RiskMitigationinNewsvendorNetworks: ResourceDiversiﬁcation ... · MANAGEMENT SCIENCE...

MANAGEMENT SCIENCEVol. 53, No. 8, August 2007, pp. 1269–1288issn 0025-1909 eissn 1526-5501 07 5308 1269

informs ®

doi 10.1287/mnsc.1070.0700©2007 INFORMS

Risk Mitigation in Newsvendor Networks:Resource Diversification, Flexibility,

Sharing, and HedgingJan A. Van Mieghem

Kellogg School of Management, Northwestern University, Evanston, Illinois 60208,[email protected]

This paper studies how judicious resource allocation in networks mitigates risk. Theory is presentedfor general utility functions and mean-variance formulations and is illustrated with networks featuring

resource diversification, flexibility (e.g., inventory substitution), and sharing (commonality). In contrast tosingle-resource settings, risk-averse newsvendors may invest more in networks than risk-neutral newsvendors:some resources and even total spending may exceed risk-neutral levels. With normally distributed demand,risk-averse newsvendors change resource levels roughly proportionally to demand variance, while risk-neutralagents adjust only proportionally to standard deviation.Two effects explain this operational hedge and suggest rules of thumb for strategic placement of safety

capacity and inventory in networks: (1) Risk pooling suggests rebalancing capacity toward inexpensive resourcesthat serve lower-profit variance markets. This highlights the role of profit variance (instead of demand variance)in risk-averse network investment. (2) Ex post revenue maximization suggests rebalancing capacity towardsubstitutable flexible but away from shared capacity when markets differ in profitability. Capacity imbalanceand allocation flexibility thus mitigate profit risk and truly are operational hedges.

Key words : risk aversion; mean-variance; capacity; inventory; investment; balance; real options; network designHistory : Accepted by Paul H. Zipkin, operations and supply chain management; received March 25, 2004. Thispaper was with the author 13 months for 3 revisions. Published online in Articles in Advance July 20, 2007.

1. IntroductionThis paper studies how resource allocation in net-works can mitigate risk exposure. It presents the-ory and insight on how risk attitude and networkconfiguration drive the strategic placement of safetycapacity and inventory for operational hedging. Thenetworks considered here are designed and man-aged by a single expected utility maximizer. Designinvolves the sizing of resources, which include inven-tories as well as capacities, and management meansprocessing to best fill market demands. Timing fol-lows a two-stage recourse model: Resources are sizedex ante when the demand vector is uncertain but itsprobability distribution is known, while processingoccurs after observing demand.Sections 2 and 3 present the model, theory, and

general results in terms of statistical quantities thatallow for computation by simulation. These propo-sitions hold for any portfolio of real options withgeneral network topology and utility functions. Tobring that theory to life, however, the remaining sec-tions focus on newsvendor networks, which are linearrecourse models that feature parsimony, tractability,and effectiveness in yielding insights into planningunder uncertainty. Sections 4 through 7 each analyzea specific network in two steps. First, observations are

made from a numerical study and intuitive explana-tions are proffered. Second, the insights are general-ized as properties, which are statements for a specificnetwork under certain conditions that are specifiedand proved analytically in the appendix.After reviewing the single-resource case in §4,

special attention is devoted to the three canonicalnewsvendor networks shown in Figure 1. All threeserve two markets and are building blocks for generalnetworks. The dedicated network features a dedicatedresource for each market and pure diversification ben-efits that provide a natural or passive hedge: While itstwo resources lack operational dependence, the net-work profit has less variability than the sum of theindividual resource profits. Aversion to financial vari-ability thus induces resource investment dependence.The other two networks add a third resource, eitherin series or in parallel, that features operational flex-ibility benefits: demand pooling and ex post revenuemaximizing allocation provide an active hedge thatincreases value by exploiting upside variations. Inserial networks, each market requires some upstreamdedicated work that is followed by a shared resourcerequirement. Examples are disk drive and computermanufacturing, where a common set of computers

