterogeneity is high. In the random yield model, increasing cost

heterogeneity can reduce the attractiveness of improvement, and improvement can be favored over dual

sourcing if the reliability heterogeneity is high. A combined strategy (improvement and dual sourcing) can

provide significant value if suppliers are very unreliable and/or capacity is low relative to demand.

1. IntroductionIn the McKinsey & Co. Global Survey of Business Executives (2006), 65% of respondents reported

that their firm’s supply chain risk had increased over the past five years. Moreover, the survey

identified supplier reliability as one of the top three supply chain risks of concern to companies

during their most recent strategic/operational planning cycle. Echoing this finding, a 2008 survey by

the consulting company PRTM found that “companies named on-time delivery of critical products

as well as overall product/supply availability as major risks when globalizing their supply chain,”

(Global Supply Chain Trends 2008-2010, Sixth Annual Survey by PRTM. 2008).

A common, albeit implicit, thread in the growing academic literature on supply risk is that firms

can take operational actions to mitigate their risk, e.g., by dual sourcing or backup sourcing, but

cannot or do not take actions to reduce the underlying delivery risk posed by a particular supplier.

See for example Tomlin (2006), Babich et al. (2007), Dada et al. (2007), and Federgruen and Yang

(2007a). In practice, firms can and do take actions to improve supplier reliability in lieu of or in

addition to sourcing from multiple suppliers:

1

2

Companies have developed numerous ways to minimize disruption related to quality and deliv-

ery issues. Increasing the frequency of on-site audits is the most commonly cited approach,

followed by physical deployment of their company’s resources within the supplier’s location,

increased inspection, and increased supplier training. Other risk mitigation strategies men-

tioned frequently include consistent dual sourcing strategies.

- p. 8, Global Supply Chain Trends 2008-2010, Sixth Annual Survey by PRTM. 2008.

Both Honda and Toyota devote significant resources to improving supplier performance in cost,

quality, and order fulfillment reliability (Handfield et al. 2000, Liker and Choi 2004, Sheffi 2005):

“of the 310 people in Honda’s purchasing department, fifty are engineers who work exclusively with

suppliers,” Handfield et al. (2000) p.44. Supplier improvement is practiced by other automotive

companies, e.g., Daimler, BMW, and Hyundai (Handfield et al. 2000, Wouters et al. 2007), and

many non-automotive companies such as Heineken, Intel, Kimberly Clark, and Siemens (Handfield

et al. 2000, Wouters et al. 2007, SCQI 2009). Some companies single source and work with the sup-

plier to improve performance, e.g. Altera, Daewoo and National Computer Resources (Morris 2006,

Handfield et al. 2000), whereas other companies dual source and work with one or both suppliers

to improve performance, e.g., Honda and Toyota (Liker and Choi 2004). Still other companies dual

source but do not collaborate with suppliers on process improvement (Krause and Ellram 1997,

Krause 1999), with US firms lagging Japanese firms in supplier improvement efforts (Krause et al.

2007).

This paper examines both the process improvement strategy, i.e., exerting effort to increase sup-

plier reliability, and the dual sourcing strategy in the context of a single-product newsvendor with

unreliable suppliers. We explore both the random-capacity and random-yield models of supplier

reliability and propose a model of process improvement in which the firm can exert effort to improve

the reliability of a supplier. The effort may or may not succeed but if it does, then the supplier is

more reliable in the sense that the delivered quantity (for any given order quantity) is stochastically

larger after improvement. As both the improvement and dual sourcing strategies are observed in

practice, the objective of this work is not to determine which strategy is inherently superior, but

rather to identify the circumstances that favor a particular strategy and to determine if and when

firms should deploy both strategies simultaneously. A particular focus is the exploration of whether

and how characteristics of the supply base (cost structure, reliability, degree of heterogeneity in

supplier cost structures and reliabilities, etc.) influence the strategy preference.

In addition to fully characterizing the optimal improvement effort and sourcing quantities, we

establish a number of important managerial insights, including the following. If the two suppliers

3

differ in at most one dimension, e.g., cost or reliability, then increasing heterogeneity (weakly)

increases the expected profit for both the improvement and dual sourcing strategies. If supply

uncertainty is of the random capacity type, then (all else being equal) improvement is preferred

over dual sourcing as supplier cost heterogeneity increases but dual sourcing is favored if reliability

heterogeneity is high. In contrast, if supply uncertainty is of the random yield type, then (all else

being equal), cost heterogeneity can favor dual sourcing and high reliability heterogeneity can favor

improvement. For both the random capacity and random yield models, if suppliers are identical,

dual sourcing is more likely to be preferred in a low-cost supply base but improvement is more likely

to be preferred in a low-reliability supply base. Deploying the combined strategy of improvement

and dual sourcing can add significant value if capacity and reliability are both low.

The rest of this paper is organized as follows. Section 2 surveys the relevant literature, and §3presents the underlying model. For the random capacity model, we characterize and contrast the

improvement and dual-sourcing strategies in §4 and §5 respectively. We consider the combined

strategy in §6. Section 7 examines the random yield model. Section 8 concludes. All proofs can be

found in the appendix.

2. Literature

There exists a large stream of literature that studies unreliable supply. This literature models

reliability in three different but related ways: random capacity, e.g., Ciarallo et al. (1994), Erdem

(1999), random yield, e.g., Gerchak and Parlar (1990), Parlar and Wang (1993), Anupindi and

Akella (1993), Agrawal and Nahmias (1997), Swaminathan and Shanthikumar (1999), Federgruen

and Yang (2007a), and random disruption, e.g., Parlar and Perry (1996), Gurler and Parlar (1997),

Tomlin (2006), Babich et al. (2007). Financial default is another element of supply risk and has

recently been explored by Babich et al. (2007) and Swinney and Netessine (2008).

In the random capacity model, the delivered quantity is the lesser of the order quantity and the

realized supplier capacity (and the capacity is independent of the order size.) In the (proportional)

random yield model, the delivered quantity is a random fraction of the order quantity and the

supplier’s capacity is typically assumed to be infinite. Therefore, the random capacity model differs

from the random yield model primarily in two aspects: whether the delivered quantity is directly

proportional to the order quantity and whether capacity is limited. In the random disruption

model, a supplier is either active or inactive and a firm can only place orders when the supplier is

active. The random disruption model can be seen as a special case of the proportional random yield

model where the realized yield is either 100% or 0%. We note that Dada et al. (2007) investigate

4

a supplier selection problem using a more general model of supplier reliability. Their focus is to

identify supplier rankings in selecting a subset of suppliers. All the above papers treat a supplier’s

reliability as exogenous and, therefore, the improvement strategy is not considered.

There is an extensive empirical and case-based literature on supplier development, e.g., Leenders

and Blenkhorn (1988), Krause (1997), Krause et al. (1998), Krause (1999), Handfield et al. (2000),

Krause et al. (2007), Wouters et al. (2007), and references therein. Handfield et al. (2000) (p.37)

define supplier development as “any activity that a buyer undertakes to improve a supplier’s per-

formance and/or capability to meet the buyer’s short-term or long-term supply needs.” Delivery

reliability (quantity and time) is an important motive for supplier development: “The need for

supplier development resulted from supplier problems that threatened to delay, or even bring to a

standstill, the buying firm’s production,” Krause et al. (1998) p.45. Krause (1997) classifies devel-

opment activities into three types: (a) enforced competition, i.e., multiple sourcing; (b) incentives,

i.e., a promise to the supplier of current or future benefits such as a higher price or volume, and

(c) direct involvement in which the buying firm expends “resources to help the supplier increase

its performance,” Krause (1997) p.15. A recent working paper by Federgruen and Yang (2007b)

explores enforced competition in a random yield setting where suppliers compete on the mean

and variability of production yield. As part of their exploration of joint marketing and inventory

decisions, Liu et al. (2009) investigate the value of higher reliability and examines a case in which

a retailer pays a higher unit price for higher reliability. This is an example of incentives. Neither

Federgruen and Yang (2007b) or Liu et al. (2009) review the supplier development literature or

adopt the classification or terminology of Krause (1997) but we use the classification to position

the research. Our paper explores direct involvement as the mechanism of developing supplier’s

delivery reliability performance. There is empirical evidence that direct involvement may be a more

effective mechanism for improving reliability: in their study of supplier development in the U.S.,

Krause et al. (2007) (p.540) found that “performance outcomes in quality, delivery and flexibility

appear to depend more on direct involvement supplier development than cost performance out-

comes.” We note that Zhu et al. (2007) studies a buying firm’s quality improvement effort at its

supplier. Their focus is to contrast supplier- and buyer-initiated quality improvement efforts in a

single supplier and buyer setting with deterministic demand. They conclude that buyer-initiated

quality improvement is important to achieve higher product quality.

Our paper is also related to the process improvement literature, but the focus is quite different

from the main thrust of that body of work, namely, the design and control of improvement efforts.

Typically the process improvement literature can be categorized into two different streams. The

5

first stream primarily focuses on finding the optimal control policies to improve the production

process while minimizing operating cost, e.g., Porteus (1986), Fine and Porteus (1989), Marcellus

and Dada (1991), Dada and Marcellus (1994), Chand et al. (1996), where process improvement is

typically measured in effective capacity (Spence and Porteus 1987), amount of defects (Marcellus

and Dada 1991), or general cost of failures (Chand et al. 1996). The second stream of the literature

focuses on the interaction of process improvement with the firm’s knowledge creation and learning

curve, e.g., Fine (1986), Zangwill and Kantor (1998), Carrillo and Gaimon (2000), Terwiesch and

Bohn (2001), Carrillo and Gaimon (2004). This stream establishes theoretical foundations for

the evaluation of process improvement benefits. None of the above process improvement papers

considers diversification or dual sourcing strategies.

In summary, the existing supply-uncertainty literature typically focuses on sourcing or inventory

strategies for managing supply risk rather than on reliability improvement. The supplier devel-

opment literature suggests that supplier reliability can be improved and that direct involvement,

i.e., exertion of effort by the buying firm, may be the most effective mechanism. The process

improvement literature focuses on the relative effectiveness of certain process improvement efforts

or particular policies for process improvement, but does not typically consider a firm’s procurement

strategy. Our work contributes to the existing literature by exploring and comparing both dual

sourcing and direct-involvement reliability improvement.

3. Model

In this section, we first introduce the basic features of the model, then describe our model of supplier

reliability and process improvement, and conclude by formulating the firm’s problem. Throughout

the paper we adopt the convention that y+ = max(0, y), E[·] is the expectation operator, and ∇x

denotes a partial derivative with respect to x. We occasionally also use ′ to denote derivatives when

there is no possibility of ambiguity. The terms increasing, decreasing, larger, and smaller are used

in the weak sense throughout this paper.

3.1. Basic features

We study a newsvendor model in which the firm sells a single product over a single selling season.

Let F (·) and f(·) denote the distribution and density function of the demand X, respectively. Also,

let r, v, and p denote the product’s per-unit revenue, salvage value and penalty cost (for unfilled

demand), respectively. The firm can source from two suppliers i = 1,2. Suppliers are unreliable

in that the quantity yi delivered by supplier i is less than or equal to the quantity qi ordered by

the firm. Reliability is further defined below. For a given order quantity qi and realized delivery

6

quantity yi, the firm incurs a total supplier-i procurement cost of (ηiqi + (1− ηi)yi)ci, where ci

is the supplier-i unit cost and 0≤ ηi ≤ 1 is the supplier-i committed cost. The committed cost ηi

reflects the fact that, as discussed in Tomlin and Wang (2005), firms at times incur a fraction of

the procurement cost for undelivered product.

3.2. Supplier reliability and process improvement

We consider two models of supplier reliability in this paper – random capacity and random yield.

For ease of exposition, we focus attention on the random-capacity model and then later, in §7,

establish that many key results can also be proven for the random-yield model.

Let Ki denote supplier i’s design capacity, i.e., the maximum production it can achieve. Supplier

i’s effective capacity is less than or equal to its design capacity due to, for example, random

production technology malfunctions, raw material shortages, or utility interruptions. Let ξi ≥ 0

denote supplier i’s realized capacity loss. Then supplier i’s effective capacity is (Ki− ξi)+. For a

given order quantity qi, supplier i’s delivery quantity is then given by yi = minqi, (Ki− ξi)+. The

firm can exert effort to improve a supplier’s reliability. A natural model of reliability improvement

is one in which the effective capacity is (first-order) stochastically larger after improvement.1 To

formalize this, we associate a reliability index ai with supplier i. For a given value of ai, we let

Gi(·, ai) and gi(·, ai) denote the distribution and density function of the capacity loss ξi, respectively.

An increase in the reliability index, say from ai to ai implies increased reliability in the sense

that Gi(·, ai)≤Gi(·, ai). We assume that the capacity-loss distribution2 is continuous and that the

capacity losses are independent.

Let supplier i’s initial reliability index be given by a0i . The firm can exert effort, e.g., knowledge

transfer or equipment investment (Krause 1997, Krause et al. 1998), to increase supplier i’s reliabil-

ity index. However, improvement efforts can and do fail (Krause et al. 1998, Krause 1999, Handfield

et al. 2000). If the firm exerts an effort level of zi ≥ 0, then supplier i’s capability improves to

ai(zi)≥ a0i with probability θi and remains at a0

i with probability 1− θi. Unless otherwise stated,

we assume that the firm’s improvement cost is linear in its effort, i.e., it costs the firm mizi to

exert effort zi to improve supplier i. All results in the paper extend directly to the case where

1 In their investigation of how supplier reliability influences order quantities in the two-supplier case, Dada et al.(2007) assume that a higher reliability is associated with a stochastically higher effective capacity, although they donot use this exact language. See p. 22.

2 If the supplier capacity loss ξi is exponentially distributed with parameter λ, reliability improvement corresponds toan increase in λ. For a normal distribution with parameter (µ,σ), reliability improvement corresponds to a decreasein µ. For a Weibull distribution with parameter (α,β), reliability improvement corresponds a decrease in β. For auniform distribution (0, b), reliability improvement corresponds to a decrease in b. In each of the above four cases,the reliability index ai is given by λ, 1/µ, 1/β, and 1/b, respectively.

