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MANUFACTURING & SERVICE

OPERATIONS MANAGEMENT

Vol. 7, No. 1, Winter 2005, pp. 3757issn 1523-4614 eissn 1526-5498 05 0701 0037

informs

doi 10.1287/msom.1040.0063 2005 INFORMS

On the Value of Mix Flexibility and Dual Sourcing inUnreliable Newsvendor Networks

Brian Tomlin, Yimin WangKenan-Flagler Business School, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3490

{[email protected], [email protected]}

We connect the mix-flexibility and dual-sourcing literatures by studying unreliable supply chains that pro-duce multiple products. We consider a firm that can invest in product-dedicated resources and totallyflexible resources. Product demands are uncertain at the time of resource investment, and the products can dif-fer in their contribution margins. Resource investments can fail, and the firm may choose to invest in multipleresources for a given product to mitigate such failures.

In comparing a single-source dedicated strategy with a single-source flexible strategy, we refine the commonintuition that a flexible strategy is strictly preferred to a dedicated strategy when the dedicated resources arecostlier than the flexible resource. We prove that this intuition is correct if the firm is risk neutral or if theresource investments are perfectly reliable. The intuition can be wrong, however, if both of these conditionsfail to hold, because there is a resource-aggregation disadvantage to the flexible strategy that can dominatethe demand pooling and contribution-margin benefits of the flexible strategy when resource investments areunreliable and the firm is risk averse.

We investigate the influence that resource attributes, firm attributes, and product-portfolio attributes have onthe attractiveness of various supply-chain structures that differ in their levels of mix flexibility and diversifica-tion, and we investigate the influence these attributes have on the optimal resource investments within a givensupply-chain structure. Our results indicate that the appropriate levels of diversification and flexibility are verysensitive to the resource costs and reliabilities, the firms downside risk tolerance, the number of products, theproduct demand correlations and the spread in product contribution margins.

Key words : reliability; flexibility; dual sourcing; loss aversion; risk

History : Received: June 15, 2004; accepted: December 7, 2004. This paper was with the authors 1 month for2 revisions.

1. IntroductionIt is well established, both in the literature andin practice, that resource flexibility is advantageousfor firms that sell multiple products with uncertaindemand, and that dual sourcing is advantageous forfirms that face uncertainty in supply. Research to date,however, has studied these supply-chain strategies inisolation: The mix-flexibility literature has assumed

perfectly reliable supply, and the dual-sourcing liter-ature has focused on single-product problems.Consider a firm selling multiple products that dif-

fer in their contribution margins (sales price less vari-able costs) and have uncertain demands. The firmmight invest in product-dedicated resources only, or,alternatively, it might invest in one single flexibleresource that can produce all products. The dedicatedresources might be cheaper but the flexible strategy

offers a demand-pooling benefit and a contribution-margin option benefit (Van Mieghem 1998). There is,however, a resource-aggregation disadvantage to theflexible strategy that to the best of our knowledgehas been ignored in the mix-flexibility literature. Asingle failure in the flexible strategy leaves the firmwith no productive resource, whereas with the dedi-cated strategy all resources must fail for the firm to

be in similar straits. Supply uncertainty, by which wemean that the realized resource investment may dif-fer from the target investment, should therefore influ-ence the firms preference for either a dedicated orflexible strategy. The firm, of course, is not limited tothese two strategies, and it may want to consider dualsourcing for one or more products to protect itselfagainst supply uncertainties. The goal of this paperis to simultaneously study mix flexibility and dual

37


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Tomlin and Wang: On the Value of Mix Flexibility and Dual Sourcing in Unreliable Newsvendor Networks38 Manufacturing & Service Operations Management 7(1), pp. 3757, 2005 INFORMS

Figure 1 Four Network Structures

Dual-source flexible

(DF)

Single-source dedicated

(SD)

Resource Product

Dual-source dedicated

(DD)

Single-source flexible

(SF)

1

2

N

1

2

N

N

1

2

N

2

1

sourcing to provide insight into effective supply-chaindesign in the presence of supply and demand uncer-tainties.

Figure 1 illustrates four canonical network struc-tures for a firm that sells N products. Circles represent

products, squares represent resources, and arcs repre-sent the ability of a resource to fulfill demand for aproduct. SD represents a single-source dedicated net-work, SF represents a single-source flexible network,DD represents a dual-source dedicated network, andDF represents a dual source flexible network. We notethat these structures represent possible sourcing deci-sions rather than actual decisions, and so a firm with aDD or a DF network may choose to single source oneor more products, even though it could dual sourcethem.

The dual-sourcing networks (DD and DF) offerdiversification benefits that are advantageous in thepresence of unreliable resource investments. The flex-ible strategies (SF and DF) offer demand-pooling andcontribution-margin benefits that are advantageous inthe presence of demand uncertainty. The demand-pooling benefit arises only if demands are not per-fectly positively correlated. The contribution-margin

benefit arises only if product contribution margins

differ. In the presence of supply uncertainty, thereis a resource-aggregation disadvantage to a flexibleresource. In general, the desirability of any of the fournetworks will be influenced by resource investmentcosts, resource reliabilities, product contribution mar-

gins, demand correlations, and the firms attitudestoward risk.

Mix flexibility, whereby a resource has the abilityto produce multiple products, has been investigatedin the operations literature as a design strategyfor firms that sell multiple products with uncer-tain demand. Hereafter, we will simply use the termflexibility rather than mix flexibility. Such literaturehas primarily focused on single-period (newsvendortype) investments in dedicated and totally flexibleresources, and that is the focus of this paper. We referthe reader to Jordan and Graves (1995), Graves andTomlin (2003), and Muriel et al. (2004) for treatmentsof partial flexibility. For single-period investments indedicated and totally flexible resources, Van Mieghem(2004a) establishes that component commonality andresource flexibility are distinctions without a differ-ence; the problems can be shown to be mathemati-cally equivalent. As such, we use the term resourcewith the understanding that the resource in question


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Tomlin and Wang: On the Value of Mix Flexibility and Dual Sourcing in Unreliable Newsvendor NetworksManufacturing & Service Operations Management 7(1), pp. 3757, 2005 INFORMS 39

might be inventory or capacity. The resource may beproduced in house or may be provided by an outsidesupplier.

The archetypal total-flexibility model (e.g., Fine andFreund 1990) is a single-period, N product, uncertain-demand model in which a risk-neutral firm can investin N dedicated resources and one totally flexibleresource. Investment costs are linear and there are nofixed costs. Gupta et al. (1992), Li and Tirupati (1994,1995, 1997), and Van Mieghem (1998, 2004b) all inves-tigate variations on this theme. The paper of mostdirect relevance is Van Mieghem (1998). In that model,the firm sells two products that differ in contribu-tion margins. This difference in contribution marginsmakes flexibility valuable even if demands for both

products are perfectly positively correlated, a resultthat contradicted the prevailing intuition. Our workrelaxes two implicit assumptions (reliable investmentsand risk neutrality) of these models. There is a bur-geoning literature on non-risk-neutral decision mak-ers in the single-product newsvendor context, e.g.,Eeckhoudt et al. (1995), Agrawal and Seshadri (2000),Schweitzer and Cachon (2000), Caldentey and Haugh(2004), and Chen et al. (2003). As far as we are aware,the only mix-flexibility paper (other than this one) torelax the risk-neutrality assumption is Van Mieghem(2004b). That paper and our paper can be seen ascomplementary, in that the research questions beingaddressed differ, as do the treatments of non-risk-neutral decision makers. Van Mieghem (2004b) inves-tigates how risk aversion influences the flexibilityinvestment levels in perfectly reliable newsvendornetworks by using concave-increasing utility func-tions (to investigate the directional influence of riskaversion) and a mean-variance approach (to investi-gate the magnitude of the influence). In contrast, weinvestigate flexibility and dual sourcing in unreliablenewsvendor networks and, in doing so, allow for non-risk-neutral firms by considering both loss aversion(Kahneman and Tversky 1979) and the ConditionalValue-at-Risk (CVaR) measure (Rockafellar and Urya-sev 2000, 2002).