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Van Mieghem: Risk Mitigation in Newsvendor Networks1270 Management Science 53(8), pp. 1269–1288, © 2007 INFORMS

Figure 1 Three Canonical Building Blocks for Multiresource Networks That Are Analyzed in Detail

(b) Serial network (c) Parallel network

x2

x1

FlexibleK3

x3

D1

D2

x4

DedicatedK1

DedicatedK2

DedicatedK1

x1 x1D1

D2Dedicated

K2

x2 x2

(a) Dedicated network

DedicatedK1

DedicatedK2

D1

D2

SharedK3

perform final burn-in and test routines. The serial net-work also is a core model for pure component com-monality. The parallel network is a dedicated networkaugmented with auxiliary flexible capacity; it alsomodels inventory substitution and resource redun-dancy. It allows a tailored response to uncertaintywhere dedicated capacity mainly fills base demand,while flexible capacity supplies variable demand. Eco-nomically, the shared resource is a complementaryreal asset, while the flexible resource is a substitute.Risk exposure (or simply risk) refers to the undesir-

able consequence of a random prospect. In newsven-dor models, the random prospect is typically calleddemand and is modeled by an exogenous probabil-ity distribution. The operational consequence of ran-dom demand is a likely, but undesirable mismatchbetween supply and demand manifested as over-age or underage. The expected cost of overages andunderages is called mismatch cost, following Cachonand Terwiesch (2006). The financial consequence ofrandom demand is profit risk: profit variability riskas well as a decrease in expected profit.Risk attitude describes how a decision maker per-

ceives risk. Risk-averse agents prefer the expectedvalue over the random variable. While risk-neutralnewsvendors only care about the mismatch cost, risk-averse agents also care about profit variability risk.Traditionally, an increase in risk is equated with anincreased mean-preserving spread (e.g., see Gollier2001 for general definitions). For univariate normalrandom variables, this is equivalent to an increasein variance while keeping the mean constant. Whiledemand variance impacts risk exposure (and will beused in our graphs), the remainder will illustratethat profit variances are the natural descriptors ofrisk exposure in networks. They summarize marketand network interactions and capture the importantimpact of demand correlations, an increase of whichtypically increases profit risk.1

This paper establishes that risk attitude and net-work structure fundamentally change resource alloca-tion. In contrast to single-resource settings, risk-averse

1 Corbett and Rajaram (2006) show that intuition from multivari-ate normals extends beyond normal distributions. For example, fora broad class of distributions, aggregation of inventories is morevaluable as demands are less-positively dependent.

newsvendors may invest more in networks thanrisk-neutral newsvendors: some resources and eventotal spending may exceed risk-neutral levels. Withnormally distributed demand, risk-averse newsven-dors change resource levels roughly proportionally todemand variance (i.e., levels are quadratic in standarddeviation), while risk-neutral agents adjust only pro-portionally to standard deviation.These findings are explained in terms of hedging.

Hedging is the action of a decision maker to mit-igate a particular risk exposure. Operational hedg-ing is risk mitigation using operational instruments.This definition is deliberately general to include risk-neutral agents as well as univariate settings. Holdingexcess assets such as stock or capacity reserves by arisk-neutral newsvendor is interpreted as operationalhedging because it mitigates mismatch costs. Thisillustrates that operational hedging impacts expectedprofits; operational flexibility can even exploit riskand add value. Yet, the standard financial motivationfor hedging is mitigation of profit variability risk,2

which is the main topic of investigation in this paper.Hedging by “betting on two horses” or “not holdingall eggs in one basket” presumes access to at least tworisks whose counterbalancing effect is to reduce totalrisk. With multiple risks (demands) and multiple bas-kets (resources), newsvendor networks are a naturalvehicle to study how operational instruments reducetotal risk and may create value.The analysis of the three canonical networks iden-

tifies three types of operational hedging that are sum-marized in Table 1. Risk mitigation through purediversification or demand pooling steers the portfo-lio mix toward assets supplying lower-profit vari-ance markets. These need not be the lower demandvariance markets, which highlights the importance ofprofit variance to understanding risk-averse network