7

the firm’s improvement cost is convex increasing in zi. Consistent with the process improvement

literature, we assume ai(zi) is concave increasing in zi, with ai(0) = a0i . In effect, then, there are

declining returns to improvement efforts. Note that we focus exclusively on the reliability benefit of

process improvement and ignore any additional benefits such as unit procurement cost reductions

or improved payment terms. Additional side benefits from process improvement will only serve to

increase its attractiveness as a strategy.

In summary, suppliers are unreliable in that they suffer a random capacity loss. Supplier i’s

capacity loss, ξi, depends on its reliability index ai which in turn depends on the firm’s improvement

effort zi.

3.3. Problem Formulation

We now describe the firm’s problem. The firm first decides its improvement efforts z = (z1, z2)

and then, after observing the success or failure of these efforts, determines the order quantities

q = (q1, q2). The firm’s problem can therefore be formulated as a two-stage stochastic program. In

the second stage, knowing the realized supplier reliability indices ar = (ar1, a

r2), which determine

the distribution functions for the capacity losses (ξ1, ξ2), the firm’s objective is to determine q≥ 0

so as to maximize its expected profit, i.e., Eξ(ar),X [π(q)], where

π(q) =−∑

i

(ηiqi +(1− ηi)yi)ci + rmin

x,

∑i

yi

+ v

(∑i

yi−x

)+

− p

(x−

∑i

yi

)+

, (1)

x is the realized demand and yi is supplier i’s realized delivery quantity, i.e., yi = minqi, (Ki−ξi)+.We can rewrite (1) in a more compact form by defining ψk ≡−ηkck/(r + p− v) and φk ≡ (r + p−(1− ηk)ck)/(r + p− v). Then,

π(q) = (r + p− v)

∑

i

ψiqi +∑

i

φiyi−(∑

i

yi−x

)+− px. (2)

Letting Π2 (q;ar) denote the second-stage expected profit, i.e., Eξ(ar),X [π(q)], and taking expecta-

tions over the capacity losses ξ = (ξ1, ξ2) and demand X, we have

Π2 (q;ar) = (r + p− v)

∑i

ψiqi +Eξ(ar)

[∑i

φiyi

]−Eξ(ar),X

(∑i

yi−X

)+

− pEX [X]. (3)

Let Π∗2 (ar) denote the optimal second-stage expected profit as a function of the realized reliability

indices, i.e., Π∗2 (ar) = maxq≥0 Π2 (q;ar). Then, the firm’s first-stage expected profit (as a function

of improvement effort), is given by

Π1 (z) =2∑

i=1

−mizi + θ1θ2Π∗2 (a1 (z1) , a2 (z2))+ θ1(1− θ2)Π∗

2

(a1 (z1) , a0

2

)

8

+(1− θ1)θ2Π∗2

(a0

1, a2 (z2))+(1− θ1)(1− θ2)Π∗

2

(a0

1, a02

), (4)

and the firm’s first-stage problem can be written as maxz≥0 Π1 (z). Instead of framing the first-

stage problem in terms of the effort vector z, it is analytically more convenient to frame it in terms

of the reliability indices a(z) = (a1 (z1) , a2 (z2)). We therefore rewrite (4) as

Π1 (a) =2∑

i=1

−mizi (ai)+ θ1θ2Π∗2 (a1, a2)+ θ1(1− θ2)Π∗

2

(a1, a

02

)

+(1− θ1)θ2Π∗2

(a0

1, a2

)+(1− θ1)(1− θ2)Π∗

2

(a0

1, a02

), (5)

where zi (ai) is the effort level associated with the reliability index ai. Note that zi (ai) is a convex

increasing function of ai because ai (zi) is a concave increasing function of zi.

4. Pure Strategies

As discussed in §1, some firms, e.g., Honda and Toyota, engage in both dual sourcing and process

improvement, whereas other firms engage in either dual sourcing or process improvement but not

in both. In this section and the next, we focus on the two pure strategies: (i) dual sourcing without

improvement and (ii) single sourcing with improvement. We first characterize and investigate each

strategy in isolation (§4) and, then, in §5 compare and contrast the strategies to develop insights as

to when a particular strategy is preferred. (We note that because we ignore the supplier-competition

benefits of dual sourcing, there is additional value to dual sourcing that we do not capture.) In

§6 we will return to our general model formulation in which the firm can engage in both process

improvement and dual sourcing.

4.1. Dual sourcing

The firm makes no process improvements in the pure dual-sourcing strategy and, therefore, we

are left with the second-stage problem in which the firm determines the order quantities q =

(q1, q2) to maximize Π2 (q;ar), which is given by (3), and where ar = (a01, a

02). Before characterizing

the optimal solution, we present a technical property that will be useful in establishing several

important directional results in subsequent sections.

Lemma 1. Π2 (q;ar) is submodular in q.

Lemma 1 tells us that, everything else being equal, an increase in qi results in a decrease in the

optimal q∗j , where j denotes the complement of i, i.e., j = 2(1) if i = 1(2). Despite the fact that we

are maximizing a submodular function that is not in general jointly concave, we can establish the

necessary and sufficient condition for the optimal procurement quantity vector q∗.

9

Theorem 1. There exists an optimal procurement vector q∗ that maximizes Π2(q;ar) such that

q∗i = 0, q∗i = K or ∇qiΠ2(q∗;ar) = 0, i = 1,2.

The above theorem uses the unimodal concept (see, for example, Aydin and Porteus (2008)) to

establish the necessary and sufficient condition for q∗. See proof of Theorem 1 for details. To the

best of our knowledge, this work is the first to establish the necessary and sufficient conditions for

the optimal dual sourcing quantities for a random-capacity model.3 While it follows from Theorem

1 and (3) that the optimal (interior) procurement quantity q∗ satisfies

ψi +Gi(Ki− q∗i , a0i )

(φi−Eξ(a0

j) [F (q∗i + yj)])

= 0, i = 1,2, (6)

closed form solutions will not typically exist in general. Using (6), we can establish the following

relationship between the optimal procurement quantity/expected profit and the reliability index.

Lemma 2. (a) The firm’s optimal procurement quantity from supplier i, q∗i , is increasing in

the supplier’s reliability index a0i . (b) The firm’s optimal expected second-stage profit, Π∗

2 (a0i ), is

increasing in the supplier’s reliability index a0i .

The firm pays ηc per unit ordered but not received. As the reliability increases, the expected

quantity of undelivered units (for a given order size q) decreases and so the firm is willing to

order more units. To further explore the firm’s dual-sourcing strategy, for the rest of this section

we assume that demand is deterministic and η = 0, i.e., the firm pays only for what is delivered.

Without loss of generality, we assume φ1G1(K1, a01) ≥ φ2G2(K2, a

02) (if not, reverse the label).

The following theorem characterizes the firm’s optimal procurement quantity q∗ as a function of

demand.

Theorem 2. Assume demand x is deterministic, η = 0, and φ1G1(K1, a01)≥ φ2G2(K2, a

02). The

firm’s optimal procurement vector q∗ is given by

x∈Ω1 ⇒ q∗1 = x, q∗2 = 0;

x∈Ω2 ⇒ q∗1 = x− q∗2 , q∗2 satisfies φ1G1(K1− (x− q∗2)) = φ2G2(K2− q∗2);

x∈Ω3 ⇒ q∗1 = x− (K2−G−12 (φ1)), q∗2 = x− (K1−G−1

1 (φ2));

3 Dada et al. (2007) analyze a related supply uncertainty problem with N suppliers but do not establish necessaryand sufficient conditions for the optimal procurement quantities. On page 21, they write that “Proposition 5 also hastechnical implications. Basically, in contrast to (10) and (11)-which establish necessary, but not sufficient, conditionsfor optimality-Proposition 5 helps establish a sufficient (but not necessary) condition for optimality.” We note thatour equation (6), see later, can be viewed as a generalization of the two-supplier case of equation (A5) in Dada et al.(2007) to allow for the committed cost. We also note that the focus of Dada et al. (2007) is quite different fromours. They mainly focus on the supplier selection problem (and do not consider improvement) while we focus oncontrasting the dual sourcing and improvement strategies.

10

x∈Ω4 ⇒

q∗1 = x− (K2−G−12 (φ1)), q∗2 = K2, if G−1

1 (φ2)≥G−12 (φ1);

q∗1 = K1, q∗2 = x− (K1−G−11 (φ2)), if G−1

1 (φ2) < G−12 (φ1).

x∈Ω5 ∪Ω6 ⇒ q∗1 = K1, q∗2 = K2.

where Ωi, i = 1, . . . ,6, partition the demand space and are given by:

Ω1 : 0≤ x≤K1−G−11

(φ2

φ1

G2(K2))

,

Ω2 : K1−G−11

(φ2

φ1

G2(K2))

< x≤K1 +K2−(G−1

1 (φ2)+G−12 (φ1)

),

Ω3 : K1 +K2−(G−1

1 (φ2)+G−12 (φ1)

)< x≤K1 +K2−min(G−1

1 (φ2) ,G−12 (φ1))

Ω4 : K1 +K2−min(G−11 (φ2) ,G−1

2 (φ1)) < x≤K1 +K2−max(G−11 (φ2) ,G−1

2 (φ1)),

Ω5 : K1 +K2−max(G−11 (φ2) ,G−1

2 (φ1)) < x≤K1 +K2,

Ω6 : x > K1 +K2.

The above theorem indicates that the firm’s procurement strategy is driven largely by the relative

magnitudes of demand and supplier capacities. See Figure 1. When the suppliers’ effective capacities

are high relative to demand (i.e., region Ω1), it is optimal to order a quantity equal to the demand,

but only from the most attractive supplier - the lower cost one if both suppliers have identical

reliabilities. As demand increases, dual sourcing becomes optimal. At first, in region Ω2, it is optimal

to order a total quantity equal to demand. As demand increases further, i.e., region Ω3, the firm

continues to dual source but, for additional protection, orders a total quantity larger than demand.

In other words, it hedges against capacity losses through diversification (dual sourcing) and “over”

ordering. We note that such quantity hedging, i.e., “over” ordering, never occurs in a single-supplier

random-capacity model (Ciarallo et al. 1994). Quantity hedging protects the firm against capacity

shortfall in the event that the other supplier experiences a capacity shortage and hence quantity

hedging is valuable only when there are multiple suppliers. As demand increases even further, i.e.,

regions Ω4 and Ω5, the quantity hedge is constrained by one or more of the supplier capacities, until

eventually (in region Ω6) demand is so high that it exceeds the suppliers’ maximum (i.e., design)

capacity. At this point, a quantity hedge is no longer feasible. Our numerical studies indicated that

in the case of stochastic demand a qualitatively similar progression holds.

The firm avails of dual sourcing and possibly a quantity hedge to mitigate the impact of unreliable

(and/or limited) capacity. As reliability increases, one might therefore anticipate that a firm would

be less likely to engage in these tactics. The following corollary confirms this intuition, but with a

slight caveat for dual sourcing.

Corollary 1. (a) The quantity hedge region, i.e., Ω3 ∪ Ω4 ∪ Ω5, contracts and the quantity

hedge size, i.e., [q∗1 + q∗2 −x]+, decreases as either a01 or a0

2 increases, i.e., as supplier 1’s or 2’s

11

Qua

ntity

Hed

ge

Sin

gle

Sourc

e

Dual S

ourc

e

1 2 3 4 5 6

21 KK

xDemand

* 2

* 1quan

tity

tP

rocu

rem

enq

q

Figure 1 Optimal procurement quantity as a function of demand

reliability increases. (b) The single sourcing region, i.e., Ω1, expands as a01 increases and contracts

as a02 increases.

While the quantity hedge becomes less attractive as either supplier’s reliability increases4, dual

sourcing may become more attractive if the (initially) non-preferred supplier’s reliability increases

sufficiently to compensate for its cost and/or reliability disadvantage. If both suppliers are initially

identical, then dual sourcing becomes less attractive as either supplier’s reliability increases.

4.2. Single Sourcing With Improvement

In the pure improvement strategy the firm can source from only one supplier. Because improvement

efforts are not guaranteed to succeed, the firm may wish to delay commitment to a supplier until

after observing the success or failure of its improvement efforts. We call this late commitment.

Alternatively, the firm may commit to a supplier before observing the improvement outcome. We

call this early commitment. Both early and late commitment are observed in practice. “Interviews

with purchasing managers ... indicated that supplier development is not always successful. If the

effort fails the supplier may be dropped despite the fact that the buying firm has invested in

the supplier,” Krause (1999) p.219. However, “performance improvement sought by buying firms

4 The fact [q∗1 + q∗2 −x]+ decreases as reliability increases should not be confused with Lemma 2(a) which states thatq∗i increases in supplier i’s reliability. Recall that Lemma 1 implies that an increase in q1 results in a decrease in q2

(and vice versa), and so the total order quantity q∗1 + q∗2 can decrease in the reliability.

12

are often only possible when they commit to long-term relationships with key suppliers,” Krause

et al. (2007) p.531, and so some firms prefer to commit to the supplier in advance, e.g., the diesel-

engine manufacturer Varity Corp. (Handfield et al. 2000). In what follows, we first consider early

commitment and then late commitment.

4.2.1. Early Commitment The firm selects one supplier and (after making improvement

efforts) single sources from that supplier. For the moment, let i denote the chosen supplier. After

characterizing the optimal improvement effort, we will discuss supplier selection.

Adapting (3) and (5) to the single-sourcing case, the firm’s second-stage and first-stage expected

profits functions are given by

Π2 (qi;ari ) = (r + p− v)

(ψiqi +φi Eξi(ar

i ) [yi]−Eξi(ari ),X

[(yi−X)+

])− pEX [X],

and

Π1 (ai) =−mizi (ai)+ θiΠ∗2 (ai)+ (1− θi)Π∗

2

(a0

i

), (7)

respectively, where Π∗2 (ai) = maxqi≥0 Π2 (qi;ai). The second-stage single-sourcing problem, i.e.,

maxqi≥0 Π2 (qi;ai), can be viewed as a special case of the dual-sourcing problem characterized in

§4.1, but with the other (not selected) supplier having an infinite procurement cost. All results of

§4.1 therefore apply to this second-stage problem. Now, we are in a position to characterize the

firm’s optimal investment effort.