Unreliable, single-product, single-resource prob-lems have been widely studied in the yield anddisruption literatures. In contrast, unreliable supplychains (multiple resources or multiple products, or

both) have received less attention. Dual-sourcing

strategies have been investigated in the context ofrandom yield (Gerchak and Parlar 1990, Parlar andWang 1993, Anupindi and Akella 1993, Agrawal and

Nahmias 1997, Swaminathan and Shanthikumar 1999,Dada et al. 2003, Tomlin 2004b), random disruptions(Parlar and Perry 1996, Grler and Parlar 1997, Tom-lin 2004a), and credit risk (Babich et al. 2004), but allthese papers assume a single product, so mix flexibil-ity is not relevant. We note that Tomlin (2004a) inves-tigates the value of volume flexibility in unreliablesupply chains.

The rest of the paper is organized as follows.Section 2 introduces the general supply chain model.In 3 we consider the SD and SF networks. In 4 weconsider the DF and DD networks, and compare them

to the single-source networks. Conclusions and direc-tions for future research are presented in 5. Proofs ofall results can be found in Appendix A.

2. The ModelWe present a general model and identify each of thesupply networks (SD, SF, DD, and DF) as instances ofthe general model. There are N products n = 1 N .The marginal contribution margin for product n is pn.We use the notational convention that p1 p2 pN. Let p = p1 pN. All vectors are assumed to

be column vectors, and denotes the transpose oper-ator. The firm can invest in nonnegative levels (Kj)of J different resources labeled j = 1 J . Let T bethe N J technology matrix with tnj = 1 indicat-ing that resource j can produce product n. DemandX = X1 XN is uncertain, with a joint densityfXx1 xN at the time of the investment decision.The demand-correlation matrix is denoted X withelement mn being the correlation coefficient for prod-ucts m and n. The marginal density for Xn is fXn xnand the cumulative distribution function is FXn xn.Realizations of demand are denoted x = x1 xN.

Resource investments are unreliable and the real-ized level Krj for resource j is stochastically propor-tional to the invested level Kj, i.e., Krj = YjKj. Inparticular, we assume a Bernoulli yield model inwhich Yj = 1 with probability j and Yj = 0 with prob-ability 1 j. We refer to j as the reliability of resourcej. There is often a Bernoulli nature to the supply pro-cess (e.g., Anupindi and Akella 1993, Parlar et al. 1995,Swaminathan and Shanthikumar 1999, Dada et al.


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2003, Tomlin 2004b). In the case of inventory, Bernoullisupply processes arise due to batch failures, accep-tance sampling, supply-chain disruptions, or sup-

plier delays (if the delivery occurs too late to servedemand). Unless otherwise stated, we assume that theYj are independent. Let Y = Y1 YJ and let a real-ization be denoted by y = y1 yJ.

Investment costs for an unreliable resource candepend on both the investment (or ordered) level andthe realized (or delivered) level. For each resource j,we assume that the firm pays jcj per unit orderedand an additional 1 jcj per unit delivered. Werefer to cj as the marginal total cost and j as themarginal committed cost. Equivalently, we can thinkof the firm paying cj per unit ordered but receiving a

rebate of 1 jcj for each unit not delivered; in thatcontext 1j is the failure rebate. Let c = c1 cJ, = 1 J, cjyj = cjj + 1 jyj, and cy =c1y1 cJyJ.

The firms investment problem can be formulatedas a two-stage stochastic program. In the secondstage, after demands and investments have been real-ized, the firm allocates production to maximize thecontribution.

rK x y = maxsn qnj0

ps (1)

s.t. sn

xn n=

1 N (2)

sn J

j=1

tnjqnj n = 1 N (3)

Nn=1

qnj yjKj j= 1 J (4)

where qnj denotes the production of product n byresource j, s = s1 sN denotes the sales of prod-ucts 1 N , constraint (2) ensures that sales do notexceed realized demand, constraint (3) ensures thatsales do not exceed production, and constraint (4)

ensures that resource usage does not exceed the real-ized level.Let w0 be the firms initial wealth and let W K be

the random gain or loss (denoted by positive or nega-tive numbers, respectively) achieved by investment K.The firms realized profit on investment K is given bywK = cyK +rK x y. The firms random termi-nal wealth is then w0 + W K. In the first stage, beforedemands and yields are realized, the firm chooses

a nonnegative investment vector K =K1 KJ tomaximize some objective function V K, where V Kdepends on the firms terminal wealth. We consider

three different types of firms: a risk-neutral firm,a loss-averse firm, and a firm concerned about down-side risk, each represented by a distinct form of theobjective function.

The risk-neutral objective function is given byVRNK = w0 + EXYW K, so the risk-neutral invest-ment problem is

VRNK = w0 + max

K0EXYW K (5)

A loss-averse decision maker (Kahneman and Tversky1979) attributes more significance to losses than togains. Schweitzer and Cachon (2000) study loss aver-sion in a classic single-product newsvendor setting

by using a piecewise-linear model that is a spe-cial case of Tversky and Kahnemans (1992) two-partpower-function model. We assume the same piece-wise linear model here; VLAK = w0 + EXYW+K WK where W+K = maxW K 0, WK =maxW K 0 and 1. Increasing loss aversionis associated with an increasing . We note thatVLAK = VRNK at = 1. The loss-averse investmentproblem is

VLAK = w0 + max

K0E

XYW

+K WK (6)

For the terminal wealth distribution associated witha resource-investment vector K, the CVaR, denotedVCVaR K, is the mean of the left -tail of the wealthdistribution. The percentile 0 1 is a param-eter that reflects the firms taste for downside risk.At = 1, the firm is risk neutral, and the com-plete wealth distribution is considered in the objec-tive. For < 1, the firm maximizes the mean ofthe wealth distribution falling below a specified per-centile level . Increasing concern with downside riskis associated with a decreasing percentile . In recent

years, the CVaR measure has gained popularity as arisk measure in the finance literature, e.g., Rockafellarand Uryasev (2000, 2002), Acerbi (2002), Acerbi andTasche (2002), and Szeg (2002). Using Theorem 10 ofRockafellar and Uryasev (2002),

VCVaR K

= w0 +maxv

v +

1

EXYminW Kv0 (7)


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i.e., for a given investment vector K, VCVaR K canbe found by solving a maximization problem. We cantherefore write the CVaR investment problem as

VCVaR K

= w0+maxK0v

v +

1

EXYminW Kv0 (8)VCVaR K is jointly concave over K 0 and v;see Rockafellar and Uryasev (2000, 2002). It is this

joint concavity property, in addition to the fact thatVCVaR K is a coherent measure of risk as defined inArtzner et al. (1999), that has given rise to its popu-larity. The optimal investment is independent of theinitial wealth for all three objective functions, and wetherefore assume that w0 = 0 without loss of generality.

In closing, we note that each of the four supplynetworks (SD, SF, DD, and DF) can be obtained asa special case of the general model presented above.For example, if N = 2, then

SD T =

1 0

0 1

SF T =

1

1

DD T =

1 1 0 0

0 0 1 1

DF T =

1 0 1

0 1 1

3. Single-Source Networks:

The Flexibility PremiumIn this section, we focus on the single-source networksSD and SF. There are N resources in the SD network,and we label these n = 1 N with resource n dedi-cated to product n. There is a single (flexible) resourcein SF, and we label this N+ 1. We focus on the coun-terbalancing effects of demand pooling and resourceaggregation by assuming p1 = p2 = = pN to elim-inate the contribution-option benefit. By eliminatingthis contribution-option benefit, we can model the SFinvestment problem as a single-product problem withdemand

XN+1 =

X1 +

X2 +

XN. We denote the den-

sity and cumulative distribution of total demand byfXN+1 xN+1 and FXN+1 xN+1, respectively.Let VSD and VSF denote the optimal objective

values for the SD and SF networks. SF is strictlypreferred if VSF > VSD , and is weakly preferredif VSF VSD . Hereafter, the term preferred should

be understood to mean weakly preferred. Clearly thefirms preference will depend on the resource costsand the reliabilities, so neither network will dominate

the other in the sense that it is preferred for all pos-sible parameters. To gain insight into what drives afirms network preference, we restrict attention in this

section to the special case where the following threeassumptions all hold.

1. The resource reliabilities are identical for allresources, i.e., 1 = 2 = = N = N+1 = , but theyield random variables are still independent acrossresources.