2 The Oxford English Dictionary Online (2004) defines hedging as “tosurround with a hedge or fence as a boundary, or for purposesof defence, or to confine or restrict movement,” while Merriam-Webster’s Collegiate Dictionary (1998) states “to protect oneself fromlosing or failing by a counterbalancing action.” “Most businessesinsure or hedge to reduce risk, not to make money” by “taking onone risk to offset another,” according to Brealey and Myers (2000,pp. 760 and 763).

Van Mieghem: Risk Mitigation in Newsvendor NetworksManagement Science 53(8), pp. 1269–1288, © 2007 INFORMS 1271

Table 1 Three Operational Instruments That Mitigate Profit VariabilityRisk and May Increase Expected Profit (Operational FlexibilityCan Exploit Risk and Create Value) by Rebalancing NetworkCapacity

Operational Portfolio of dedicated Resource Resourceinstrument resources sharing flexibility

DriverPure diversification

Demand pooling —

Ex post revenue —

maximization

ImpactMismatch cost — ↓ ↓Profit variability ↓ ↓ ↓risk

HedgingRebalance Toward assets Away from Towardcapacity mix serving lower (unique) (redundant)

profit variance shared asset flexiblemarkets asset

design. Capacity imbalance in the serial and paral-lel networks can remain even with perfectly positivecorrelations (i.e., in the absence of risk pooling). Thisisolates the contingent optimization option imbed-ded in shared and flexibile resources: they can steerand allocate production toward the higher-profit mar-ket. When markets differ in profitability, risk aversionrebalances capacity toward the (redundant) flexibleresource, but away from the (unique) shared resource.(Given that both types of resources are “product-flexible,” this means that the appropriate hedgingaction for product-flexible resources depends on theirnetwork position: increase when in parallel with ded-icated resources, decrease when in series.) Capac-ity imbalance and allocation flexibility thus mitigateprofit risk which confirms and refines their interpre-tation as operational hedges.Section 8 concludes with managerial take-aways

and discusses model limitations and extensions. Theappendix contains one key proof; all other proofsare given in the online appendix (provided in thee-companion).3

Three research areas are most related to this arti-cle: risk-averse single-resource newsvendor models,newsvendor networks, and operational hedging. Thisarticle is a natural successor to the seminal workby Eeckhoudt et al. (1995), who prove that the opti-mal level of a single-resource newsvendor is alwaysdecreasing in risk aversion for general concave util-ity functions. This article extends their ingenious prooftechnique to a newsvendor network and shows thattheir unambiguous result does not hold for networks.

3 An electronic companion to this paper is available as part ofthe online version that can be found at http://mansci.journal.informs.org/.

Other studies of risk-averse single-resource newsven-dor models include Atkinson (1979), Lau (1980), Spul-ber (1985), Anvari (1987), Lau and Lau (1999), Agrawaland Seshadri (2000a, b), Gan et al. (2005), Gaur andSeshadri (2005), Caldentey and Haugh (2006), andChod et al. (2006), with multiperiod extensions inBouakiz and Sobel (1992) and Chen et al. (2004).This article is also a natural successor to Van