Theorem 3. Let ∂2Gi(·, ai)/∂a2i ≤ 0, i.e., the supplier’s marginal reliability improvement is

decreasing in the reliability index. If ηi = 0 then the firm’s expected profit, i.e., Π1 (ai), is a concave

function of the reliability index ai. Furthermore, the optimal index a∗i satisfies

mi

θi

∂zi(ai)∂ai

=∫ Ki

Ki−q∗i

((φi−F (Ki− ξi))

∂Gi(ξi, ai)∂ai

)dξi. (8)

The assumption of ∂2Gi(·, ai)/∂a2i ≤ 0 in Theorem 3 is mild as it simply requires a (weakly) decreas-

ing marginal return on improvement efforts, and it is satisfied by, for example, the exponential,

Weibull (for α≤ 1), and uniform distributions. If ηi > 0, then concavity of the profit function can

be established if the capacity loss is uniformly distributed or for more general distributions if the

improvement cost is sufficiently convex in effort. Recall that for expositional ease we assume linear

effort cost in the paper. Closed form solutions for the optimal target reliability a∗i , or equivalently

the improvement effort z∗i , will not typically exist, but sensitivity results can be established.

13

Corollary 2. The firm’s optimal improvement effort z∗i is (a) decreasing in the improvement

cost mi, (b) increasing in the improvement success probability θi, (c) decreasing in the unit pro-

curement cost ci, (d) increasing in the committed cost ηi, and (e) increasing in the unit revenue

r.

To this point, we have characterized the optimal improvement effort if the firm selects supplier

i = 1,2 as its single source. The question remains as to which supplier, 1 or 2, the firm should select.

Numerically, this is readily solved by comparing the optimal expected profit associated with each

supplier. Analytically, we can establish some properties of the optimal selection. In what follows,

we first establish that the potential for reliability improvement can influence supplier selection but

only if the two suppliers differ in more than one dimension. We then investigate supplier selection

when suppliers differ in both cost and reliability.

Let i∗ denote the firm’s preferred supplier if reliability improvement is possible and let j∗ denote

the firm’s preferred supplier if reliability improvement is not possible.

Theorem 4. (a) If suppliers 1 and 2 differ in at most one attribute, then i∗ = j∗. (b) If suppliers

1 and 2 differ in more than one attribute, then i∗ may not equal j∗.

The second result is important from a supplier’s perspective: willingness to collaborate on improve-

ment can be an order winner even if the supplier is less competitive on other dimensions.

Now let us consider the case where the suppliers differ in their unit costs and reliabilities but

have the same committed costs and design capacities. Without loss of generality, we assume that

c1 ≤ c2, i.e., supplier 1 is cheaper than supplier 2. Define δc = c2 − c1 ≥ 0 and δa = a02 − a0

1. Also,

define δa as the threshold value of δa such that supplier 2 is preferred if and only if δa ≥ δa.

Lemma 3. All else being equal, δa is increasing in δc.

In other words, as supplier 2 becomes more expensive relative to supplier 1, the required reliability

difference for supplier 2 to be preferred also increases. In Figure 2 we present δa as a function of δc

when (a) reliability improvement is not possible, and (b) reliability improvement at supplier 2 is

possible. (Figure 2 was obtained by setting c1 = 0.1, m = 1, x = 100, K = 120, G(·, a01)∼ exp(·, a0

1),

and a01 = 2/K.)

Focusing first on Figure 2(a), in which reliability improvement is not possible, we observe a con-

vex increasing switching curve, that is, supplier 2 needs to have a an increasingly large reliability

advantage (δa > 0) to compensate for a cost disadvantage. In Figure 2(b), for which reliability

improvement is possible, we present the switching curve as a function of supplier 2’s improvement

14

0 0.1 0.2 0.3 0.4 0.5

−0.01

0

0.01

0.02

0.03

0.04

0.05

δc

δa

Prefer Supplier 2

Prefer Supplier 1

(a)

0 0.1 0.2 0.3 0.4 0.5

−0.01

0

0.01

0.02

0.03

0.04

0.05

δc

δa

Prefer Supplier 1

Prefer Supplier 2

θ2 increases from 0

to 1 in steps of 0.1

(b)

Figure 2 Supplier switching curve when improvement is not possible (a) and is possible (b)

probability. Two observations are worth noting. First, and different from (a), the switching thresh-

old δa can be negative. This means that a higher cost and lower reliability supplier may be selected

if its reliability can be improved. Second, the switching threshold δa decreases as the probability

of improvement success increases. These and other related findings can be formally established. In

particular, (i) δa > 0 if reliability cannot be improved but δa can be negative if reliability can be

improved; and (ii) ∂δa/∂m2 ≥ 0, ∂δa/∂θ2 ≤ 0, ∂δa/∂m1 ≤ 0, and ∂δa/∂θ1 ≥ 0. (Proofs omitted.) In

other words, a supplier can improve its attractiveness to the buying firm by reducing the firm’s

cost of improvement or by increasing its chances of success. A supplier can even overcome an initial

cost and reliability disadvantage if its improvement potential is sufficiently attractive.

4.2.2. Late commitment In late commitment the firm exerts improvement effort at one or

both suppliers and, after observing improvement outcomes, procures from a single supplier. By

postponing supplier selection, late commitment hedges against uncertain improvement outcomes

and, therefore, (weakly) dominates early commitment. When improvement efforts are guaranteed to

succeed, i.e., θ = 1, then late commitment offers no value over early commitment. In this subsection

we compare late and early commitment and investigate the value of late commitment, i.e., under

what circumstances does it outperform early commitment and how much value does it provide.

In late commitment, the optimal second-stage expected profit Π∗2 (ar

1, ar2) = maxΠ∗

2 (ar1) ,Π∗

2 (ar2)

where Π∗2 (ar

i ) is the optimal second stage profit if the firm single sources from supplier i given

a realized reliability index of ari . Substituting this into (5), we obtain the firm’s first stage profit

function as

ΠL1 (a) =

2∑i=1

−mizi (ai)+ θ1θ2 maxΠ∗2 (a1) ,Π∗

2 (a2)+ θ1(1− θ2)maxΠ∗

2 (a1) ,Π∗2

(a0

2

)

15

+(1− θ1)θ2 maxΠ∗

2

(a0

1

),Π∗

2 (a2)

+(1− θ1)(1− θ2)maxΠ∗

2

(a0

1

),Π∗

2

(a0

2

), (9)

where we use the superscript L to denote that it is the late-commitment profit function. Because

of the max operation in (9), ΠL1 (a) is neither concave nor unimodal in a.

Let a∗Li denote the optimal reliability index for supplier i = 1,2 in late commitment and a∗Ei

denote the optimal reliability index if supplier i = 1,2 is selected in early commitment.

Lemma 4. a∗Li ≤ a∗Ei for i = 1,2

Late commitment targets a (weakly) lower reliability index for a supplier than does early com-

mitment assuming that supplier is selected in early commitment. Equivalently, the firm exerts

(weakly) less effort on improving a supplier. The reason is that in late commitment a particular

supplier, say i, is used if and only if it is preferred in the second stage, whereas in early commit-

ment that supplier (if selected) is guaranteed to be used in the second stage. The uncertainty as

to whether a supplier will be used dampens the firm’s improvement effort.

We now turn our attention to the value provided by late commitment. We use Π∗L1 (Π∗E

1 ) to

denote the optimal late (early) commitment expected profit. Letting j denote the complement of

i, i.e., j = 2(1) if i = 1(2), it follows (almost immediately) from Lemma 4 and (9), that Π∗L1 = Π∗E

1

if Π∗2 (a∗Ei ) < Π∗

2

(a0

j

)for i = 1 or i = 2. That is, late commitment offers no value if one supplier

dominates the other in the sense that the supplier (at its currently reliability) is preferred to the

other supplier at that supplier’s optimum (early commitment) reliability. If such a dominance

exists, there is no value to postponing supplier selection as the optimal selection does not depend

on improvement outcomes. In absence of this dominance, late commitment can provide value.

Theorem 5. Let suppliers 1 and 2 be identical except for their unit costs and let c1 ≤ c2. Then

Π∗L1 > Π∗E

1 if [Π∗2 (a∗E2 )−Π∗

2 (a01)]

+>

mz(a∗E2 )

θ(1−θ).

Corollary 3. (a) If Π∗L1 > Π∗E

1 for some c1 = c1 ≤ c2, then Π∗L1 > Π∗E

1 for all c1 ≤ c1 ≤ c2. (b)

There exists θa and θb with 0 < θa < θb < 1 such that Π∗L1 = Π∗E

1 if θ < θa or θ > θb.

Theorem 5 and its corollary tell us that cost difference, improvement success probability, and

improvement cost all play a role in determining whether late commitment outperforms early com-

mitment. Let us take each in turn. As the difference in unit costs c2 − c1 decreases, the more

expensive supplier requires a smaller reliability advantage to be selected in the second stage (this

follows from Lemma 3). Therefore, the firm is less certain (in the first stage) as to which supplier

will be preferred, and so there is more value in postponing its selection. A similar argument holds

for any supplier parameter: the more similar the suppliers are the more value there is to postponing

16

selection. Late commitment is less likely to offer value when the improvement success probability

θ is very low or very high. By allowing for improvement effort at both suppliers, late commitment

hedges against an improvement failure at one supplier. This hedging benefit is more pronounced

if the probability of one success and one failure is high, as that is the event in which the hedge

is useful. Thus, the hedging benefit is low if θ is very low (very high) because the probability of

both efforts failing (succeeding) is high. As the unit improvement cost m increases, the cost of

experimentation (i.e., improve one or both suppliers before choosing which is best) increases and,

therefore, one might reasonably conjecture that the value of late commitment should decrease in

the improvement cost. We did observe this in our numerical study.

We carried out an extensive study to investigate the value of late commitment. We adopted a

uniform distribution U(0, b) for the capacity loss. Drawing from the process improvement literature,

e.g., Porteus (1986), we used a log function for reliability improvement, i.e., a(z) = a0 +log(1+ z).

Demand was uniformly distributed with a mean of 100 and standard deviation of 30. We set the unit

revenue r = 1, the salvage value v = 0, and the penalty cost p = 0. Suppliers were identical except

in their unit cost c. The design capacity was fixed at K = 120. Initial reliability was parameterized

by the expected relative capacity loss E[ξ]/K. Table 1 summarizes the values of parameters varied

in the study. There were a total of 121,500 problem instances.

Table 1 Study design for late commitment value

Parameter Valuessupplier 2 unit cost c2 0.1 to 0.9 in increments of 0.1supplier 1 unit cost c1 0.1 to c2 in increments of 0.1

improvement success probability θ 0.1 to 1.0 in increments of 0.1expected relative capacity loss E[ξ]/K 0.1 to 0.9 in increments of 0.1

fraction of committed cost η 0.0 to 1.0 in increments of 0.25unit improvement cost m 0.5, 1, 2, 4, 8, 16

The value of late commitment (relative to early commitment), VL = Π∗L1 −Π∗E

1

Π∗E1

, was 4.7% on

average, with a maximum of 90.0% and a minimum of 0.0%. In Figure 3, we show the value of late

commitment as a function of the cost difference ∆c = c2 − c1, success probability θ, improvement

cost m, and initial reliability E[ξ]/K. For example, in Figure 3(a), we plot VL for different values of

the committed cost, and VL represents the average value across all instances with that committed

cost and that cost difference. Other subfigures are similarly constructed. We see that the value

of late commitment is highest when (a) the cost difference is low, (b) the success probability

is not too high or too low, (c) the improvement cost is low, (d) the reliability is low, and (e)

the committed cost is high. Observations (a) to (c) are consistent with our earlier discussion.

17

That is, late commitment is valuable when suppliers are similar, hedging benefits are high, and

experimentation cost is low. In addition, late commitment is more valuable when suppliers are more

unreliable and the committed cost is high. At higher committed costs η, the firm is more sensitive

to low reliability because it incurs a higher cost for undelivered units. Therefore, late commitment

is more valuable as η increases because late commitment gives a higher probability of at least one

supplier having increased reliability.

0 0.2 0.4 0.6 0.8 10%

3%

6%

9%

12%

15%

18%

(b) improvement success probability θ

0 5 10 150%

3%

6%

9%

12%

15%

18%

(c) unit improvement cost m

VL, V

alue

of l

ate

com

mitm

ent

0 0.2 0.4 0.6 0.8 10%

3%

6%

9%

12%

15%

18%

(d) E[ξ]/K(→ decreasing reliability)

0 0.2 0.4 0.6 0.80%

3%

6%

9%

12%

15%

18%

(a) cost difference ∆c

VL, V

alue

of l

ate

com

mitm

ent

η decreases (1:−0.25:0)

η decreases (1:−0.25:0)

η decreases (1:−0.25:0)

η decreases (1:−0.25:0)

Figure 3 The value of late commitment

5. Dual Sourcing or Single Sourcing with Improvement?

Having investigated dual sourcing (DS) and single sourcing with improvement (SSI), we now explore

how attributes of the suppliers, e.g., cost and reliability, influence the firm’s preference for DS or

SSI. Define Π∗SSI and Π∗

DS as the optimal expected profit for the SSI and DS strategies, respectively.

(We do not consider any fixed costs associated with a given strategy as the directional effects of

such fixed costs are obvious.) Before comparing the strategies, we take a brief detour that will be

helpful for interpreting some later findings.