2. The marginal committed costs are identical forall resources, i.e., 1 = 2 = = N = N+1 = .

3. The marginal total costs are identical for the ded-icated resources, i.e., c1 = c2 = = cN = c.

We now introduce two definitions, the second ofwhich will be a key metric in much of the analysis.

Definition 1. The indifference cost cIN+1 is the valueof the flexible resources marginal total cost at whichthe firm is indifferent between the SD and SF net-works; that is, cIN+1 is the value of cN+1 such thatVSF = VSD .

Definition 2. The flexibility premium is the rela-tive difference in the marginal total costs at which thefirm is indifferent between the SD and SF networks,i.e., = cIN+1 c/c.

The flexibility premium is a useful measure of thevalue of flexibility; the firm prefers the SF network aslong as cN+1 1 + c. Put another way, > 0 implies

that the firm is willing to pay a higher price (rela-tive to the dedicated resources) for flexibility, whereas < 0 implies that the firm requires a lower price forflexibility to be preferred.

We note that although we assume that resourcereliabilities, marginal committed costs, and dedicatedmarginal total costs are identical in this section, manyof the following equations ((9)(24)) can be extendedto the nonidentical case in a straightforward manner.

3.1. Risk-Neutral Firm

This SF investment problem is an extension of the

classic single-product newsvendor model to allow forBernoulli investment failures and the specified invest-ment costs. The objective function is

VSFRNKN+1 = + 1 cN+1KN+1

+ p

KN+10

xN+1fXN+1 xN+1 dxN+1

+ KN+11 FXN+1 KN+1

(9)


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It is relatively straightforward to show that

KN+1 RN = F1

XN+11 + 1 cN+1

p (10)and

VSF RN = pKN+1

0xN+1fXN+1 xN+1 dxN+1 (11)

The SD investment problem can be modeled as Nindependent single-product problems, so

Kn RN = F1

Xn

1

+ 1 cp

n = 1 N (12)

andVSD RN = p

Nn=1

Kn0

xnfXn xn dxn (13)

For the case of N = 2 (i.e., two products), therisk-neutral flexibility premium RN is plotted as afunction of both and in Figures 2 and 3 for inde-pendent uniformly distributed U 0 1 demands.

We see that RN can be increasing or decreasing inboth and , depending on the magnitude of p/c.Observe that RN is constant for = 0 in both cases.This observation is true in general, because the opti-mal resource investment is independent of the relia-

bility (Kn RN = F1Xn 1 c/p) if investment failures arefully rebated, i.e., = 0. Observe that the flexibilitypremium is nonnegative for all combinations in

Figure 2 Flexibility Premium for Risk-Neutral Objective with p/c = 2

0.75 0.80 0.85 0.90 0.95 113%

14%

15%

16%

17%

18%

Reliability,

Flexibility

premium,

= 0

= 0.2

= 0.4

= 0.6

= 0.8

= 1.0

Figure 3 Flexibility Premium for Risk-Neutral Objective with p/c = 5

0.75 0.80 0.85 0.90 0.95 116.0%

16.5%

17.0%

17.5%

18.0%

Reliability,

Flexibilitypremium,

= 0

= 0.2

= 0.8

= 0.6

= 0.4

= 1.0

both figures; that is, SF is always preferred if cN+1 c.In fact, as the following proposition shows, this obser-vation is true in general.

Proposition 1. For any demand random vector X(i) the risk-neutral flexibility premium RN is nonnegative

for all 0 1, (ii) 0 c/p 1 c RN = 0,and (iii) RN = 0 if X = 1, i.e., pairwise perfect positivecorrelation for all products.

This proposition tells us that the SF network is pre-ferred to the SD network for all reliabilities and forall demand distributions in the case of a risk-neutralfirm facing equal resource reliabilities and costs. Thisresult was not obvious a priori (at least to the authors),

because the demand-pooling benefit of SF might beoutweighed by the resource-aggregation disadvan-tage. In fact, even when there is no demand-pooling

benefit (i.e., X = 1), the firm is still indifferent betweenSF and SD. Recall that the contribution-margin ben-efit does not exist because p1 = p2 = = pN. Theonly explanation is that resource aggregation doesnot exclusively penalize SF. Resource aggregation is

a disadvantage for SF from a downside perspectivebecause the probability of multiple resources failing inSD is less than that of the single resource failing in SF,

but an advantage from an upside perspective becausethe probability of the single resource succeeding inSF is higher than the probability of multiple resourcessucceeding in SD. For a risk-neutral firm, the upsideresource-disaggregation advantage, coupled with thedemand-pooling benefit of the SF network, dominates


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the downside resource-disaggregation disadvantage,resulting in SF being preferred to SD. Allowing forasymmetric contribution margins only increases the

advantage of the SF network.Risk neutrality implies a utility function that has a

constant marginal return to wealth. Resource aggrega-tion influences the distribution of the terminal wealth,so it is natural to ask whether alternate wealth pref-erences would change the result that SF is preferredto SD. We now address this question by consideringthe loss-averse and CVaR objective functions.

3.2. Loss-Averse Firm

We assume that N = 2 in this section, but the resultsfor the SF network extend in an obvious fashion to

the case where there are more than two products.Recall that, using our convention, the total demandis labeled X3 and the flexible resource is labeled K3when N = 2 (because N+ 1 = 3).

The SF investment problem is an extension ofthe loss-averse single-product newsvendor model toallow for Bernoulli investment failures and the speci-fied investment costs. The objective function is

VSFLAK3 = EX3 Y3 W3K3

+ 1EX3 Y3

W3K3

W3K3 < 0 (14)

where W3K3 is the random profit realized by aninvestment of K3. Therefore,

VSFLAK3= + 1 c3K3

+ p

K30

x3fX3 x3 dx3 + K31 FX3 K3

+ 1

1 c3K3 +

c3K3FX3 c3K3/p

+ pc3K3/p

0x3fX3 x3 dx3

(15)

and the first and second derivatives are

dVSFLAK3

dK3= p1 FX3 K3 1c3FX3 c3K3/p

+ 1 c3 (16)

d2VSFLAK3

dK23=

pfX3 K3 1c3

fX3

c3K3

p

c3p

0 (17)

The optimal flexible investment K3 is therefore givenby

FX3 KSF 3 LA + 1c3p FX3c3KSF

3 LAp

= 1 + 1 c3

p (18)

and the resulting objective value is

VSF LA = p

KSF 3 LA0

x3fX3 x3 dx3

+ 1c3/pKSF 3 LA

0x3fX3 x3 dx3

(19)

We note that Equations (14) to (19) extend directly to

the case where N > 2, with N+ 1 replacing 3. We alsonote that Equation (18) collapses to the risk-neutraloptimal investment (10) when = 1.

Closed-form solutions for KSF 3 LA and VSF

LA will notexist in general. We have, however, been able toobtain closed-form solutions for the case of X1 and X2having independent uniform distributions.

Proposition 2. Let X1 and X2 have independentU 0 1 distributions. If

p/2 1c3/p3 + 1 c3

then

KSF 3 LA =

21 + 1 c3/3p

1 + 1c3/p3

Otherwise,

KSF 3 LA =

1 +

1

12

1

c3p

3 1

2 +

+ 1 c3p

1/2

1

2 1c3

p 3

11

We now turn to the SD network. Let wnxn yn Kn be the realized profit generated by product n froman investment of Kn in dedicated resource n, and letWnKn be the random profit. Then

wnxnynKn = +1yncKn

+pminxnynKn (20)


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The random total profit for the SD network is

WSDK1 K2 =

W1K1 +

W2K2. The probability of a

loss depends on the realization of

X1,

X2, Y1, and

Y2.

Because losses and gains are weighted differently, theloss-averse SD network cannot be decomposed intotwo single-product loss-averse problems as was pos-sible in the risk-neutral case. The loss-averse objectivefunction is

VSDLA K1 K2 = EXYWSDK1 K2 + 1 EXYWSDK1 K2 WSDK1 K2 < 0

(21)

where

EXYWSDK1K2=2

n=1+1cKn +pLXn Kn+Kn1FXn Kn

(22)

LXn z =z

0xnfXn xndxn (23)

EXYWSDK1K2 WSDK1K2


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Figure 5 Flexibility Premium for Loss-Averse Objective with p/c = 2

and = 08

1 2 3 4 5 6 7 8 9 10

50%

40%

30%

20%

10%

0%

10%

20%

Loss-aversion coefficient,

Flexibilitypremium,

= 0.75

= 0.80

= 0.85

= 0.90

= 0.95

= 1.00

Flexibility premium negative,

i.e., flexible resource needs

to be cheaper to break even

The insight that the flexible resource may have tobe cheaper than the dedicated resources for the firmto prefer SF to SD does not hinge on the choice ofa loss-averse objective function. Qualitatively similarresults can be shown to hold under the CVaR objec-tive; that is the flexibility premium can be negative.Specific propositions and results for the CVaR objec-tive can be found in Appendix B.