Mieghem and Rudi (2002), who define and analyzenewsvendor networks under expected profit maxi-mization. Flexibility in risk-neutral parallel networkswas first studied by Fine and Freund (1990) with a dis-crete math-programming model and by Van Mieghem(1998) with a newsvendor network model. Otherrelated newsvendor network studies of the risk-neutral parallel network include Bassok et al. (1999),Hale et al. (2000), Rudi (2000), Netessine et al. (2002),VanMieghem (2004), Bish andWang (2004), and Goyaland Netessine (2007). The risk-neutral serial networkwas studied in Harrison and Van Mieghem (1999) andextended in Van Mieghem (2003). As far as we areaware, the only other paper on risk-averse newsven-dor networks is Tomlin and Wang (2005), which com-plements this one in terms of research question andtreatment of risk attitude. They investigate flexibilityand dual sourcing in unreliable newsvendor networksand consider both loss aversion and conditional value-at-risk. With unreliable resources and risk aversion,inherent redundancy in a dedicated network can makeit the preferred strategy to a flexible resource even ifthe latter is cheaper.This article also relates to the literature on opera-

tional hedging, a term promulgated by Huchzermeierand Cohen (1996). They provided a valuation modeland numerical evidence that embedded real optionslike contingent supply and production switchingreduce downside risk in the presence of exchange-rateuncertainty. Ding et al. (2007) review and add recentanalytical advances on joint operational and financialhedging of exchange-rate risk. Hedging in that settingof price uncertainty also may lead to an increase incapacity, similar to our finding under demand uncer-tainty. Operational hedging by means of flexibility andcapacity imbalance in newsvendor networks underdemand uncertainty was studied in Harrison and VanMieghem (1999) and extended in Van Mieghem (2003).Boyabatlı and Toktay (2004) survey and critically dis-cuss papers on operational hedging, most of whichassume expected profit maximization. Hedging obvi-ously requires the presence of uncertainty but its stan-dard objective is to reduce risk, not to make money.This paper shows that risk aversion magnifies theseoperational constructions, establishing that they mit-igate risk and strengthening their interpretation asoperational hedges.


2. Model2.1. Decision ProblemConsider a firm that has n different real assets or“means of processing,” which we will call resources.We adopt the notation of Van Mieghem (2003) anddenote its resource portfolio by the nonnegative resourcevector K ∈ n

+, whose ith component represents thelevel of resource i available for processing during theperiod. The resulting operating profit gained at theend of the period is a random variable that is a func-tion of the available resources. Let K denote thisoperating profit function, where is a sample pointin the sample space . The operating profit functionis concave in the resource vector K, reflecting the nat-ural assumption of decreasing marginal returns frominvestment. The financial investment cost to installresource levels K is denoted by C0K. As usual, C0is assumed to be convex to guarantee a well-behavedconcave optimization problem. A typical economicassumption, however, is that C0 exhibits economies ofscale and slightly concave functions or the addition ofa fixed cost often does not pose a problem.The research problem is to decide on the resource

vector K given a probability distribution P on thesample space . The risk-averse decision maker hasvon Neumann-Morgenstern preferences and maxi-mizes expected utility of terminal wealth. (This canmodel the behavior not only of a small firm, but alsoof an individual decision maker in a large firm. Afterall, decisions ultimately are made by a few individualswho may be risk averse. Shortcomings of the utilityapproach are discussed in the conclusion.) The agenthas an initial endowment or wealth W0 and can bor-row and lend without limitations at the risk-free inter-est rate r . Let W = 1+ rW0 and CK= 1+ rC0Kdenote the initial endowment and resource costs inend-of-period monetary units. Let V denote the netfuture value of the firm’s resource portfolio: V K=K − CK. Under these assumptions, a risk-averse investor will choose a resource vector Ku thatmaximizes expected future utility UKW, where

UKW= ƐuV K+W (1)

and u· is strictly increasing (such that the investorprefers “more over less”) and concave (such that theinvestor is “risk averse” and prefers the expectedvalue over the risky outcome) on . Thus, this isan investment model of a risk-averse agent who caninvest in a portfolio with one riskless asset and n riskyassets.

2.2. Technical AssumptionsThe only essential requirements are concavity: Thefunction V · is concave on n

+ for almost every ,and u· is concave on . The analysis below presents

expressions that allow computationally efficient, sim-ulation-based optimization, also known as infinites-imal perturbation analysis (IPA). These expressionsrequire the interchange of differentiation and integra-tion, which is typically justified by a monotone orbounded convergence theorem. This requires certaintechnical conditions that, in one way or another,bound the derivatives as shown in the appendix.