18

The fundamental challenge in a random-supply setting is that the delivered quantity can differ

from the ordered quantity. For the purposes of this discussion, consider any general supply process

that transforms an order q into a delivery y (q). Let us define ρ (q) = y (q)/q as the “yield” for

a given order quantity q. The buying firm faces two challenges: a mean effect and a variability

effect. The mean effect refers to the fact that E [ρ (q)]≤ 1.5 Absent variability, i.e., when ρ (q) is a

deterministic function of q, then the firm can scale its order quantity to ensure it receives exactly

the quantity it desires. However, if the firm faces a positive committed cost, then this scaling can

be expensive as the firm pays for units not received. The variability effect refers to the fact that the

firm is uncertain about the quantity it will receive. This uncertainty results in the firm having to

deal with overage or underage costs even if demand is deterministic. For a deterministic demand,

random yield model with no committed cost, Agrawal and Nahmias (1997) prove that the buying

firm is better off when the mean yield is higher and the yield variance is lower. While it may not be

generally true that variance fully captures the variability effect, it is helpful to frame our discussion

of DS and SSI in terms of how each strategy influences the mean and variance of ρ (q), denoted

as µρ (q) and σ2ρ (q) respectively. In a random capacity setting, both DS and SSI increase µρ (q)

(proof omitted) and reduce σ2ρ (q) (numerically observed) relative to the “original” single supplier

system and, therefore, improve the firm’s performance by mitigating the mean/variability effects

of random capacity.6 DS achieves this by splitting the order between suppliers, whereas SSI does

it by reducing (in a stochastic sense) the supplier’s capacity loss. For later use, we note that if the

two suppliers have identically distributed effective capacities, then the more the order split departs

from an equal division, the less mitigation benefit DS achieves.

5.1. Nonidentical Suppliers

Firms within a given industry may exhibit heterogeneity in their capabilities and performance

due to (among other things) “their history or initial firm endowment,” Barney (2001) p.647, or

their geographic location (Hazra and Mahadevan 2006). Arguing that the internet has facilitated

global sourcing, Hazra and Mahadevan (2006) suggest that supply base heterogeneity has increased

in recent years and that supplier capacity and cost structures are two primary dimensions of

heterogeneity. In this paper, we say that two suppliers are more heterogeneous if they differ more

on a particular attribute. For example, define c1 = c − ∆c and c2 = c + ∆c, then the suppliers

5 In theory a random supply process may result in E [ρ (q)] > 1 but it is natural to consider E [ρ (q)] ≤ 1. The logicpresented can be adapted to the case in which E [ρ (q)] > 1.

6 In comparing DS to the original single-supplier system which is limited to q ≤ K, it is appropriate to comparethe mean and variability for q ≤K. In this case DS increases the mean and reduces the variability. For q > K, DSincreases the mean but may increase the variability because the single source supply system can provide at most K.

19

become more heterogeneous as ∆c increases (0≤∆c < c). We refer to ∆c as the cost heterogeneity

parameter. Analogously to ∆c, define ∆η, ∆a, and ∆K as the committed cost, reliability, and

capacity heterogeneity parameters, respectively.

Theorem 6. Assuming both suppliers are identical except in the pertinent heterogeneity param-

eter, then (a) Π∗SSI is increasing in ∆c, ∆K, ∆η and ∆a, (b) Π∗

DS is increasing in ∆c and ∆η.

Also, Π∗DS is increasing in ∆a if G(·, a) is uniformly distributed.7

One supplier becomes more attractive and one becomes less so as the heterogeneity parame-

ter increases. Being a single-sourcing strategy, SSI benefits as heterogeneity increases because its

preferred supplier becomes more attractive. That DS should benefit from heterogeneity is less obvi-

ous as there is a tension between the increasing attractiveness of one supplier and the decreasing

attractiveness of the other. Focusing on the unit cost, the firm can reduce its average unit cost by

directing more of its order to the lower cost supplier but doing so reduces the mean/variabilility

mitigation benefit associated with dual sourcing. However, the average cost benefit outweighs the

mitigation disadvantage. A similar argument holds for the committed cost.8

More interesting, perhaps, than the effect of supplier heterogeneity on strategy performance is

its effect on strategy preference. We examine this question both numerically and analytically. We

constructed 18,750 base case instances for the study. The underlying study was the same as that

described in §4.2.2 but with the base case parameter values in this study described in Table 2.

For each base case, we studied the impact of heterogeneity in cost, reliability, design capacity, and

committed cost. For each heterogeneity type, we did the following for all 18,750 base case instances:

we created eleven different heterogeneity values (for example, varying the ratio of ∆c/c from 0%

to maximum possible values min(99%, r/c− 1)) and solved for the optimal dual and improvement

strategies at each heterogeneity value.9 For any given heterogeneity value, let %SSI denote the %

of base cases in which SSI was strictly preferred, i.e., Π∗SSI > Π∗

DS. Define %DS similarly and define

%Ind as the % of cases in which the firm was indifferent, i.e., Π∗SSI = Π∗

DS. For each heterogeneity

type, Figure 4 presents %SSI, %DS and %Ind as a function of the pertinent heterogeneity value.

We discuss each heterogeneity type in turn.

7 Recall that for a uniform (0, b) distribution, the reliability index a is given by 1/b. Numerically, we observed thatΠ∗DS was increasing in ∆a even if G(·, a) follows some other types of distributions, e.g., the Weibull distribution.Numerically we observed that Π∗DS was also increasing in ∆K .

8 If supplier 1 and supplier 2 differ in more than one parameter, then increasing heterogeneity can hurt dual sourcingeven if increasing heterogeneity makes supplier 1 unambiguously more attractive and supplier 2 unambiguously lessattractive. Examples available from authors.

9 When exploring committed cost heterogeneity, we excluded the η = 0 and η = 1 cases as we cannot have heterogeneityabout these extreme values.

20

Table 2 Study design for heterogeneous and homogeneous suppliers

Parameters Valuesfraction of committed cost η 0, 0.25, 0.5, 0.75, 1.0

improvement success probability θ 0.1, 0.3, 0.5, 0.7, 0.9unit cost c 0.1, 0.3, 0.5, 0.7, 0.9

unit improvement cost m 0.5, 1, 2, 4, 8design capacity K 20, 50, 80, 120, 160, 200

expected relative capacity loss E[ξ]/K 0.1, 0.3, 0.5, 0.7, 0.9

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

(b) reliability heterogeneity (∆a / a)

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

(c) capacity heterogeneity (∆K / K)

% in

stan

ces

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

(d) committed cost heterogeneity (∆η / η)

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

(a) cost heterogeneity (∆c / c)

% in

stan

ces

% SSI% DS% Ind

% SSI% DS% Ind

% SSI% DS% Ind

% SSI% DS% Ind

Figure 4 Heterogeneous supplier results

We see from Figure 4(a), that SSI is increasingly preferred as cost heterogeneity increases. This

is confirmed analytically by the following theorem.

Theorem 7. If both suppliers are identical except in ci, then Π∗SSI −Π∗

DS is increasing in ∆c.

A partial explanation for this result is that while both strategies benefit from the cheaper supplier,

the benefit for DS is offset somewhat by the more expensive supplier. To fully understand this

result, let us consider the mean/variability mitigation benefit of each strategy. As cost heterogeneity

increases, (i) DS directs a higher fraction of its total order to the lower cost supplier and this

21

reduces its mitigation benefit, and (ii) SSI exerts more improvement effort (see Corollary 2) and this

increases its mitigation benefit. The net effect is that SSI is increasingly preferred as heterogeneity

increases.

As reliability heterogeneity increases, (i) the mitigation benefit of DS decreases (numerically

observed), and (ii) SSI exerts less improvement effort (numerically observed) and this decreases

its mitigation benefit. Thus, the net effect of increasing reliability heterogeneity does not unam-

biguously favor DS or SSI. Under certain conditions, we can establish that increasing reliability

heterogeneity favors DS:

Theorem 8. Assume G(·, a) is uniformly distributed, demand equals x with probability 1, and

θ = 1. If both suppliers are identical except in the reliability ai, then Π∗SSI−Π∗

DS is locally decreasing

in ∆a (i.e., ∂Π∗SSI∂∆a

<∂Π∗DS∂∆a

) if −m∂z(a∗1)

∂∆a< φ∆a

2a(2K −x)2.

However, Π∗SSI −Π∗

DS is not in general decreasing in ∆a. We see from Figure 4(b) that %SSI is not

monotonic decreasing in reliability heterogeneity. Observe that DS is more likely to be preferred

than SSI at a high reliability heterogeneity. At a high heterogeneity, SSI exerts little effort in

improving the already highly reliable supplier and DS procures only a small quantity from the

highly unreliable supplier. While SSI incurs the improvement cost, DS confronts supply risk from

the highly unreliable supplier. However, in a random capacity setting, a highly unreliable supplier

does not necessarily translate to a high supply risk because supply risk depends on the order size:

an unreliable supplier can deliver a very small quantity with a high probability because the realized

capacity loss has to be very high to impact a small order. Thus, the improvement cost disadvantage

of SSI is more significant that the supply risk of DS, and so DS is more likely to be preferred

(albeit not by a very large amount) when reliability heterogeneity is high. In the extreme case of

reliability heterogeneity, i.e., when one supplier is perfectly reliable, DS weakly dominates SSI.

We see from Figure 4(c), that capacity heterogeneity favors SSI. Increasing capacity heterogeneity

reduces the mean/variability mitigation benefits for both DS and SSI. However, the reduction in

the mitigation benefit is more rapid for DS than for SSI and, therefore, SSI is increasingly preferred.

(These statements can be analytically established for the mean effect and were numerically observed

for the variability effect.) In the extreme case of capacity heterogeneity, i.e., when one supplier has

zero capacity, SSI weakly dominates DS.

As committed cost heterogeneity increases, (i) DS directs a higher fraction of its total order

to the lower committed-cost supplier and this reduces its mitigation benefit, and (ii) SSI exerts

less improvement effort (see Corollary 2) and this reduces its mitigation benefit. The net effect

22

is therefore unclear. We see from Figure 4(d) that committed cost heterogeneity has only a weak

effect but that SSI seems to be increasingly preferred as heterogeneity increases.

To this point we have assumed that suppliers differed only on a single dimension and so increasing

heterogeneity made one supplier unambiguously better. When suppliers differ on multiple dimen-

sions, increasing heterogeneity may not favor one supplier. For example, cost and reliability may

be negatively correlated: as the cost difference increases so does the reliability difference with the

lower cost supplier becoming less reliable. In our numerical studies, we observed that cost hetero-

geneity (which favors SSI) had a stronger effect than reliability heterogeneity (which favors DS at

high heterogeneity).

5.2. Identical suppliers

We now explore how supplier attributes influence the firm’s strategy preference (DS or SSI) assum-

ing a homogenous (i.e., identical) supply base. That is, we investigate if and how the firm’s pref-

erence changes as a particular attribute, e.g., supplier cost, is changed simultaneously for both

suppliers. For example, both suppliers might be impacted by increases in energy prices and pass

this price increase on to the firm. As we assume suppliers are identical in this subsection we sup-

press the supplier subscript i = 1,2 on parameters. Π∗SSI increases in the success probability θ and

decreases in the improvement cost m. Therefore, all else being equal, SSI is favored as θ increases

or m decreases. Both Π∗SSI and Π∗

DS decrease in the cost c and committed cost η but increase in the

design capacity K and reliability index a. We conducted a numerical study to explore the impact

on strategy preference. The underlying study was the same as used in the heterogeneous study (see

Table 2) so there were a total of 18,750 problem instances. Analogous to the heterogeneous study,

Figure 5 presents %SSI, %DS and %Ind as a function of the unit cost c, reliability (as measured

by E[ξ]/K), design capacity K and committed cost η.

We see from Figure 5(a) that DS becomes less preferred as the unit cost c increases. The firm’s

order quantity decreases as c increases. Numerically we observed that SSI was better at mitigating

the mean/variability effect at lower order quantities and DS was better at higher order quantities.

This is consistent with the finding that higher unit costs tend to favor SSI. At very high unit

costs, the total order quantity is very low and supply risk is negligible in a random capacity model

(assuming the upper bound on the capacity loss is less than the design capacity), and the firm will

be indifferent between SSI and DS. We did observe this in the numerical study.

We see from Figure 5(b) that SSI is preferred at low reliabilities (as measured by E[ξ]/K) but

SSI is preferred at high reliabilities. This is consistent with our numerical observation that SSI was

23

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(b) E[ξ]/K (→ decreasing reliability)

0 50 100 150 2000%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(c) design capacity K

% in

stan

ces

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(d) committed cost η

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(a) unit cost c

% in

stan

ces

% SSI% DS% Ind

% SSI% DS% Ind

% SSI% DS% Ind

% SSI% DS% Ind

Figure 5 Homogenous supplier results

better at mitigating the mean/variability effect at lower reliabilities and DS was better at higher

reliabilities.

We see from Figure 5(c) that the preference for DS decreases as the design capacity increases.

If E[ξ]/K < 0.5, supply risk decreases as K increases as there is an increasingly large guaranteed

minimum capacity. The firm is then indifferent between DS and SSI for sufficiently high K. If

E[ξ]/K ≥ 0.5, supply risk does not disappear even at very high capacities and SSI is weakly preferred

to DS.

We see from Figure 5(d) that SSI is increasingly preferred as the committed cost η increases.

SSI exerts more improvement effort as η increase (see Corollary 2) and this increases its mitigation

benefit. One mechanism DS uses to manages supply risk is the quantity hedge but this is more

expensive as the committed cost increases. The combined effect is to make SSI increasingly preferred

as η increases.

24

6. Combined Strategy

As discussed in §1, some firms, e.g., Honda and Toyota, engage in both dual sourcing and process

improvement, and we now turn our attention to the general problem in which the firm can com-

bine improvement (of one or both suppliers) with dual sourcing. The following lemma partially

characterizes the firm’s expected profit function, Π1 (a), which was given earlier by (5).

Lemma 5. Π1 (a) is a submodular function in a.

Although Π1 (a) can be component-wise unimodal in its reliability index a, it is not in general

jointly concave.