3.3. Flexibility Premium inPerfectly Reliable Supply Chains

The reader may have noticed that the loss-averse flex-ibility premium LA is nonnegative everywhere for = 1 in Figures 4 and 5. A similar phenomenon isalso observed for the CVaR objective. Such numeri-cal results suggest that the flexibility premium isalways nonnegative for = 1. This suggestion is con-firmed by the following proposition. Define U1 asthe set of utility functions that are nondecreasing inwealth, i.e., more is (weakly) better.

Proposition 4. Let the firm have an initial wealth ofw0. Let X have any joint distribution. Let 1 = 2 = =N = N+1 = 1. Then (i) 0 for all utility functionsu1 U1, (ii) in particular 0 for the loss-averse objective

function and the CVaR objective function.

Proposition 4 is a quite general result because itmakes no assumptions on the demand distributionand only very mild assumptions on the utility function

(that it be nondecreasing in wealth). Clearly, U1 con-tains all concave increasing utility functions; the com-mon model for risk aversion. In fact U1 contains all

(locally and globally) risk-averse or risk-seeking utilityfunctions as long as uw 0 everywhere.

It is commonly accepted intuition that a flexiblestrategy is preferable to a dedicated strategy if theinvestment costs are equal. This intuition manifestsitself in the literature in the assumption that the unitcost for the flexible resource is higher than for dedi-cated resources. Using Propositions 1 and 4, we canestablish when the common intuition is in fact valid.

Remark 1. For the case of identical resources (mar-ginal total costs, marginal committed costs, and reli-abilities), the SF network is preferred to the SD net-

work if either the resource investments are perfectlyreliable (i.e., = 1) or the firm is risk neutral. If neithercondition holds, then the SD network can be strictlypreferred to the SF network.

Thus, the common intuition is valid if either the riskneutrality or the perfect reliability assumption holds,

but can be incorrect if neither condition holds.

4. Dual-Source Networks: DD and DFWe now consider the DD and DF networks. Wechoose to call these dual-source networks because the

firm can invest in dual resources for any given prod-uct. We do not, however, force the firm to invest inall available resources, so a single-source strategy forone or more products may be optimal.

4.1. Two-Product DF Network

We label the resources as follows: resource 1 is ded-icated to product 1, resource 2 is dedicated to prod-uct 2, and resource 3 is totally flexible. This DFnetwork is an extension of Van Mieghem (1998), here-after referred to as VM98, in that we allow for unre-liable resource investments (with costs that are lin-ear in both the target investment and the realizedinvestment) and non-risk-neutral objective functions.The model collapses to that of VM98 if all reliabilitiesequal one and the objective value is VRNK. VM98restricts attention to c3 > maxc1 c2, because other-wise the flexible resource clearly dominates at leastone of the dedicated resources. We consider all c3 0,

because in our more general model investing in thededicated resources may be optimal even if c3


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minc1 c2. There are eight possible structures to theoptimal solution (corresponding to different combina-tions of positive resource investments), and each of

the eight structures can be optimal depending on themodel parameters.

Let us consider the risk-neutral problem. For anyfeasible investment vector K =K1 K2 K3, the partialderivatives with respect to Kj are given by

VDFRN K

K1= c11 +111

+1

p1

13P1+3P 2

+3p2

12P3+2P 4

VDFRN K

K2

= c22 +122

+2p213P5+3

11P6+P 7

+1

P 4+P 8+P 9

VDFRN K

K3= c33 +133

+3p1

11P10+1P 2

+3p2

1112P11+112

P3+P 12+112P 6

+12P 4+P 9 (25)where each of the k, k = 1 12, is a demand-space region, and the union of the k cover thedemand space. We note that because of the Bernoullifailures, the problem does not lend itself to a strictpartitioning (disjoint k) of the demand space as inFigure 1 of VM98. Expressions for the P k can befound in Appendix C. VDFRN K can be shown to be

jointly concave in K1 K2 K3. Let VDF

RN denote theoptimal objective value and KDFj RN the optimal invest-ment level for resource j= 1 2 3. An interior solution

K

DF

1 RN > 0 K

DF

2 RN > 0 K

DF

3 RN > 0 is optimal if and onlyif the three derivatives in (25) all equal zero for someK1 > 0 K2 > 0 K3 > 0. If such a solution exists, itis unique. If such a solution does not exist, then oneor more of the Kj RN must equal zero and so at leastone of the products is single sourced. Closed-formsolutions for K1 RN K

2 RN K

3 RN do not exist for the

VM98 model and so will not exist for our model. Wecan, however, characterize the directional influence of

prices, marginal total costs, marginal committed-costs,reliabilities, and failure rebates on VDF RN and K

DF j RN .

We first extend Propositions 3 and 4 of VM98 to

characterize the directional influence of marginal totalcosts and prices on VDF RN and K

DF j RN when resource in-

vestments are unreliable.

Proposition 5. For a risk-neutral firm, (i) VDF RN is anonincreasing convex function of the marginal total-cost

vector c and the marginal committed-cost vector , and a

nondecreasing convex function of the contribution-margin

vector p (ii) if the marginal demand density fXn x, n =1 2, is either log-concave or log-convex, then the direc-tional sensitivity of the optimal investment vector KDF RNwith respect to p, c and is given by the following matri-

ces (e.g., KDF 1 RN/c3 0)

pKDF RN =

0 0 0

0 0 0

cKDF RN =

0 0 0

0 0 0

0 0 0

KDF

RN = 0 0 0

0 0 00 0 0

Actual expressions for the sensitivities are available

on request, but are very involved and not particu-larly insightful. Proposition 5 proves that the perfect-reliability results of VM98 for the c and p vectorsstill hold for an unreliable network. As discussed inVM98, the presence of the flexible resource means thatthe optimal investment level for the resource dedi-cated to product n is influenced by the resource ded-icated to product 3 n, n = 1 2. We also see thatthe marginal total costs and committed costs have thesame directional influence. VM98 does not addressmarginal committed costs because such costs are irrel-evant in a perfectly reliable network. We note that theassumption of log-concavity or log-convexity for themarginal demand densities is a mild one that is met

by the Uniform, Normal, Weibull, Gamma, Pareto,and Logistic distributions, as well as by many others.


11/21


Proposition 6. For a risk-neutral firm, (i) VDF RN is anondecreasing convex function of the reliability vector ;

(ii) if the marginal demand densities fXn x are uniform,

then the directional sensitivity of the optimal investmentvector KDF RN with respect to is given by the following

matrix:

KDF RN =

0 0 0

0 0 0

0 0 0

As one would expect, the firm benefits from

increasing resource reliabilities. We again see a cross-dependence for the dedicated resources because of thepresence of the flexible resource. An increase in thereliability of a dedicated resource j = 1 2 increases

the investment in that resource j. This decreases theinvestment in the flexible resource, and as a conse-quence increases the investment in the other dedi-cated resource 3 j.

4.2. Two-Product DD Network

We now consider the DD network in which thereare four dedicated resources (j = 1 2 dedicated toproduct 1 and j= 3 4 dedicated to product 2). Thereare 16 possible structures to the optimal solution(corresponding to different combinations of positiveresource investments); each of the structures can beoptimal depending on the model parameters.