2.3. NotationLet K and 2K denote the mean and varianceof the value of the resource portfolio K expressedin end-of-period monetary units: K = ƐV Kand 2K = ƐV K − K2 = Ɛ2K −ƐK2. It will be useful to denote the marginaloperating profit KK by K, the marginalresource cost CK by cK, and the Hessian of theexpected value 2

KƐV K by HK.

2.4. Newsvendor NetworksTo illustrate the general theory, we will use newsven-dor networks, a tractable class of linear recourse mod-els introduced in Van Mieghem and Rudi (2002) anddefined via three data sets: (1) Demand data: whenresults apply only to newsvendor networks, the sam-ple point will be replaced by the demand vec-tor D which has continuous probability measure P .(2) Financial data: activity vector x yields gross mar-gin m′x, where m equals price minus any marginalprocessing and transportation cost. The unit capacityinvestment costs are cK while inventory S incurs unitpurchasing and holding costs cS and cH; and unmetdemand incurs shortage cost cP. (3) Network data: theinput-output matrices RS and RD and the capacity con-sumption matrix A. Here RS ij denotes the amount ofinput stock i consumed per unit of activity j , RD ij isthe amount of output i per unit of activity j , and Akj isthe amount of resource k capacity consumed per unitof activity j .Let v = m + R′

DcP + R′ScH denote the net value vec-

tor associated with the various processing activities.Denote an ex post optimal activity vector byx∗KSD. It solves the linear program maxv′x sub-ject to x ≥ 0, RSx ≤ S RDx ≤ D and Ax ≤ K.The operating profit function then is KSD =v′x∗KSD − c′PD − c′HS, which is concave for anynewsvendor network. For notational simplicity andwithout loss of generality, we will treat all resourcesas capacities so that the relevant resource levels are K.For the three networks in Figure 1, we will label mar-ket or product 1 as the more profitable one: v1 ≥ v2.Also, i and i (i = 12) will denote the mean andstandard deviation of market-i demand Di and thecorrelation coefficient between D1 and D2. (To avoidconfusion with the mean and standard deviation ofvalue, the demand mean and standard deviation will


always have a subscript.) As usual, let and denotethe standard normal cumulative distribution and den-sity function.

3. General Theory and ResultsThis section reviews relevant concepts and summa-rizes theory and results for general resource portfo-lio problems. These results are applied and extendedto specific newsvendor networks in the remainingsections.

3.1. Optimality Conditions and Results forGeneral Utility Functions

Proposition 1. The expected utility function UKWis concave inK for any W .

Given that adding risk aversion does not destroyconcavity, the optimization problem remains wellbehaved with sufficient first-order conditions:

Proposition 2. An interior optimal investment Ku fora risk-averse utility function u solves the necessary and suf-ficient condition UKu= 0, where

UK= ƐK− cKu′V K+W (2)

Let Kn denote an optimal resource vector for therisk-neutral case. (The superscript n is mnemonic for“risk-neutral” and also for “newsvendor.”) In the risk-neutral setting, the utility function has no curvature sothat u′ is a constant and (2) simplifies to

U nKn= ƐKn− cKn= 0 (3)

in agreement with Proposition 1 in Van Mieghem andRudi (2002). Condition (3) is the multidimensionalgeneralization of the familiar critical fractile condi-tion. In a newsvendor network, the operating profitis the maximum of an underlying linear program,as discussed earlier. Then, there exists a partition of=⋃

j jK such that ƐK = ∑j jP jK,

where j is the constant shadow vector of the resourceconstraint Ax≤K in event jK. The optimality con-dition