Our primary objective is to explore the value of deploying a combined strategy. We denote the

optimal expected profit for the combined strategy as Π∗COM . Recall that Π∗

DS and Π∗SSI denote the

optimal expected profits for the pure strategies, dual source and single-source with improvement,

and so Π∗COM ≥maxΠ∗

SSI ,Π∗DS. We define Ψ = (Π∗

COM −maxΠ∗SSI ,Π∗

DS)/maxΠ∗SSI ,Π∗

DS as

the value of the combined strategy, i.e., Ψ is the relative increase in expected profit as compared

to the best (for the given problem instance) of the pure strategies. Unless otherwise stated, Π∗SSI

refers to the early commitment optimal profit. We numerically investigated the value of the com-

bined strategy. Because the experimentation/hedging benefit of the late commitment strategy is

embedded in the combined strategy, and this value is highest when suppliers are identical (§4.2.2),

we focused on identical suppliers in our study. As suppliers become less similar, the value of the

combined strategy will decrease. The underlying study was the same as used in §5 (see Table 2)

and so there were 18,750 problem instances.

The value of the combined strategy, Ψ, was 24.3% on average, with a maximum of 100.0%, and

a minimum of 0.0%. (Using late commitment rather than early commitment, these numbers were

11.7%, 100.0%, and 0.0% respectively.) Because Ψ has such a large range, the interesting question

is when is the value high and when is it low. The factors that had the most pronounced impact

were the design capacity K and the reliability. Using both (a) early and (b) late commitment for

Π∗SSI , Figure 6 presents the value of the combined strategy as a function of K and the reliability (as

measured by the expected relative capacity loss E[ξ]/K). Focusing on (a), we see that if reliability

is very high, i.e., E[ξ]/K = 0.1, then the combined strategy offers negligible value.10 At a moderate

reliability, i.e., E[ξ]/K = 0.3, the combined strategy is beneficial when design capacity is low relative

10 If suppliers are perfectly reliable then the combined strategy cannot outperform dual sourcing and so it has novalue. However, because our model does not consider the supplier-competition benefits of dual sourcing nor the unit-cost reduction benefits of improvement efforts, it understates the potential value in practice. Therefore, a combinedstrategy may be useful in highly reliable supply chains if unit cost is influenced by competition and improvement.

25

to mean demand (which was set to 100 in this study). In this case, the buying firm wants to

access as much effective capacity as possible and therefore wants to dual source and improve both

suppliers. At lower reliabilities, i.e., E[ξ]/K = 0.5,0.7,0.9, the combined strategy retains value even

at design capacities that are twice the mean demand. Focusing on (b), we see a qualitatively similar

finding but with a lower value for combined strategy because late commitment dominates early

commitment.

0 50 100 150 2000%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(a) design capacity K

Ψ, V

alue

of c

ombi

ned

stra

tegy

0 50 100 150 2000%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(b) design capacity K

early commitment late commitment

E[ξ]/K decreasing(0.9 → 0.1)

E[ξ]/K decreasing(0.9 → 0.1)

Figure 6 The value of the combined strategy

In both the dual sourcing and combined strategies, the firm can, but does not have to, source

from both suppliers. When suppliers are identical, Theorem 2 and Corollary 1 imply that the

firm will source from both suppliers at one pair of reliabilities if it does so at any pair of higher

reliabilities. Therefore, if suppliers are identical, the combined strategy sources from both suppliers

only if the dual sourcing strategy does. In our study, dual sourcing used both suppliers in 88.7%

of the instances and the combined strategy did so in 71.1% of instances.

7. Random Yield

We now consider random-yield type of supply uncertainty and adopt the commonly-used model

in which the delivered quantity is stochastically proportional to the order quantity, i.e., yi =

minqi, ξiqi, where ξi now represents supplier-i’s yield rather than its capacity loss.11 The random

yield version of our model is identical to that formulated in §3 but with yi = minqi, ξiqi replacing

11 ξi has a nonnegative support. If the support has an upper bound of 1,yi = minqi, ξiqi can be written as yi = ξiqi.

26

yi = minqi, (Ki− ξi)+ and Gi(·, ai) now referring to the yield distribution (for a given reliability

index ai) rather than the capacity-loss distribution. A natural model of reliability improvement for

random yield is one in which the yield is (first-order) stochastically larger after improvement and,

therefore, an increase in the reliability index, say from ai to ai implies Gi(·, ai)≥Gi(·, ai). Using

this random yield model (hereafter RY), we replicated our earlier analysis of the random capacity

model (hereafter RC). We also replicated all our numerical studies.12 For the sake of brevity, we

focus our discussion here on the key results and, in particular, focus on some important differences

between RC and RY. Appendix B contains full statements and proofs of all theoretical results

referred to in this section.

In RC, both dual sourcing (DS) and single sourcing with improvement (SSI) mitigate the mean

and variability effects of supply risk. However, in RY, (i) DS mitigates the variability effect but

has no impact on the mean effect; and (ii) SSI mitigates the mean effect and (depending on the

yield distribution) may mitigate or amplify the variability effect (proofs omitted).13

Similar to RC, the RY dual-sourcing quantity problem is well-behaved. In fact the expected

profit is jointly concave in the order quantities if Gi has support over [0,1] (see Theorem 9) and

not simply jointly unimodal as for RC.14 Similar to RC, Π∗DS is increasing in a supplier’s reliability

index ai (Lemma 6). Different from RC, however, the quantity procured from a supplier is not

necessarily increasing, nor even monotonic, in ai, and can strictly decrease in ai (Lemma 7).

Similar to RC, the improvement problem is concave subject to certain restrictions (Theorem 10).

Different from RC, however, improvement effort can decrease as the unit cost decreases (Lemma 8).

With regard to late versus early commitment, the findings on the directional impact of improvement

cost, success probability, reliability, supplier cost difference, and committed cost were similar to

those for RC. The value of late commitment was, however, lower in RY, with an average value of

1.4% (compared to 4.7% for RC), a maximum of 87.0% and a minimum of 0.0%.

We next consider the impact of supplier heterogeneity in RY. As with RC, both Π∗DS and Π∗

SSI

increase in heterogeneity (Theorem 6). With regard to the impact of heterogeneity on strategy

12 Each study was the same as that described for the random capacity model except that we assumed a uniformlydistributed random yield rather than a uniformly distributed a capacity loss. The expected yield loss values, i.e.,1− E [ξ] were set to the same values as the expected relative capacity loss E[ξ]/K values in the random capacitystudies. As we did not want the design capacity constraint to obscure the comparison of RC and RY, in our numericalstudies we adopted a random yield model with a design capacity constraint of K, i.e., yi = minqi, ξiqi and qi ≤Ki.

13 SSI mitigates the variability effect if the yield has a uniform (a,1) distribution but amplifies the variability effectif the yield has a Weibull (α,β) distribution (in which case the reliability index is a = β).

14 Parlar and Wang (1993) established joint concavity for a dual-sourcing, random-yield newsvendor when the firmpays fully for the quantity ordered, i.e., η = 1. The joint-concavity result extends readily to the case of η≤ 1.

27

preference, there are some key differences between RC and RY. For RC, increasing cost hetero-

geneity favors SSI (Theorem 7) but for RY increasing cost heterogeneity can favor DS (Theorem

12). The reason for the difference is as follows. In RC, the mean/variability mitigation benefit of

SSI (DS) increases (decreases) in the cost heterogeneity. In RY the mitigation benefit of SSI can

decrease in cost heterogeneity because improvement effort can decrease as cost decreases. Therefore

SSI is sometimes favored as cost heterogeneity increases. Across our 18,750 base case instances,

Π∗SSI −Π∗

DS decreased in cost heterogeneity in only 0.9% of the instances (these were instances

with high θ, high η and low E[ξ]/K), and so the aggregate view presented in Figure 7(a) shows

%SSI increasing in cost heterogeneity.

In RC, high reliability heterogeneity favored DS. We observed the opposite in RY, that is, high

reliability heterogeneity favored SSI (see Figure 7(b)); however, as noted for the random capacity

model, in the extreme case of reliability heterogeneity, i.e., when one supplier is perfectly reliable,

DS weakly dominates SSI. The fundamental distinction between RC and RY that drives this

difference is that the supply risk, i.e., the probability that delivered quantity is less than the ordered

quantity, decreases as the order quantity decreases in RC but remains constant in RY. Therefore,

while a firm can obtain a small quantity with little or no risk of less than full delivery from a

low-reliability supplier in RC, even a small quantity has a high risk of less than full delivery in

RY. Thus, the low-reliability supplier offers negligible diversification value in RY, and this makes

improvement preferred when reliability heterogeneity is high. There is some anecdotal support for

this finding in the experience of Xilinx who moved from a dual source to a single source strategy

due to the large difference in yield performance of its two chip suppliers: “Xilinx originally planned

to use IBM as the secondary source for its most advanced 90nm-based FPGA (field programmable

gate array) - Spartan 3. However, as UMC continues to widen its production volume and yield

advantages over IBM, Xilinx has outsourced all of its orders for 0.13-micron and more advanced

processes to UMC, according to a high-ranking executive at Xilinx in Taiwan,” Yu and Lu (2004).

With regard to the impact of parameters when suppliers are identical, we found two key dif-

ferences between RC and RY. First, the influence of the unit cost c was highly dependent on the

value of the committed cost η in RY. (In RC, DS was less preferred as c increased.) At η = 1 DS

was preferred as c increased but at η = 0 DS was less preferred as c increased. Second, as can be

seen by comparing Figures 5(d) and 8(d), the committed cost η had a much stronger impact in

RY. Because DS mitigates the variability effect but not the mean effect in RY, DS is very sensitive

to the committed cost as it makes scaling the order quantity (to account for the mean yield loss)

expensive.

28

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

(b) reliability heterogeneity (∆a / a)

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

(c) capacity heterogeneity (∆K / K)

% in

stan

ces

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

(d) committed cost heterogeneity (∆η / η)

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

(a) cost heterogeneity (∆c / c)

% in

stan

ces

% SSI% DS% Ind

% SSI% DS% Ind

% SSI% DS% Ind

% SSI% DS% Ind

Figure 7 Heterogeneous supplier results

With regard to the value of the combined strategy in RY, the average value was 10.2% (as

compared to 24.3% for RC), with a maximum of 100.0% and a minimum of 0.0%. Using late

commitment, these numbers were 6.1%, 100.0% and 0.0% respectively. As with RC, the value of

combined strategy was highest when capacity and reliability were low. The two semiconductor

competitors Xilinx and Altera (who both deal with random-yield suppliers) offer an interesting

contrast in supplier strategies: “One place where Xilinx takes a distinctively different tack from

arch-rival Altera is in their foundry strategy. Xilinx uses multiple foundries - today Toshiba and

UMC – claiming that multiple foundries give a hedge against process problems and increase volume

production capability. Altera, on the other hand, claims that working with a single fab - TSMC –

gives them an edge because they can focus their engineering efforts on the capabilities and char-

acteristics of a single supplier, allowing them to converge faster on a high-yield, high-performance

design with each process generation,” (Morris 2006). The Xilinx strategy is consistent with our

findings that the combined strategy is valuable if capacity and reliability are a concern. Altera’s

view suggests that single sourcing might make improvement efforts more efficient (a perspective

not captured in our model) which would lower the value of a combined strategy.

29

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(b) expected yield loss E[ξ] (→ decreasing reliability)

0 50 100 150 2000%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(c) design capacity K

% in

stan

ces

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(d) committed cost η

0 0.2 0.4 0.6 0.8 10%

10%

20%

30%

40%

50%

60%

70%

80%

90%

(a) unit cost c

% in

stan

ces

% SSI% DS% Ind

% SSI% DS% Ind

% SSI% DS% Ind

% SSI% DS% Ind

Figure 8 Homogenous supplier results

8. Conclusions

By relaxing the common assumption in the supply-risk literature that the buying firm does not

influence the reliability of its supply base, this research addresses an important strategy observed in

practice: the exertion of effort to improve a supplier’s reliability performance. We explore a model

in which a buying firm can improve a supplier’s reliability, dual source or deploy both strategies.

Our work identifies a number of interesting managerial findings. For random capacity, improvement

is preferred over dual sourcing as supplier cost heterogeneity increases but dual sourcing is favored

if reliability heterogeneity is high. For random yield, however, cost heterogeneity can favor dual

sourcing and high reliability heterogeneity can favor improvement. If suppliers are homogenous,

low reliability favors improvement whereas low cost or capacity favors dual sourcing. Deploying a

combined strategy can add significant value if capacity and reliability are both low.

We hope this work provides a foundation for future research in the area of reliability improve-

ment. For example, future work could explore the supplier competition benefits of dual sourcing.

Also, if competing firms share suppliers, then one firm’s improvement efforts might spill over and

benefit their competitor. Another extension is to consider a general number of suppliers. While

30

there is anecdotal support for some of our findings, it would be of interest to empirically test

some of the observations to develop a richer understanding of supply risk strategies. We note that

because “most chip makers don’t speak publicly about yield problems” (Weil 2004), it might be

difficult to obtain data on supplier reliability performance.

Acknowledgements

The authors would like to thank the associate editor and the three referees for their comments and

suggestions, which significantly improved this paper.

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33

SUPPLEMENTSAppendix A: Proofs

Without loss of generality, we scale p = 0 and r +p−v = 1 in all our proofs. For notational ease, we suppress

the dependence of ξ on ar in all proofs.

Proof of Lemma 1. The lemma statement is true if and only if ∇2qiqi

Π2(q;ar)−∇2qiqj

Π2(q;ar)≤ 0 for any

i 6= j. Taking the derivative of (3) with respect to qi, we have

∇qiΠ2(q;ar) = ψi + φi Eξ

[dyi

dqi

]−Eξ

[dyi

dqi

F

(∑i

yi

)]. (10)

Recalling that yi = minqi, (Ki− ξi)+, we have

dyi

dqi

= 1 if ξi ≤Ki− qi, anddyi

dqi

= 0 if ξi > Ki− qi. (11)

Substituting (11) into (10), we have

∇qiΠ2(q;ar) = ψi + Gi(Ki− qi, a

ri ) (φi−Eξ [F (qi + yj)]) . (12)

Using (12), we have

∇2qiqi

Π2(q;ar) =−gi(Ki− qi, ari ) (φi−Eξ [F (qi + yj)])−Gi(Ki− qi, a

ri )Eξ [f (qi + yj)] , (13)

∇2qiqj

Π2(q;ar) =−Gi(Ki− qi, ari )Gj(Kj − qj , a

rj)f (qi + qj) . (14)

Noting that Gi(Ki − qi, ari )Gj(Kj − qj , a

rj)f(qi + qj) ≤ Gi(Ki − qi, a

ri )Eξ [f(qi + yj)], the lemma statement

then follows immediately.