This model is an extension of the single-period ver-sion of Model 1 in Anupindi and Akella (1993), here-after referred to as AA93. Setting X2 = 0 (i.e., singleproduct), 1 = 2 = 0 and V K = VRNK recoversthe AA93 model. For any feasible investment vec-tor K =K1 K2 K3 K4, the partial derivatives (for therisk-neutral objective function) with respect to Kj aregiven by

VDDRN K

Kj= cjj + 1 jj

+ j3j1 FX1 K1 + K2

+ 13jFX1 Kj j = 1 2 (26)VDDRN K

Kj= cjj + 1 jj

+ j7j1 FX2 K3 + K4

+ 17jFX2 Kj j = 3 4 (27)

VDDRN K can be shown to be jointly concave inK1 K2 K3 K4. Let V

DD RN denote the optimal objec-

tive value and KDDj RN the optimal investment level forresource j= 1 4. AA93 proved that there was noclosed-form solution for the optimal investment levels

in their model, and so there will not be a closed-formsolution to our more general model. We can, how-ever, still say something about the directional influ-ence of the costs, reliabilities, and prices on these opti-mal values.

Proposition 7. For a risk-neutral firm, (i) VDD RN is anonincreasing convex function of the marginal total-cost

vector c and the marginal committed-cost vector , and a

nondecreasing convex function of the contribution-margin

vector p and the reliability vector ; (ii) the directionalsensitivity of the optimal investment vector KDD RN with

respect to p, c, , and is given by the following matrices

pKDD RN =

0 0 =0 =0

0 0 =0 =0

cKDD RN =

0 0 =0 =0

0 0 =0 =0

=0 =0 0 0

=0 =0 0 0

KDD RN =

0 0 =0 =0

0 0 =0 =0

=0 =0 0 0

=0 =0 0 0

KDF RN =

0 0 =0 =0

0 0 =0 =0

=0 =0 0 0

=0 =0 0 0

For the DF network, the investment level for dedi-cated resource for product n = 1 2 was influenced bythe costs and reliabilities of the dedicated resource forproduct 3 n, because of the coupling effect of theflexible resource. This proposition tells us that in theDD network, investment levels for a resource dedi-cated to product n = 1 2 are influenced by the otherdedicated resource for product n, but not by the ded-icated resources for product 3 n.


12/21


4.3. Numerical Studies

Sections 4.1 and 4.2 established the directional influ-ence of costs, reliabilities, and prices on the optimal

profits and absolute investment levels within a givennetwork structure, but they did not speak to either theinfluence of model parameters on the relative invest-ment levels within a network structure or the relativeattractiveness of the different network structures.

We address such questions through two numericalstudies. For the case of discrete demand distributions(i.e., when probabilities are characterized by scenariosrather than densities), the firms investment problemcan be formulated as a scenario-based stochastic lin-ear program. Formulations for the general model (foreach of the three objective functions) are presented

in Appendix E. We used this approach in the twonumerical studies.

The first study was designed to address two typesof questions. First, for a given network structure, wewant to understand how the investment mix (i.e., per-centage invested in each resource) is influenced bydemand correlation, contribution-margin difference,reliability, and investment criterion. These were cho-sen because they are key drivers of the attractivenessof flexibility and dual sourcing. Second, we want tounderstand the relative value of the various networkstructures. For example, consider a firm that currentlyoperates an SD network. How much benefit does itget by moving to a dual-sourcing network? Whichdual-sourcing network is preferred, and under whatcircumstances? We fixed the number of products atN = 2 in this study. The second study was designedto investigate the relative performance of the four net-works as the number of products increases.

We now describe the design and discuss the resultsfor the first study. The demand distribution for eachproduct was characterized by 200 demand scenarios.The demand scenarios were drawn randomly froma bivariate normal distribution. Because the demandvariance was not a focus of this study, we fixed thedemand mean and standard deviation to be 100 and30 respectively, for both products. We varied the cor-relation coefficient () from 1 to 1 in increments of1/3, giving us a total of seven demand distributionsthat varied in their correlation coefficient. The actualcorrelations were 100, 069, 038, 004, 031,066, and 100. The contribution margin for product 2

was fixed at p2 = 10. The relative contribution mar-gin (p1/p2) was varied from 10 to 12 in incrementsof 005, giving us a total of five contribution-margin

ratios. In 4.1 and 4.2, we analytically characterizedthe influence of a change in the reliability of a sin-gle resource, so we chose to investigate the influenceof changes in the overall network reliability in thenumerical studies. To that end, we assume that j = for all resources (resource failures are still indepen-dent) and vary from 02 to 10 in increments of 01,giving us a total of nine supply-chain reliabilities. =01 was not used because it can be shown that allresources will have zero investment at this reliabilitylevel. We chose nine different objective functions: riskneutral, loss aversion ( = 2 3 4 5), and CVaR ( =

095 09 085 08). All other model parameters wereheld constant across problem instances because theywere not the focus of the study. The marginal com-mitted cost was fixed at = 02 for all resources. Themarginal total costs were fixed at cj = 5 for all dedi-cated resources and at cj = 6 for all flexible resources.A full factorial set of tests was run for each network,so a total of 7 5 9 9 = 2835 instances was solvedfor each of the four networks. The optimal solutionand objective value for each problem instance wasstored in a database that is available from the authorson request.

The investment-mix question is particularly rele-vant for the DF network. For the risk-neutral objec-tive, Tables 1 to 3 present the percentage invested inthe flexible resource,

%KDF 3 RN =KDF 3 RN

KDF 1 RN + KDF 2 RN + K

DF 3 RN

where the numbers presented are medians for %KDF 3 RNtaken across the study instances. For example, there

Table 1 Influence of Correlation on Percent

Investment in the Flexible Resource

in the DF Network

%KDF 3 RN

100 38

069 38

038 35

004 33

031 31

066 30

100 29


13/21


Table 2 Influence of Relative Contribution

Margin on Percent Investment in the

Flexible Resource in the DF Network

p1/p2 %KDF

3 RN

1.00 30

1.05 32

1.10 33

1.15 34

1.20 36

were 315 risk-neutral instances for the DF network,so there were a total of 45 instances for each of theseven correlations. As can be seen, the median per-centage invested in the flexible resource increased inthe relative contribution margin and decreased in the

demand correlation. Even for perfectly positively cor-related demands, the median investment in the flex-ible resource was 29%. This number is driven bytwo factors. First, as shown by VM98, an asymmet-ric contribution margin can make the flexible resourceattractive even if = 1. Second, the flexible resourceoffers a diversification benefit in unreliable networks.For those instances with perfectly positively corre-lated demand and identical contribution margins (i.e.,no flexibility benefit), the average flexible investmentwas 0% when = 1 (i.e., perfectly reliable), but was18% when = 08. This demonstrates the importanceof the diversification benefit that the flexible resourceprovides in the DF network when investments areunreliable. Reliability itself has a somewhat complexinfluence on %KDF 3 RN. For a given supply-chain reli-ability, the expected marginal investment cost cj +1 is 20% higher for the flexible resource than

Table 3 Influence of Reliability on Percent

Investment in the Flexible Resource

in the DF Network

%KDF 3 RN

0.2 22

0.3 38

0.4 38

0.5 37

0.6 35

0.7 32

0.8 27

0.9 23

1.0 17

Table 4 Influence of Correlation on Percent of Instances in Which DF

Is Better, Worse, or Equal to DD

VDF RN > VDD

RN VDF

RN < VDD

RN VDF

RN = VDD

RN

100 8222 1778 000

069 7778 2222 000

038 7111 2889 000

004 6444 3556 000

031 4444 5556 000

066 1778 8222 000

100 000 8889 1111

for a dedicated resource (because c1 = c2 = 5, c3 = 6).At low reliabilities, this higher cost outweighs anyflexibility benefits provided by the flexible resource

because the benefits only arise in the unlikely event

that the resource investment succeeds. As reliabil-ity increases, there is initially a dramatic increase in%KDF 3 RN because the flexibility and diversification ben-efits of the flexible resource become significant, butas the network reliability continues to improve thereis a significant decrease in %KDF 3 RN as the diversifica-tion benefit becomes less important (and nonexistentat = 1). While the numbers in Tables 1 to 3 are pre-sented for the risk-neutral objective, the same obser-vations for the influence of correlation, relative mar-gin, and reliabilities held for the loss-averse and CVaRobjective functions.

We compared the optimal objective value across thenetworks for each problem instance. Given the studydesign, the DF network can never underperform theSD or SF networks, but can underperform the DD net-work. The DF network outperformed the SD network(i.e., VDF RN > V

SD RN ) in 93.65% of the 315 risk-neutral

instances and the performance was equivalent in theother 6.35% of the 315 instances. The median expected-profit improvement was 14.63% over the 93.65% ofcases for which VDF RN > V

SD RN , indicating that a firm

can gain significant advantage by moving from an SD

Table 5 Influence of Relative Margin on Percent of Instances in Which

DF Is Better, Worse, or Equal to DD

p1/p2 VDF

RN > VDD

RN VDF

RN < VDD

RN VDF

RN = VDD

RN

1.00 3810 6032 159

1.05 4444 5397 159

1.10 5079 4762 159

1.15 5714 4127 159

1.20 6508 3333 159


14/21


Table 6 Influence of Network Reliability on Percent of Instances in

Which DF Is Better, Worse, or Equal to DD

VDF RN > VDD

RN VDF

RN < VDD

RN VDF

RN = VDD

RN

0.2 000 10000 000

0.3 857 9143 000

0.4 4286 5714 000

0.5 5714 4286 000

0.6 6286 3714 000

0.7 6857 3143 000

0.8 6571 3429 000

0.9 6857 3143 000

1.0 8571 000 1429

network to a DF network. The DF network offers bothflexibility and diversification benefits, so the question

arises as to which type of benefit is really driving thesuperior performance. We investigated this questionby creating a network with the same diversificationbenefit but no flexibility benefit. This was done bycreating a special DD network in which the failuresof resources 2 and 3 were perfectly positively corre-lated (i.e., if one failed, so did the other) and havingtheir marginal total costs be identical to the flexibleresource cost (i.e., c2 = c3 = 6). This network outper-formed the SD network in 66.67% of the instances, andin those instances the median expected profit improve-ment was 3.48%. Comparing these results with those

for the DF network, we see that, whereas the diver-sification benefit of the DF network is significant, theflexibility benefit is more significant.

Although the DF network provides some diversi-fication, the (independent failure) DD network pro-vides more diversification but no flexibility. For therisk-neutral objective, the DF network outperformedthe DD network (VDF RN > V

DD RN ) in 51.11% of the

instances, underperformed (VDF RN < VDD

RN ) in 47.30%of the instances, and performed equally in 1.59% ofthe instances. Such data might suggest that a firm

Table 7 Influence of Loss Aversion on Percent of Instances in Which

DF Is Better, Worse, or Equal to DD

VDF LA > VDD

LA VDF

LA < VDD

LA VDF

LA = VDD

LA

1 5111 4730 159

2 4146 5679 174

3 3750 6071 179

4 3500 6321 179

5 3383 6429 188

is somewhat indifferent between these two networks, but this is not the case. As Tables 4 to 7 demon-strate, the network preference is highly driven by the

demand correlation, the relative contribution margin,the resource reliability, and the investment criterion.The DF network is much more attractive at lowerdemand correlations, higher relative margins, andhigher reliabilities, whereas the DD network is muchmore attractive at higher demand correlations, lowerrelative margins, and lower reliabilities. The attrac-tiveness of the DF network increases as either thedemand correlation decreases or the relative contri-

bution margin increases because the demand-poolingand contribution-margin benefits of flexibility arehigher in such circumstances. The DF network is less

attractive at low reliabilities because the extra level ofdiversification provided by the DD network is very

beneficial if resources are very unreliable. The firmsinvestment criterion is also a key driver of networkpreference. As the firm becomes more loss averse,it prefers the DD network in a higher percentage ofinstances.

In the second study, we investigated the influenceof the number of products on the relative perfor-mance of the four networks. This was done by fix-ing all other parameters and varying the number ofproducts from two to five. Demand for each prod-

uct was assumed to be independent (as the influenceof correlation was established above) and was char-acterized by 200 demand scenarios randomly drawnfrom a normal distribution with mean and stan-dard deviation of 100 and 30, respectively. Productcontribution margins were assumed to be identical(as the influence of contribution-margin differenceswas established above) and equal to 10. Resource

Table 8 Influence of the Number of Products on the Relative Difference

in Profit Between the DF and DD Networks

N= 2 N= 3 N= 4 N= 5

0.3 1287 903 588 368

0.4 456 045 408 667

0.5 143 390 746 1012

0.6 024 512 861 1104

0.7 017 522 851 1078

0.8 057 494 774 988

0.9 137 480 713 879

1.0 248 514 679 838


15/21


Table 9 Drivers of a Firms Preference for a Flexible or Dedicated Strategy

Element Attribute Influence on preference Reason

Product portfolio Demand correlations Preference for DF increases as demands become

more negatively correlated.

Negative correlation increases the demand-pooling

benefit of the flexible resource in the DF

network.

Contribution margins Preference for DF increases as the spread in

contribution margins increases.

Wider margin range increases the

contribution-margin option benefit (VM98) of

the flexible resource in the DF network.

Number of products Preference for DF increases as the number of

products increases.

The demand-pooling benefit of the flexible resource

increases as the number of products increases.

Resources Reliabilit ies Preference for DF decreases as resource

investments become less reliable.

Higher probability of resource failures increases

the diversification benefits of the DD network.

Firm Risk tolerance Preference for DF decreases as firm becomes

more concerned about downside risk.

In an unreliable network, the extra diversification

provided by the DD network lowers the firms

downside risk.

costs were assumed to be the same as in the abovestudy. We chose a risk-neutral objective (as the influ-ence of non-risk-neutral objectives was establishedabove) and solved the investment problem for eachof the four network structures for network reliabilities = 03 10 and number of products N = 2 3 4 5.We then calculated the relative network performanceas and N vary. The relative performance for the DFand DD networks (100 VDF RN V

DD RN /V

DD RN ) can

be found in Table 8. For any given value of , therelative performance of the DF network improves asthe number of products increases. The reason for this

is that the demand-pooling benefit of the flexibilityresource increases with the number of products.

5. ConclusionsIn this paper, we bridged the mix-flexibility anddual-sourcing literatures by studying four canonicalsupply-chain design strategies. Comparing the SDand SF networks, we identified the critical roles thatrisk tolerance and resource reliabilities play in the rel-ative attractiveness of the two networks. We refinedthe prevailing intuition that an SF-type network ispreferable to an SD-type network if a flexible resourceis no more costly than a dedicated resource. In par-ticular, we proved that the intuition is valid if eitherthe resource investments are perfectly reliable or thefirm is risk neutral, but the intuition can be wrongif neither condition holds. All things being equal(resource costs and reliabilities), a dedicated strategycan actually be strictly better than a flexible strategy.In fact, the dedicated strategy can be strictly better

even if dedicated resources are more expensive thana flexible resource.

We provided analytical results for the directionalinfluence of prices, marginal costs (total and commit-ted), and reliabilities on the optimal expected profitand resource levels for both the DD and DF networks.In contrast to the DD network, optimal dedicated-resource levels in the DF network are dependent onresources that are dedicated to other products, a resultthat suggests that flexible supply chains may not lendthemselves to decentralized design as easily as doproduct-dedicated supply chains.

A comprehensive numerical study was undertakento investigate how the attributes of three key supply-chain elementsnamely, product portfolio, resources,and the firminfluence the desirability of a givendesign strategy. As one would expect, the desirabil-ity of a dual-sourcing network (DD versus SD orDF versus SF) increases as supply-chain reliabilitydecreases. The story is more nuanced when comparingsingle-sourcing networks (SD versus SF) or dual-sourcing networks (DD versus DF). Table 9 summa-rizes the key results for such comparisons. We notethat, whereas Table 9 is framed in terms of dual-sourcing networks, a similar story holds for single-sourcing networks.

We conclude by identifying two dimensions notconsidered in this paper. First, a firms enthusiasm forresource diversification might be dampened by scaleeconomies in resource investments or by coordinationcosts in dealing with multiple suppliers for thesame product. Second, supply-chain design may be a


16/21


decentralized rather than a centralized endeavor, andthis raises the question of how to coordinate resource-investment decisions. We hope that future research

will further refine the insights provided in this paperby addressing these and other considerations.

AcknowledgmentsThe authors would like to thank the senior editor and thetwo referees for their comments and suggestions, which sig-nificantly improved this paper.