Proof of Theorem 1. We prove the theorem statement for the one- and two-supplier cases. If there is only

one supplier then it is straightforward to verify (using (12) and (13)) that Π2(q;ar) is unimodal in q and

therefore (a) and (b) follow immediately from the unimodal property.

Next, we consider the two-supplier case. For any given q1, Π2(q|q1;ar) reduces to a univariate function of

q2. For a fixed q1, the two supplier case then has the same property of the single supplier case, i.e., there exists

a unique q∗2(q1) such that q∗2(q1)∇q2Π2(q∗2(q1)|q1;ar) = 0. Note that q∗2(q1) maximizes Π2(q|q1;ar). Instead of

maximizing Π2(q;ar) directly, we maximize the induced function Π2(q1, q∗2(q1);ar). By the envelope theorem,

∂2Π2(q1, q∗2(q1);ar)

∂q21

=∇2q1q1

Π2(q1, q∗2(q1);ar)+∇2

q1q2Π2(q1, q

∗2(q1);ar)

dq∗2(q1)dq1

. (15)

Setting (12) equal to zero and applying the implicit function theorem, we have

dq∗2(q1)dq1

=−C2

L2

, (16)

where C2 = G2(K2 − q2, ar2)G1(K1 − q1, a

r1)f (q1 + q2) and L2 = g2(K2 − q2, a

r2) (φ2−Eξ [F (y1 + q2)]) +

G2(K2− q2, ar2)Eξ [f (y1 + q2)]. Substituting (13), (14) and (16) into (15), we have

∂2Π2(q1, q∗2(q1);ar)

∂q21

=−L1 +C2

2

L2

,

where L1 =−g1(K1−q1, ar1) (φ1−Eξ [F (q1 + y2)])−G1(K1−q1, a

r1)Eξ [f (q1 + y2)]. Note that Π2(q1, q

∗2(q1);ar)

is unimodal in q1 if and only if −L1L2 + C22 < 0. If q∗2(q1) = 0 then the above reduces to a single variable

problem. If q∗2(q1) > 0, then by (12) and (13) L2 > C2. Analogously, we have L1 > C2. The theorem statement

then follows.

34

Proof of Lemma 2. Part a). Setting (12) equal to zero and applying the implicit function theorem, we

have

dq∗i (ari )

dari

=1|A|

∣∣∣∣∣− ∂hi

∂ai

∂hi

∂qj

− ∂hj

∂ai

∂hj

∂qj

∣∣∣∣∣ ,

where |A| denote the determinant of the hessian of Π2(q;ar) and hi and hj denote the first order derivatives

with respect to qi and qj , respectively. Using (12) to (14), one can verify that ∂hi/∂ari ≥ 0 and ∂hj/∂ar

i < 0.

Recognizing that |A|> 0 and all elements in A are negative, we have dq∗i (ari )/dar

i ≥ 0. Part b). Using (3), we

have

∂Π∗2(a

r)∂ar

i

= φi

dEξ[yi]dar

i

−∂ Eξ,X

[(∑

k yk −X)+]

∂ari

. (17)

One can show that

dEξ[yi]dar

i

=∫ Ki

Ki−q∗i

∂Gi(ξi, ari )

∂ari

dξi, and (18)

∂ Eξ,X

[(∑

k yk −X)+]

∂ari

= Eξ

[∫ Ki

Ki−q∗i

∂Gi(ξi, ari )

∂ari

F (Ki− ξi + yj)dξi

]. (19)

Substituting (18) and (19) into (17), we have

∂Π∗2(ar)

∂ari

= φi

∫ Ki

Ki−q∗i

∂Gi(ξi, ari )

∂ari

dξi−Eξ

[∫ Ki

Ki−q∗i

∂Gi(ξi, ari )

∂ari

F (Ki− ξi + yj)dξi

]

≥ φi

∫ Ki

Ki−q∗i

∂Gi(ξi, ari )

∂ari

dξi−Eξ

[∫ Ki

Ki−q∗i

∂Gi(ξi, ari )

∂ari

F (q∗i + yj)dξi

]

= φi

∫ Ki

Ki−q∗i

∂Gi(ξi, ari )

∂ari

dξi−Eξ [F (q∗i + yj)]∫ Ki

Ki−q∗i

∂Gi(ξi, ari )

∂ari

dξi

= (φi−Eξ [F (q∗i + yj)])∫ Ki

Ki−q∗i

∂Gi(ξi, ari )

∂ari

dξi ≥ 0. (20)

Note that (20) follows from the fact that ∂Gi(·, ari )/∂ar

i ≥ 0, and φi−Eξ [F (q∗i + yj)]≥ 0 by (12).

Proof of Theorem 2. Because η = 0, (6) can be simplified to

Gi(Ki− q∗i )(φi−Eξj

[F (qi + yj)])= 0, i = 1,2. (21)

To simplify notation we suppress the dependence on reliability and simply write Gi(·) = Gi(·, ai). We now

prove the theorem statement by considering different regions of the demand. First note that for any given

ε > 0, when q1 + q2 = x − ε both equations in (21) are strictly positive. We must have φ1G1(K1 − q1) =

φ2G2(K2− q2) unless φ1G1(K1−x)≥ φ2G2(K2)⇒ x≤K1−G−11

(φ2φ1

G2(K2)). Next consider the case when

q1 + q2 > x. Using (6), the optimal procurement quantity is given by

q∗1 = x− (K2−G−12 (φ1)), q∗2 = x− (K1−G−1

1 (φ2)). (22)

For q1 + q2 > x to hold, we must have x > K1 +K2−G−11 (φ2)−G−1

2 (φ1). However, for (22) to be an interior

solution, we must have φ1−G2(K2− (x−K1))≤ 0⇒ x≤K1 +K2−G−12 (φ1), and φ2−G1(K1− (x−K2))≤

0 ⇒ x ≤ K1 + K2 −G−11 (φ2). Therefore, we must have x ≤ K1 + K2 −min(G−1

1 (φ2),G−12 (φ1)) for firm to

35

dual source without hitting the capacity limit from one of the suppliers. Note that when the firm utilize the

fully capacity of one supplier, the total procurement quantity q∗1 + q∗2 > x. Otherwise, suppose q∗i < Ki and

q∗j = Kj (i = 1,2; j = 3− i), then we have q∗i + Kj = (x−Kj + G−1j (φi)) + Kj = x + G−1

j (φi) > x. The case of

both suppliers’ capacities are fully utilized can be similarly proved. The last region in the theorem statement

follows directly from the above analysis. Not all demand regions Ω1 to Ω6 exist for all parameter values. For

example, if the two suppliers are identical, then Ω1 will not exist. The optimal ordering quantity in each

region directly follows from the above analysis.

Proof of Corollary 1. (a) and (b) follow directly from Theorem 2 and fact that G−1i (·) decreases in ai.

Proof of Theorem 3. Using (7), we have

∂Π1(ai)∂ai

=−mi

dzi(ai)dai

+ θi

∂Π∗2(ai)

∂ai

, (23)

∂2Π1(ai)∂a2

i

=−mi

d2zi(ai)da2

i

+ θi

∂2Π∗2(ai)

∂a2i

, (24)

where

∂2Π∗2(ai)

∂a2i

=∂2Π2(qi;ai)

∂a2i

∣∣∣∣qi=q∗

i(ai)

+dq∗i (ai)

dai

∂2Π2(qi;ai)∂qi∂ai

∣∣∣∣qi=q∗

i(ai)

.

Note that by (12) ∂2Π2(qi;ai)

∂qi∂ai|qi=q∗

i(ai) = ∂Gi(Ki−q∗i ,ai)

∂ai(φi − F (q∗i )), which equals to 0 when ηi = 0 and q∗i

satisfies the first order condition (12). In addition,

d2zi(ai)da2

i

=− 1(a′i(z))2

a′′i (z)a′i(z)

. (25)

Substituting (25) and (20) into (24), and recalling that a′i(z)≥ 0 and a′′i (z)≤ 0, we have ∂2Π1(ai)/∂a2i ≤ 0.

Note that (8) can be obtained by setting (23) equal to zero.

Proof of Corollary 2. The corollary statements follow directly by applying the implicit function theorem

on (8).

Proof of Theorem 4. Part a. We prove the theorem statement by contradiction. If the two suppliers are

identical except for one parameter, the firm will always choose the more desirable supplier for its single-

sourcing supplier. Suppose c1 < c2 but the firm choose to make investment z∗ in supplier 2. Since the two

suppliers are identical except for this parameter, the optimal expected profit satisfies Π1(a∗2|c2) < Π1(a∗1|c1)

because ∂Π1(a∗)/∂c < 0. But this contradicts with the fact the firm find it optimal to invest in supplier

2. The theorem statement for other parameters can be analogously proved. Part b. We prove the theorem

statement by constructing a particular example. Assume demand X is deterministic, η = 0, and θ = 1.

Suppose suppliers 1 and 2 differ only in cost and reliability: i.e., c1 < c2 and a1 < a2 = ∞. Note that

φi Eξi[yi] = (r − ci)

∫ K

K−qiGi(ξ, ai)dξ. If improvement is not possible, the firm will choose to source from

supplier 2 if and only if

(r− c2)X > (r− c1)∫ K

K−X

G1(ξ, a1)dξ. (26)

Since X >∫ K

K−XG1(ξ, a1)dξ, it is easy to find parameter values such that c1 < c2 but (26) holds, i.e., the

firm sources from supplier 2. Now, if improvement is possible the firm will either source from supplier 2

36

or improve and source from supplier 1. Using (8), the optimal reliability after improvement is a1 = a01 +

r−c12m

∫ K

K−X

∂G1(ξ,a1)

∂a1dξ. Assume Gi(·, ai) is uniformly distributed with support greater than or equal to K, we

obtain a closed form solution as

a1 = a01 +

12m

12(r− c1)(K2− (K −X)2). (27)

Let ∆ denote the LHS - RHS of (26). If the optimal expected profit with supplier 1 is greater than ∆, then

the firm will source from supplier 1 (after improvement). Substituting (27) into (7), we have

Π1(a∗1) =− 14m1a2

1

[12(r− c1)(K2− (K −X)2)

]2

+[a1 +

12m1

12(r− c1)(K2− (K −X)2)

]× 1

2m1

12(r− c1)(K2− (K −X)2).

If Π1(a∗1) minus the RHS of (26) is greater than ∆, then the theorem statement is true. Note this can be

simplified to 12m1

(1− 1

2a21

)[12(r− c1)(K2− (K −X)2)

]2> ∆, which can be easily satisfied if a1 >

√2/2 and

m is small.

Proof of Lemma 3. Let Π1(a∗i ) denote the optimal expected profit associated with sourcing from supplier

i, where Π1(a∗i ) is given by (7). Let v = Π1(a∗2)−Π1(a∗1) = 0. By the implicit function theorem (IFT), we

have ∂δa/∂δc =−(∂v/∂δc)/(∂v/∂δa). By (7), ∂v/∂δc ≤ 0 and ∂v/∂δa ≥ 0. The lemma statement then follows.

Proof of Lemma 4. By (9), we have ∂ΠL1 (a)

∂ai= −mi

dzi(ai)

dai+ θiθj

∂∂ai

maxΠ∗2 (ai) ,Π∗

2 (aj) + θi(1 −θj) ∂

∂aimax

Π∗

2 (ai) ,Π∗2

(a0

j

). Because ∂

∂aiΠ∗

2 (aj) = 0 and ∂∂ai

Π∗2 (ai) ≥ 0 (Lemma 2), we have ∂ΠL

1 (a)

∂ai≤

−midzi(ai)

dai+ θiθj

∂∂ai

Π∗2 (ai) + θi(1− θj) ∂

∂aiΠ∗

2 (ai) = −midzi(ai)

dai+ θi

∂∂ai

Π∗2 (ai) = ∂ΠE

1 (a)

∂ai. Therefore, because

ΠE1 (a) is decreasing in ai for all ai > a∗E

i , ΠL1 (a) is also decreasing in ai for all ai > a∗E

i . Thus, a∗Li ≤ a∗E

i .

Proof of Theorem 5. Supplier 1 is the optimal selection for early commitment15 and

Π∗E1 =−mz

(a∗E1

)+ θΠ∗

2

(a∗E1

)+(1− θ)Π∗

2

(a01

). (28)

Π∗L1 ≥ΠL

1 (a∗E) because a∗E is a feasible improvement vector for late commitment. Now,

ΠL1

(a∗E

)=

2∑i=1

−mizi

(a∗E

i

)+ θ1θ2 max

Π∗

2

(a∗E1

),Π∗

2

(a∗E2

)+ θ1(1− θ2)max

Π∗

2

(a∗E1

),Π∗

2

(a02

)

+(1− θ1)θ2 maxΠ∗

2

(a01

),Π∗

2

(a∗E2

)+(1− θ1)(1− θ2)max

Π∗

2

(a01

),Π∗

2

(a02

).

Using the fact that suppliers are identical except that c1 < c2, we have Π∗2 (a∗E

1 ) > Π∗2 (a∗E

2 ) and Π∗2 (a∗E

1 ) >

Π∗2 (a0

2). Substituting these into ΠL1 (a∗E), we have,

ΠL1

(a∗E

)=−mz

(a∗E1

)−mz(a∗E2

)+ θΠ∗

2

(a∗E1

)

+ (1− θ)θmaxΠ∗

2

(a01

),Π∗

2

(a∗E2

)+(1− θ)2Π∗

2

(a01

). (29)

Comparing (28) and (29) and rearranging terms, Π∗L1 > Π∗E

1 if [Π∗2 (a∗E

2 )−Π∗2 (a0

1)]+

>mz(a∗2)

θ(1−θ).