Appendix A. ProofsProof of Proposition 1. (i) 0 VSF RN V

SD RN 0 at

cN+1 = c. Using Equations (10), (11), (12), and (13), we thenhave

0

F1XN+1

0 xN+1fXN+1 xN+1 dxN+1

N

n=1

Kn0

xnfXn xn dxn 0 (A-1)

where = 1 + 1 c/p. Now,F1X

0 xfxdx =ESX , where ESX is the -expected shortfall for acontinuous random variable X as defined in Acerbi andTasche (2002, hereafter AT02). Note that 0 1. By sub-stituting this into (A-1), rearranging the terms, and usingthe fact that XN+1 =Nn=1 Xn, we then have

RN 0 Nn=1

ES

Xn ES

Nn=1

Xn

0

Now ESX is subadditive (see AT02), i.e.,

ES

Nn=1

Xn Nn=1

ESXnand so RN 0.

(ii) It is straightforward to show that the flexible-onlyresource level and the dedicated-only resource levels willbe positive if and only if > c/p 1 c. Therefore,

c/p 1 c VSF RN VSD

RN = 0

(iii) 1n = 1 for n = 2 N . Therefore, for each for n =2 N we have Xn = a1nX1 + b1n for some constants a1nand b1n with a1n > 0. Using the positive-homogeneity andtranslation-invariance properties of the expected shortfallmeasure, we have ESaX + b = aESX b; see AT02,Proposition 3.1(iii) and (iv). We then have

Nn=1

ESXn = ESX1 + Nn=2

a1nESX1 b1n=

1 +

Nn=2

a1n

ESX1 N

n=2

b1n

Furthermore,N

n=1Xn = 1 +Nn=2 a1nX1 +Nn=2 b1n and so

ESN

n=1 Xn

=

1 +

N

n=2a1n

ESX1

N

n=2b1n =

N

n=1ES

Xn

Therefore, RN = 0. Proof of Proposition 2. If X1 and X2 are independent

and identically distributed (i.i.d.) U 0 1, then X3 = X1 + X2has a triangular distribution over 0 2 and so FX3 x = x

2/2for 0 x 1 and FX3 x = 2x x

2/2 1 for 1 < x 2. Prooffollows from application of Equation (18).

Proof of Proposition 3. Both resources are identical,so an optimal solution will have KSD 1 LA = K

SD 2 LA. We there-

fore can restrict attention to investments of the form K1 =K2 = K, and K

must be less than or equal to one becausethe demand is U 0 1. Using Equations (21)(24) and theGy1 y2 K1 K2 equations in Appendix D, one can derive thefollowing expressions for VSDLA KK. If p > 2c, then

VSDLA KK

=2

p c +1K pK2

21

12cK

+1c1K2

2p+

22c3

3p3K3

(A-2)

and if p 2c, then

VSDLA KK

= 2

p c +1K pK2

21

12cK

+1c1K2

2p+25c6

p

4

2c3

3p3K3

2c

p

2

2c2

p

K2

(A-3)

VSDLA KK is a concave (cubic) function in K (for both p/cregions) and so the first-order condition is sufficient foroptimality. For both p/c regions the first-order conditionis quadratic with one single nonnegative root. This rootis the optimal investment level given in the propositionstatement.

Proof of Proposition 4. (i) Let KSD = K1 KN be

the optimal SD investment for some u1 U1. Consider thefollowing feasible SF investment: KN+1 =

Nn=1 K

n. Let

WSD

and WSF be the profit random variables for the SD and SFnetwork using these strategies, with distributions denotedby FWSD w and FWSF w. For any demand realization x =x1 xN, the SD and SF networks will have the follow-ing terminal wealths:

wSDKSD = w0 + pN

n=1

minxn Kn c

Nn=1

Kn (A-4)

wSF N

n=1

Kn

= w0 + p min

Nn=1

xnN

n=1

Kn

c

Nn=1

Kn (A-5)


17/21


so

wSF

N

n=1Kn

wSDKSD

= p

min

Nn=1

xnN

n=1

Kn

Nn=1

minxnKn

0 (A-6)

Therefore, FWSF w FWSD w, so WSF first-order stochasti-cally dominates WSD. Therefore,

EuSD1 KSD E

uSF1

Nn=1

Kn

for all utility functions u1 U1 (see Equation (4) in Levy1992) and so

EuSD1 KSD EuSF1 KN+1

Therefore 0.(ii) The loss-averse objective function VLAK is identical

to an expected utility objective function where the utilityfunction is piecewise-linear increasing (with the breakpointat w0). Such a utility function is in the U1 set, so the loss-averse result follows directly from Proposition (4). WSF first-order stochastically dominates WSD, so WSF second-orderstochastically dominates WSD. Recalling that CVaR is theaverage value of the profit falling below the -percentilelevel, we can use Theorem 3 in Levy (1992) to state that

VSDCVaRK1 KN VSFCVaR Nn=1

KnTherefore,

VSDCVaRK1 K

N V

SFCVaRK

N+1

so 0 for the CVaR objective function. Proof of Proposition 5. Proof of this proposition fol-

lows the same structure as that of VM98 Propositions 3and 4. VDFRN K is nonincreasing convex in c = c1 c2 c3 onthe convex set +3 for each K

+3 . V

DF RN c is therefore non-

increasing convex in c as maximization preserves convex-ity. A similar argument holds for =

1

2

3. VDF

RNK is

nondecreasing convex in p = p1 p2 on the convex set p p2 0 for each K

+3 . V

DF RN p is therefore nondecreas-

ing convex in p as maximization preserves convexity. TheHessian matrix for VDFRN K is

h11 h12 h13

h21 h22 h23

h31 h32 h33

where

h11 = 1p13I4K1K2K3+13I4K1K20

+I6K1K20+p223I2K1K2K3+I5K1K2K3+123I2K10K3

+I5K10K3I4K1K2K3

+p1 p23I6K1K2K3

h12 = h21 = 123I2K1K2K3

h13 = h31 = 13

p1I4K1K2K3+p22I2K1K2K3

+12I2K10K3I4K1K2K3

+p1 p2I6K1K2K3

h22 =2p213I1K1K2K3+I2K1K2K3+I3K1K2K3+113I20K2K3+I30K2K3

+13I1K1K20+I3K1K20

h23 = h32 = 23p2

1I1K1K2K3+I2K1K2K3

+11I20K2K3

h33 = 3

p11I4K1K2K3+11I40K2K3

+p212I1K1K2K3+I2K1K2K3

+112I1K10K3+I2K10K3

I4K1K2K3+112I20K2K3

+1112I200K3I20K2K3

+p1 p21I6K1K2K3+11I60K2K3

where the line integrals IkK1 K2 K3 k = 1 6 are givenin Appendix C and are similar to the Ik expressions in theproof of Proposition 1 in VM98. Because we have to general-ize for the case of investment failures, we write the functionin terms of IkK1 K2 K3 rather than simply Ik.

Proof of the gradients of Kj with respect to c, , andp follows from use of the Implicit Function Theorem (IFT)for each of the eight possible optimal solution structures.Application of the IFT requires detailed algebra that is avail-able from the authors on request. We note that the interior

solution structure K

1 > 0 K

2 > 0 K

3 > 0 is by far the mostcumbersome, and the only one in which we needed the suf-ficient condition of log-convexity or log-concavity for theproof.

Proof of Proposition 6. Proof of this proposition fol-lows the same structure as that of Proposition 5. VDFRN K isnondecreasing convex in = 1 2 3 on the convex set0 j 1, j= 1 3, for each K

+3 . V

DF RN K is therefore

nondecreasing convex in as maximization preserves con-vexity. Proof of the gradients of Kj with respect to follows


18/21


from use of the IFT for each of the eight possible optimalsolution structures. Application of the IFT requires detailedalgebra that is available from the authors on request. We notethat the interior solution structure K1 > 0 K

2 > 0 K

3 > 0

is by far the most cumbersome and the only one in whichwe used the sufficient condition of i.i.d. uniform demandsfor the proof.

Proof of Proposition 7. VDDRN K is nonincreasing con-vex in c (and in ) on the convex set +3 for each K

+3 .

VDD RN K is therefore nonincreasing convex in c (and in )as maximization preserves convexity. VDDRN K is nondecreas-ing convex in p (and in ) on the convex set p p2 0(0 j 1, j= 1 3) for each K

+3 . V

DF RN K is there-

fore nondecreasing convex in p (and in ) as maximiza-tion preserves convexity. VDDRN K is separable in K1 K2and K3 K4, i.e., the problem decomposes into two single-product problems. The resources dedicated to product i arethus independent of resources dedicated to product 3 i,

i = 1 2. Proof of the gradients of Kj follows from use ofthe IFT for each of the four possible optimal solution struc-tures for the single-product problem; the detailed algebra isavailable from the authors on request.

Appendix B. The Flexibility Premium for theCVaR Measure

Chen et al. (2003) prove that K = F1X p c/p v is theoptimal investment for a perfectly reliable single-productnewsvendor problem under a CVaR objective, where v isthe salvage value (assumed to be zero in our work). Weextend this result to allow for Bernoulli investment failures.Recall that

XN+1 =

X1 +

X2 +

XN and cN+1 is the marginal

total cost for the flexible resource.

Proposition 8. For the SF network, the optimal investmentlevel for the CVaR objective is

K3CVaR = F1

XN+1

1

c3p

+

1

cN+1

p11

(B-1)

Proof. Available from the authors on request. We note that at = 1 (B-1) gives the perfect-reliability

investment of Chen et al. (2003) for the case of zero salvagevalue, and it also gives the risk-neutral optimal investment(10) at = 1.

Using Equation (7), the SD investment problem can beformulated as

VSDCVaR

= maxK10KN0v

v+ 1 EXYmin WSDK1KNv0 (B-2)As with loss aversion, the SD investment problem cannot bedecomposed into N single-product problems. For the two-product case, one can show that

EXYminWSDK1K2v0=1112G00K1K2v +112G10K1K2v

+112G01K1K2v +12G11K1K2v (B-3)

where the Gy1 y2 K1 K2 v expressions can be found inAppendix D. Numerical results (available from the authorson request) prove that the flexibility premium CVaR can benegative, so SD can be strictly preferable to SF even whenthe flexible resource costs the same or less than the ded-icated resources, a result that echoes the loss-averse case.In the case of CVaR, the downside resource-disaggregationbenefit of the dedicated strategy is amplified by the fact thatthe left tail of the profit distribution is always factored in,whereas the right tail is not, with the result that SD can out-perform SF.

Appendix C. Expressions for P k and LineIntegrals IkK1 K2 K3

P 1 =

K1

0fXx1 x2 dx2 dx1

P 2 =

K1+K3

0

fXx1 x2 dx2 dx1

P 3 =K1+K3

K1

K1+K3x1

fXx1 x2 dx2 dx1

P 4 =K1+K3

K1

K1+K2+K3x1

fXx1 x2 dx2 dx1

P 5 =

0

K2

fXx1 x2 dx2 dx1

P 6 =K3

0

K2+K3x1

fXx1 x2 dx2 dx1

P 7 =

K3

K2

fXx1 x2 dx2 dx1

P 8 = K1+K3

K2

fXx1 x2 dx2 dx1

P 9 =K1

0

K2+K3

fXx1 x2 dx2 dx1

P 10 =

K3

0

fXx1 x2 dx2 dx1

P 11 =K3

0

K3x1

fXx1 x2 dx2 dx1

P 12 =K1

0

K3

fXx1 x2 dx2 dx1

I1K1 K2 K3 =K1

0fXx1 K2 + K3 dx1

I2K1 K2 K3 = K1+K3K1 fXx1 K1 + K2 + K3 x1 dx1I3K1 K2 K3 =

K1+K3

fXx1 K2 dx1

I4K1 K2 K3 =K2

0fXK1 + K3 x2 dx2

I5K1 K2 K3 =

K2+K3

fXK1 x2 dx2

I6K1 K2 K3 =

K2

fXK1 + K3 x2 dx2


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Appendix D. Gy1y2 K1 K2 and Gy1y2 K1 K2 vExpressions

We present the expressions for the general case wherethe marginal committed costs () and marginal total costs(c) can differ for resources 1 and 2. The Gy1y2 K1 K2expressions found in Equation (24) in the loss-averse sec-tion are given by Gy1y2 K1 K2 = Gy1y2 K1 K2 v = 0, whereGy1y2 K1 K2 v are given below. We present the more gen-eral Gy1y2 K1 K2 v expressions because they are used inthe CVaR analysis.

G00K1K2v =

0 for v < c11K1 +c22K2

c11K1 +c22K2 +v

for v c11K1 +c22K2

G10K1K2v =

pLX2

c11K1 +c2K2 +v

p

c2K2 +c11K1 +v

FX2

c11K1 +c2K2 +v

p

for v p c2K2 c11K1

pLX2 K2+K21FX2 x2

c2K2 +c11K1 +v

for v>pc2K2 +c11K1

G01K1K2 =

pLX2

c11K1 +c2K2 +v

p

c2K2 +c11K1 +vFX2

c11K1 +c2K2 +v

p

for v p c2K2 c11K1

pLX2 K2+K21FX2 x2

c2K2 +c11K1 +v

for v>p c2K2 +c11K1v minp c1K1 c2K2p c2K2 c1K1

G11K1K2 = p

NX1X2

c1K1 +c2K2 +v

p

+OX1X2

c1K1 +c2K2 +v

p

c1K1 +c2K2 +vMX1X2

c1K1 +c2K2 +v

p p c2K2 c1K1 maxp c1K1 c2K2p c2K2 c1K1

G11K1K2

= pLX1

c1K1 p c2K2 +v

p

+pLX2 K2

+K21FX2 K2FX1 c1K1 p c2K2 +vp c1K1 +c2K2 +vFX1

c1K1 p c2K2 +v

p

+p

NX1X2

c1K1 p c2K2 +v

p

c1K1 +c2K2 +v

p

NX1 X2

K1

c1K1 +c2K2 +v

p

+p

OX1X2

c1K1 p c2K2 +v

p

c1K1 +c2K2 +v

p

OX1X2

K1

c1K1 +c2K2 +v

p

c1K1 +c2K2

MX1X2

c1K1 p c2K2 +vp

c1K1 +c2K2 +v

p

MX1 X2

K1

c1K1 +c2K2 +v

p

+

pK1 c1K1 c2K2 vFX2

c2K2 p c1K1 +v

p

+pLX2

c2K2 p c1K1 +v

p

1FX1 K1


20/21


where

MX1 X2 w1 w2 =

w2

w1

FX2 w2 x1fX1 x1 dx1

NX1 X2 w1 w2 =w2

w1

x1FX2 w2 x1fX1 x1 dx1

OX1 X2 w1 w2 =w2

w1

LX2 w2 x1fX1 x1 dx1

MX1 X2 w = MX1 X2 0w

NX1 X2 w = NX1 X2 0 w

and

OX1 X2 w = OX1 X2 0w

Appendix E. Linear Program Formulations forInvestment Problem

There are a total of M= 2J possible yield scenarios. A yieldscenario m = 1 M is defined by a vector ym1 y

mJ

where ymj 0 1, with ymj = 0 denoting failure. The prob-

ability of scenario m is denoted by m. Let there be I dif-ferent demand scenarios, each defined by a demand vectorxi1 x

iN, with probability

i, i = 1 I . If the demandscenarios are randomly generated from some joint distribu-tion X, then i = 1/I for i = 1 I . By assumption, yieldsare independent of demands and so we have a total of MIyield-demand scenarios, each specified by an mi pair withan associated probability of mi = mi.

Risk-neutral linear program (LP) formulation:

max

Ii=1

Mm=1

miwmi

subject to

wmi =N

n=1

pnsmin

Jj=1

cjj+1jymj Kj

m = 1M i = 1I (E-1)

smin xin n =1N m = 1M i = 1I (E-2)

smin J

j=1

tnjqminj n =1N m =1M i = 1I (E-3)

Nn=1

qminj ymj Kj j= 1J m =1M i = 1I (E-4)

Kjsmin q

minj 0 n =1N j= 1J

m = 1M i = 1I (E-5)

Loss-averse LP formulation:

maxI

i=1

Mm=1

miwmi + wmi

subject to

wmi+ wmi = wmi m = 1M i = 1I (E-6)

wmi+

wmi

0 m = 1M i = 1Iand (E-1), (E-2), (E-3), (E-4), (E-5). (E-7)

CVaR LP formulation:

max

v +1

Ii=1

Mm=1

mizmi

subject to

zmi wmi vmi m = 1 M i = 1 I (E-8)

zmi 0 m = 1 M i = 1 I

and (E-1), (E-2), (E-3), (E-4), (E-5) (E-9)

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