15 Although suppliers are identical except for c, to aid exposition in the proof we keep the i subscript on the initialreliability index a0

1 even though a01 = a0

2. The subscript indicates which supplier is selected in the second stage.

37

Proof of Corollary 3. (a) follows from Theorem 5 and fact that Π∗2 (a0

1) is decreasing in c1 while Π∗2 (a∗E

2 )

and z (a∗2) are constant in c1. (b) follows from Theorem 5 and fact that [Π∗2 (a∗E

2 )−Π∗2 (a0

1)]+ is finite while

mz(a∗2)

θ(1−θ)→∞ as θ→ 0 or θ→ 1.

Proof of Theorem 6. We prove the theorem statement for the unit procurement cost. The theorem state-

ments associated with other system parameters can be analogously proved. First consider part (a). Because

suppliers are identical except for the unit procurement cost c, the improvement strategy selects the cheaper

supplier (c −∆c) and we label this supplier as supplier 1. We need the following property to prove the

theorem statement.

Property: ∂Π∗2(a

r)/∂∆c is increasing in ar1.

Proof: By Lemma 2, dq∗1/dar1 ≥ 0. By Lemma 1, Π2(q;ar) is a submodular function in q. It follows that

∂q∗2/∂ar1 ≤ 0. (Note this can also be directly verified by applying the implicit function theorem to (10).)

Using (1) and substituting c1 = c−∆c and c2 = c + ∆c, we have

∂Π∗2(a

r)∂∆c

= η(q∗1 − q∗2)+ (1− η) (Eξ1 [y1]−Eξ2 [y2]) . (30)

From above, dq∗1/dar1 ≥ 0 and dq∗2/dar

1 ≤ 0. Also, Eξ1 [y1] is increasing in ar1 and Eξ2 [y2] is constant in ar

1. We

therefore have ∂Π∗2(a

r)/∂∆c is increasing in ar1.

We are now in a position to prove the theorem statement. Note that

∂Π∗SSI

∂∆c

= θ∂Π∗

2(a1)∂∆c

+ (1− θ)∂Π∗

2(a01)

∂∆c

≥ ∂Π∗2(a

01)

∂∆c

= ηq∗1 +(1− η)Eξ[y1]≥ 0,

where the inequality follows from the above property. Part (b) of the theorem can be similarly proved by

noting that ∂Π∗DS/∂∆c equals ∂Π∗

2(ar)/∂∆c, which is non-negative as established above.

Proof of Theorem 7. We prove the theorem statement by leveraging results in Theorem 6. Consistent with

the proof of Theorem 6, we label the cheaper supplier (c−∆c) as supplier 1.

∂Π∗SSI

∂∆c

= θ∂Π∗

2(a1)∂∆c

+(1− θ)∂Π∗

2(a01)

∂∆c

≥ ∂Π∗2(a0

1)∂∆c

= ηq∗1 + (1− η)Eξ[y1]

≥ η(q∗1 − q∗2)+ (1− η) (Eξ[y1]−Eξ[y2]) =∂Π∗

DS

∂∆c

,

where the first inequality follows from the property proved in Theorem 6, i.e., ∂Π∗2(a1)/∂∆c is increasing in

a1.

Proof of Theorem 8. Recall that for the improvement strategy, the objective function is Π∗SSI = ΠSSI(a∗1) =

−mz(a∗1) + θΠ∗2(a

∗1)+ (1− θ)Π∗

2(a01). Hence, by the envelope theorem we have

∂Π∗SSI

∂∆a

=∂ΠSSI(a∗1)

∂a01

=−m∂z(a∗1)∂a0

1

+(1− θ)∂Π∗

2(a01)

∂a01

=−m∂z(a∗1)∂a0

1

,

where the last step follows from the assumption that θ = 1. Note that because capacity loss is uniformly

distributed, ΠSSI(a1) is inherently concave in a1 and therefore a∗1 is upper bounded. For reasonable forms of

improvement function z(a1),∂z(a∗1)

∂a01

is also bounded above. For the dual sourcing strategy, derivative of the

objective function with respect to ∆a is

∂Π∗DS

∂∆a

=∂Π∗

2(a01, a

02)

∂a01

− ∂Π∗2(a

01, a

02)

∂a02

.

38

Note that by Theorem 2, the firm will not use quantity hedge when demand satisfies x≤K1+K2−(G−11 (φ2)+

G−12 (φ1)). Note that although Theorem 2 assumes η = 0, this will not affect our analysis here since a positive

η only serves to reduce the optimal order quantity. Since the two suppliers are identical except for reliability,

we simply assume x < 2K.

∂Π∗DS

∂∆a

= φ

(q∗1

∂G(K − q∗1, a01)

∂a01

+∫ K

K−q∗1

(K − ξ)∂g(ξ, a0

1)∂a0

1

−q∗2∂G(K − q∗2, a

02)

∂a02

−∫ K

K−q∗2

(K − ξ)∂g(ξ, a0

2)∂a0

2

).

Substituting the uniform distribution U(0,1/a) into the above expression, we can simplify it to

∂Π∗DS

∂∆a

=φ

2(2K −x)(q∗1 − q∗2) > 0, (31)

where the inequality follows from the fact that x < 2K and q∗1 > q∗2. By Theorem 3, we know that q∗2

satisfies φ1G1(K1− (x− q∗2)) = φ2G2(K2− q∗2). Utilizing the uniform distribution, we can explicitly solve q∗2:

q∗2 = (a+∆a)x−2∆aK

2a. It follows that q∗1 = x− q∗2 = (a−∆a)x+2∆aK

2a. Substituting q∗1 and q∗2 into (31), we have

∂Π∗DS

∂∆a

=φ∆a

2a(2K −x)2 > 0,

Therefore, if −m∂z(a∗1)

∂a01

< φ∆a

2a(2K−x)2, then Π∗

SSI −Π∗DS decreases in ∆a. Note that the above condition is

valid since ∂z(a∗1)

∂a01

is bounded. The above condition is more likely to hold when m is small, K is large, (and

θ is large), and ∆a is large.

Proof of Lemma 5. Using (20), we have

∂2Π∗2(ar)

∂ai∂aj

=−∇ajEξ

[∫ Ki

Ki−q∗i

∂Gi(ξi, ai)∂ai

F (Ki− ξi + yj)dξi

]+

dq∗i (aj)daj

∂2Π∗2(ar)

∂ai∂qi

+dy∗j (aj)

daj

∂2Π∗2(ar)

∂ai∂yj

=−∫ Kj

Kj−q∗j

∫ Ki

Ki−q∗i

∂Gj(ξj , aj)∂aj

∂Gi(ξi, ai)∂ai

f (Ki− ξi + Kj − ξj)dξidξj

+dq∗i (aj)

daj

∂Gi(Ki− q∗i , ai)∂ai

(φi−Eξ F (qi + yj))

− dy∗j (aj)daj

∫ Ki

Ki−q∗i

∂Gi(ξi, ai)∂ai

∫ Kj−q∗j

0

f(Ki− ξi + qj)dGj(ξj , aj)dξi. (32)

Note that the first term in (32) is non-positive because ∂Gi(·, ai)/∂ai ≥ 0. The second term is non-positive

because dq∗i (aj)/daj ≤ 0 (by Lemma 1 and Lemma 2(a)), and the third term is non-positive because

dy∗j (aj)/daj ≥ 0. The lemma statement follows because ∂2Π1(a)/∂ai∂aj is a positive, linear combination of

∂2Π∗2(a

r)/∂ai∂aj .

Appendix B: Random Yield - Theorem Statements and Proofs

In this appendix we present the analytical results referenced in the main paper and the associated proofs.

B.1. Dual Sourcing

Theorem 9. If the random yield distribution Gi is bounded by 1, then the firm’s expected second-stage

profit, Π2(q;ar), is jointly concave in q.

39

Proof. Adapting (3) to the random yield model, we have

∂Π2(q;ar)∂qi

= ψi +φi

dEξ[yi]dqi

−∂ Eξ,X

[(∑

k yk −X)+]

∂qi

. (33)

Note that dEξ[yi]/dqi =∫ 1

0ξidGi(ξi, a

ri ), and ∂ Eξ,X

[(∑

k yk −X)+]/∂qi =

∫ 1

0ξi Eξ F (ξiqi + yj)dGi(ξi, a

ri ).

Substituting these two expressions into (33), we have

∂Π2(q;ar)∂qi

= ψi +∫ 1

0

ξi(φi−Eξ F (ξiqi + yj))dGi(ξi, ari ), i = 1,2. (34)

The second order derivatives with respect to q are therefore given by ∂2Π2(q;ar)/∂q2i =−∫ 1

0

∫ 1

0ξ2

i f(ξiqi +

ξjqj))dGj(ξj , arj)dGi(ξi, a

ri ), and ∂2Π2(q;ar)/∂qi∂qj = −∫ 1

0

∫ 1

0ξiξjf(ξiqi + ξjqj))dGj(ξj , a

rj)dGi(ξi, a

ri ).

Clearly, the Hessian is symmetric and all elements are negative. By Proposition 2 of Parlar and Wang (1993),

the determinant of the Hessian is non-negative. The lemma statement then follows.

Lemma 6. If the random yield distribution Gi(ξi, ari ) satisfies ∂Gi(ξi, a

ri )/∂ar

i =−ρξigi(ξi, ari ), where ρ is

a real, non-negative scaler, then the firm’s optimal expected second-stage profit, Π∗2(a

ri ), is increasing in the

supplier’s reliability index ari .

The assumption that ∂Gi(ξi, ari )/∂ar

i = −ρξigi(ξi, ari ) is a mild one, as it is satisfied by many common

distributions such as uniform u(0, a), exponential, and Weibull distributions.

Proof. Adapting (3) to the random yield model, we have

∂Π∗2(ar)

∂ari

= φi

dEξ[yi]dar

i

−∂ Eξ,X

[(∑

k yk −X)+]

∂ari

. (35)

Note that Eξ[yi] =∫ 1

0ξiq

∗i dGi(ξi, a

ri )+ (1−Gi(1, ar

i ))q∗i = q∗i

(1− ∫ 1

0Gi(ξi, a

ri )dξi

). It follows that

dEξ[yi]dar

i

=−q∗i

∫ 1

0

∂Gi(ξi, ari )

∂ari

dξi. (36)

For the second term in (35), note that Eξ,X

[(∑

k yk −X)+]

= Eξ

[(1−Gi(1, ar

i ))∫ q∗i +yj

0(q∗i + yj −x)dF (x)+

∫ 1

0

∫ ξiq∗i +yj

0(ξiq

∗i + yj −x)dF (x)dGi(ξi, a

ri )

], which can be simplified to Eξ

∫ q∗i +yj

0(q∗i + yj − x)dF (x) −

Eξ

∫ 1

0Gi(ξi, a

ri )q

∗i F (ξiq

∗i + yj)dξi. It follows that

∂ Eξ,X

[(∑

k yk −X)+]

∂ari

=−Eξ

[∫ 1

0

∂Gi(ξi, ari )

∂ari

q∗i F (ξiq∗i + yj)dξi

]. (37)

Substituting (36) and (37) into (35), we have

∂Π∗2(a

r)∂ar

i

=−φiq∗i

∫ 1

0

∂Gi(ξi, ari )

∂ari

dξi + Eξ

[∫ 1

0

∂Gi(ξi, ari )

∂ari

q∗i F (ξiq∗i + yj)dξi

]

=−q∗i

∫ 1

0

∂Gi(ξi, ari )

∂ari

[φi−Eξ F (ξiq∗i + yj)]dξi

= ρq∗i

∫ 1

0

ξigi(ξi, ari ) [φi−Eξ F (ξiq

∗i + yj)]dξi ≥ 0. (38)

Note that (38) follows from the assumption that ∂Gi(·, ari )/∂ar

i = −ρξigi(ξi, ari ), and

∫ 1

0ξi[φi −

Eξ F (ξiq∗i + yj)]dGi(ξi, a

ri )≥ 0 by the fact that q∗i satisfies the first order condition (see (34)). We note that

if ηi = 0 then (38) equals to 0, suggesting that in a random yield model, a higher reliability offers no value

when the firm pays only for what is delivered.

40

Lemma 7. For a single supplier case, if (a) Gi(ξi, ari ) ∼ U(ar

i ,1) and ψi +∫ 1

ariξi(φi − F ( ξi

ariF−1(φi +

ψi

ari)))dGi(ξi, a

ri )≤ 0, or (b) Gi(ξi, a

ri )∼U(0, ar

i ) and ψi +∫ ar

i

0ξi(φi−F ( ξi

ariF−1(φi + ψi

ari)))dGi(ξi, a

ri )≥ 0, then

∂q∗i /∂ari ≤ 0.

Note that the condition in part (a) is guaranteed to hold, for example, when ηi = 0. The condition in part

(b) is satisfied, for example, when ψi

ari

+φi ≤ 0 (e.g., when the unit cost ci is high). In addition, if demand is

deterministic, then dq∗i /dari ≤ 0 (see Lemma 8).

Proof. Part (a). By (34), the optimal procurement quantity q∗i satisfies

v = ψi +∫ 1

ari

ξigi(ξi, ari )(φi−F (ξiq

∗i ))dξi = 0. (39)

Applying the IFT to (39), we have dq∗i /dari =−(∂v/∂ar

i )/(∂v/∂qi). By concavity of Π2(q;ar), ∂v/∂qi ≤ 0. The

lemma statement is true if ∂v/∂ari ≤ 0. Note that ∂v/∂ar

i =∫ 1

ariξi

∂gi(ξi,ari )

∂ari

(φi−F (ξiq∗i ))dξi−ar

i gi(ari , a

ri )(φi−

F (ari q∗i )) =− 1

1−ari(ψi +ar

i (φi−F (ari qi))), which is negative if q∗i ≤ 1

ariF−1(φi + ψi

ari). Substituting this expres-

sion into (39) obtains the stated condition stated in part (a). Part (b). The proof is analogous to part (a)

and therefore is omitted.

B.2. Single Sourcing with Improvement

Theorem 10. (i) Let γ = 1 − 2(1 − ai)ψi − (1 − a2i )φi. If demand is deterministic and (a) Gi(·, ai) ∼

U(ai,1) and (√

γ − γ)2 ≥ (1 − ai)2(ψi + aiφi)2, or (b) Gi(·, ai) ∼ U(0, ai), then Π1 (ai) is concave in the

reliability index ai. Furthermore, the optimal index a∗i satisfies

mi

θi

dzi(ai)dai

=

12φiq

∗i − (q∗i −x)2

2(1−ai)2q∗i, if Gi(·, ai)∼U(ai,1);

12(φi− 1)q∗i + x2

2a2iq∗

i, if Gi(·, ai)∼U(0, ai).

(40)

(ii) If η = 0 and Gi(ξi, ari ) satisfies ∂Gi(ξi, a

ri )/∂ar

i = −ρξigi(ξi, ari ), where ρ is a real, non-negative scaler,

then the firm’s optimal expected profit Π1 (ai) is constant in the reliability index ai.

Proof. Part (i) First we note that both (24) and (25) in Theorem 3 apply to the random yield case. Consider

the second term in (25):

∂2Π∗2(ai)

∂a2i

=∂2Π2(qi;ai)

∂a2i

∣∣∣∣qi=q∗

i(ai)

+dq∗i (ai)

dai

∂2Π2(qi;ai)∂qi∂ai

∣∣∣∣qi=q∗

i(ai)

.

By Lemma 8 (see analysis of (41)), the above expression is non-positive under appropriate technical assump-

tions. Note that (40) can be obtained by adapting (23) to the random yield case and setting it equal to zero.

Part (ii). Follows from proof of Lemma 6. See note after (38).

Lemma 8. In the pure improvement strategy, assume demand is deterministic and let ψ′i = dψi

dc, φ′i = dφi

dc,

and γ = 1− 2(1− ai)ψi − (1− a2i )φi. If (a) Gi(ξi, a

ri )∼ U(ar

i ,1) and (√

γ − γ)2 ≥ (1− ai)2(ψi + aiφi)2 and

γ ≤ (ψi +aiφi)(2ψ′iφ′

i+1−ai), or (b) Gi(ξi, a

ri )∼U(0, ar

i ) and ai ≤ 2ψiψ′iψiφ′

i+2(1−φi)

, then the optimal improvement

effort is increasing in the supplier unit cost.

41

Proof. Part (a). Note that ∂Π∗2(a

ri )/∂ar

i = φi∇ariE[yi] −∇ar

iE[(yi −X)+]. Recognizing that Gi(·, ar

i ) has

support over [ari ,1], we have E[yi] =

∫ 1

ariξiq

∗i gi(ξi, a

ri )dξi, and E[(yi −X)+] =

∫ 1

x/q∗i(ξiq

∗i − x)gi(ξi, a

ri )dξi. It

follows that∇ariE[yi] =

∫ 1

ariξiq

∗i

∂gi(ξi,ari )

∂ari

dξi−ari q∗i gi(ar

i , ari ), and∇ar

iE[(yi−X)+] =

∫ 1

x/q∗i(ξiq

∗i −x) ∂gi(ξi,ar

i )

∂ari

dξi.

Hence ∂Π∗2(a

ri )/∂ar

i = 12φiqi− (qi−x)2

2(1−ari)2qi

. Using (7), the optimal reliability satisfies v = ∂Π1(ai)

∂ai=−mi

dzi(ai)

dai+

θi∂Π∗2(ai)

∂ai= 0. By the IFT, we have ∂a∗i /∂c =− ∂v/∂c

∂v/∂ai|ai=a∗

i. In what follows we prove ∂v/∂ai ≤ 0 and ∂v/∂c≥

0. Note that

∂v/∂ai =−mi

d2zi(ai)da2

i

+ θi

(∂2Π∗

2(ai)∂a2

i

∣∣∣∣qi=q∗

i(ai)

+dqi

dai

∂2Π∗2(ai)

∂ai∂qi

∣∣∣∣qi=q∗

i(ai)

). (41)

By (25), the first term in (41) is negative. Next we show the term in the bracket is also negative. Note∂2Π∗2(ai)

∂a2i

|qi=q∗i(ai) =− (qi−x)2

(1−ai)3q∗i, and ∂2Π∗2(ai)

∂ai∂qi|qi=q∗

i(ai) = 1

2φi− q2

i−x2

2(1−ai)2q2i. To find out dqi

dai|qi=q∗

i(ai), first note that

q∗i satisfies w = ∂Π2(qi;ai)

∂qi= ψi + 1

2(1 + ai)φi − 1

2(1−ari)(1− x2

q2i) = 0. Hence, we have dqi

dai|qi=q∗

i(ai) =− ∂w/∂ai

∂w/∂qi=

(φi + −q2i +x2

(1−ai)2q2i) (1−ai)q

3i

x2 . Combining the above together, we can simplify the second term in (41) as u =

− 1(1−ai)qi

( (qi−x)2

(−1+ai)2− q4

i (aiφi+ψi)2

x2 ). Hence, ∂v/∂ai ≤ 0 if (qi−x)2

(−1+ai)2≥ q4

i (aiφi+ψi)2

x2 ⇒ (−1+1/√

γ)2

(−1+ai)2≥ (aiφi+ψi)

2

γ2 .

Therefore, if (√

γ− γ)2 ≥ (1− ai)2(ψi + aiφi)2 then ∂v/∂ai ≤ 0. Now consider ∂v/∂c, we have

∂v/∂c = θi

(∂2Π∗

2(ai)∂ai∂c

∣∣∣∣qi=q∗

i(ai)

+dqi

dc

∂2Π∗2(ai)

∂ai∂qi

∣∣∣∣qi=q∗

i(ai)

). (42)

Note ∂2Π∗2(ai)

∂ai∂c|qi=q∗

i(ai) = 1

2q∗i φ′i. Also, dqi

dc=− ∂w/∂c

∂w/∂q, where ∂w

∂c= ψ′i + 1

2(1 + ai)φ′i and ∂w

∂q=− x2

(1−ai)q3i. Com-

bining the above together, we have ∂v/∂c = 12θiq

∗i (φ′i− (aiφi+ψi)(2ψ′i+(1−ai)φ

′i)

γ). Therefore ∂v/∂c≥ 0 if γφ′i ≥

(aiφi + ψi)(2ψ′i + (1− ai)φ′i)⇒ γ ≤ (ψi + aiφi)(2ψ′iφ′

i+1− ai).

Part (b). Note that E[yi] =∫ ar

i

0ξiq

∗i gi(ξi, a

ri )dξi, and E[(yi − x)+] =

∫ ari

x/q∗iξiq

∗i gi(ξi, a

ri )dξi. It follows that

∇ariE[yi] =

∫ ari

0ξiq

∗i

∂gi(ξi,ari )

∂ari

dξi + ari q∗i gi(ar

i , ari ) = q∗i

2, and ∇ar

iE[(yi − x)+] =

∫x/q∗

iar

i (ξiq∗i − x) ∂gi(ξi,ar

i )

∂ari

dξi +

(ari q∗i − x)gi(ar

i , ari ) = 1

2(q∗i − x2

ari2q∗

i). Hence, we have ∂Π∗2(ar

i )

∂ari

= 12q∗i (φi − 1) + x2

2ari2q∗

i2 . Using (7), the optimal

reliability satisfies v = ∂Π1(ai)

∂ai=−mi

dzi(ai)

dai+ θi

∂Π∗2(ai)

∂ai= 0. By the IFT, we have ∂a∗i /∂c =− ∂v/∂c

∂v/∂ai|ai=a∗

i. In

what follows we prove ∂v/∂ai ≤ 0 and ∂v/∂c≥ 0. We need to prove in (41) the term in the bracket is also

negative. Note ∂2Π∗2(ai)

∂a2i

|qi=q∗i(ai) =− x2

a3iq∗

i, and ∂2Π∗2(ai)

∂ai∂qi|qi=q∗

i(ai) = 1

2(φi− 1)− x2

2a2iq2

i. To find out dqi

dai|qi=q∗

i(ai),

first note that q∗i satisfies w = ∂Π2(qi;ai)

∂qi= ψi + 1

2ai(φi − 1) + 1

2x2

aiq2i

= 0. Hence, we have dqi

dai|qi=q∗

i(ai) =

− ∂w/∂ai

∂w/∂qi= ( 1

2(φi−1)− x2

2aiq2i)aiq3

i

x2 . Combining the above together, we can simplify the second term in (41) as

u = −3x4+a2i q2

i (φi−1)(a2i q2

i (φi−1)−2x2)

4a3iqix2 . Substituting q∗i (which satisfies w = ∂Π2(qi;ai)

∂qi= 0) into u, the numerator

can be simplified to − 4ψix4(3ψi+2ai(φi−1))

(2ψi+ai(φi−1))2, which is always negative. Hence, we have ∂v/∂ai ≤ 0. Now consider

∂v/∂c. With regard to (42), we have ∂2Π∗2(ai)

∂ai∂c|qi=q∗

i(ai) = 1

2q∗i φ′i. Also, dqi

dc=− ∂w/∂c

∂w/∂q, where ∂w

∂c= ψ′i + 1

2aiφ

′i

and ∂w∂q

=− x2

aiq3i. Combining the above together, we have ∂v/∂c = 1

2θiq

∗i

ai(ψiφ′i−2(φi−1))−2ψiψ′iai(ai(φi−1)+2ψi)

. Note that the

denominator is negative, so ∂v/∂c≥ 0 if ai(ψiφ′i− 2(φi− 1))− 2ψiψ

′i ≤ 0⇒ ai ≤ 2ψiψ′i

ψiφ′i+2(1−φ)

.

B.3. Nonidentical Suppliers

Theorem 11. Assuming both suppliers are identical except in the pertinent heterogeneity parameter, then

(a) Π∗SSI is increasing in ∆c, ∆η and ∆a, (b) Π∗

DS is increasing in ∆c, and ∆η. Also, Π∗DS is constant in

∆a if η = 0 and G(·, a)∼U(0, a).

42

Proof. Part (a). First note that ∂Π∗2(ai)/∂∆c = ηq∗i + (1− η)Eξ[yi]≥ 0. Because ∂Π∗

SSI/∂∆c is a positive,

linear combination of ∂Π∗2(ai)/∂∆c, it follows that ∂Π∗

SSI/∂∆c ≥ 0. The theorem statement with respect to

∆η can be analogously proved. The theorem statement with respect to ∆a follows from Lemma 6. Part (b).

First consider ∆c, we have

∂Π∗2(ar)

∂∆c

= η(q∗1 − q∗2)+ (1− η)(Eξ[y1]−Eξ[y2])≥ 0,

where the inequality follows from the fact that q∗1 ≥ q∗2 (recall that we adopt the convention that supplier 1

is the cheaper supplier). The theorem statement then follows. Now consider ∆η, we have

∂Π∗2(a

r)∂∆η

= c(q∗1 − q∗2 −Eξ[y1] + Eξ[y2]))

= c

(q∗1 − q∗2 −

∫ 1

0

ξ1q∗1dG1(ξ1, a

r1)−G1(1, ar

1)q∗1 +

∫ 1

0

ξ2q∗2dG2(ξ2, a

r2)+ G2(1, ar

2)q∗2

)

= c

(G(1, ar)(q∗1 − q∗2)−

∫ 1

0

ξ(q∗1 − q∗2)dG(ξ, ar))

= c(q∗1 − q∗2)∫ 1

0

(1− ξ)dG(ξ, ar)≥ 0.

Now consider ∆a. By assumption, Gi(·, ari )∼U(0, ar

i ) (assuming ari ≤ 1), i = 1,2. Substituting Gi(·, ar

i ) into

(38), we have

∂Π∗2(a

r)∂ar

i

=−q∗i

∫ ari

0

∂Gi(ξi, ari )

∂ari

[φi−Eξ F (ξiq∗i + yj)]dξi

=q∗iar

i

∫ ari

0

ξi [φi−Eξ F (ξiq∗i + yj)]dGi(ξi, a

ri ). (43)

Combining (43) with (34), we have ∂Π∗2(a

r)/∂ari = −ψq∗i /ar

i . It follows that ∂Π∗DS/∂∆a = −ψq∗1/ar

1 +

ψq∗2/ar2 =−ψ(q∗1/ar

1− q∗2/ar2). The theorem statement follows by recognizing that η = 0⇒ψ = 0.

Theorem 12. Let qi uniquely satisfies∫ 1

0ξiF (ξiqi)dG(ξi, a

ri ) = ψi + φi

∫ 1

0ξidGi(ξi, a

ri ). If suppliers differ

only in unit cost c and∫ 1

0

(ξi−E[ξj ])F (ξiqi)dGi(ξi, ari )≤ (ψi−ψj)+ (φi E[ξi]−φj E[ξj ]), (44)

then (a) Π∗SSI −Π∗

DS ≥ 0, and (b) if η = 1 and the conditions in Lemma 7 are also satisfied, then Π∗SSI −Π∗

DS

decreases in ∆c.

Proof. Part (a). Using (34), the optimal q∗i is upper bounded by qi. When suppliers differ only in the unit cost

(and recall we label the cheaper supplier as supplier 1), q∗1 is bounded by q1. If ∂Π2(q;ar)

∂q1|q1=q1

≥ ∂Π2(q;ar)

∂q2|q2=0,

then the optimal procurement quantity q∗ = (q1,0). Hence, when (44) holds, the dual sourcing strategy

degenerates into a single sourcing strategy (from the cheaper supplier). Because the improvement strategy

can be viewed as a relaxation of a pure single sourcing strategy, we have Π∗SSI − Π∗

DS ≥ 0. Part (b). By

(3), we have ∂Π∗SSI/∂∆c = θq∗ + (1− θ)q∗0, where q∗ and q∗0 represents the optimal procurement quantity

when improvement efforts are successful and unsuccessful, respectively. Analogously, we have ∂Π∗DS/∂∆c = q∗0

because by part (a) the dual sourcing strategy single sources from the cheaper supplier. By Lemma 7, we

have q∗ ≤ q∗0. The lemma statement then follows directly.