∑j jP jKn = c generalizes the critical frac-

tile condition of the single-resource newsvendor tohigher dimensions. It sets the likelihood of a resourcebeing a bottleneck proportional to its cost. In the risk-averse setting, Proposition 2 shows that this conditionis perturbed in the sense that it is multiplied by thestate-dependent marginal utility of money u′. Giventhat only changes from a constant u′ matter, the per-turbation impact increases proportionally to the curva-ture of u, exactly what risk aversion is expected to do.Proposition 2 directly suggests a simulation-based

gradient (steepest ascent) optimization method that iscomputationally efficient and unbiased, a character-istic of IPA. It is computationally efficient becauseone need only draw a single random sample !i"

that can be used during the entire optimizationusing the following algorithm. (This method with10000 sample demand vectors was used to pro-duce the graphs in this paper.) Choose an ini-tial value k0 and set m = 0. Compute the shadowvector kmi and its weight u′V kmi + W for each point i in the sample and compute theirweighted sum to find an unbiased estimate forUkm. Find the maximizer km+1 of U along the half-line km + tUkm, where t ≥ 0. Iterate until Ukmor km+1−km are below a tolerance.Much of the intuition behind our results will be

explained in terms of the gradient of the value vari-ance (or variance of operating profits) 2K, whichalso can be expressed in terms of the shadow vectorK. Recall that the covariance between x and yis defined as Covxy = Ɛxy′ − Ɛx Ɛy′. (Some care isneeded with dimensions: if y is scalar, then Covxyhas the dimensions of x and contains the component-wise covariances.)

Proposition 3. The gradient of the value variance is

2K= 2CovKK

and evaluated at the risk-neutral solution Kn, the followingare equivalent:

2Kn = 2ƐKn− cV Kn

= 2CovKnKn

To discuss how risk-averse managers should adjusttheir resource portfolio relative to the risk-neutralone, we need a measure of risk aversion. Pratt (1964)showed that the coefficient of absolute risk aversion,'x =−u′′x/u′x, is a simple measure of local riskaversion, while there is no simple measure of risk aver-sion “in the large.” (The coefficient of absolute riskaversion also meshes with our focus on additive risk,while relative risk aversion is better for multiplica-tive risk.) Nevertheless, comparisons of risk aversionbetween two utility functions u1 and u2 can be madesimply: u1 is (weakly) more risk averse than u2 if andonly if '1x≥ '2x for all x, which is equivalent to thestatement that u1·= hu2· for some strictly increas-ing, concave function h according to Pratt (1964, The-orem 1). We say that an optimal decision variable Ki

is “increasing in risk aversion” if Ku1i ≤ K

u2i when-

ever u2 is more risk averse than u1. This means that,when comparing two managers, the more risk-aversemanager will invest more than the other. We also willloosely say that a utility function u is moderately riskaverse if there exists an + > 0 such that if maxx 'x<+, then its optimal Ku is in an +-neighborhood of Kn.


3.2. Local Impact of Risk Aversion andCARA Utility

It is instructive to analyze the impact of a small, orlocal, increase in risk aversion, relative to the risk-neutral case. To characterize the associated optimalresource adjustment vector, we consider a parameter-ized class of utility functions and apply the implicitfunction theorem. While there is no simple measure torank general utility functions in terms of degrees ofrisk aversion, the class of utility functions with con-stant absolute risk aversion (CARA) is a very usefulexception. The condition that 'x=−u′′x/u′x be aconstant yields a differential equation that is satisfiedonly by exponential functions:

uCARAx= .e−'x +0 (4)

where . ≤ 0 and ' ≥ 0 to ensure the utility func-tion is increasing concave. (It is convenient to choose.=−0=−'−1 so that uCARAx'→ x when ' ↓ 0, aswill be assumed in later graphs.)Exponential utility functions are theoretically and

mathematically appealing. First, the scalar ' gives asimple cardinal measure of risk aversion. Second,its optimal actions are independent of the initialwealth W (under the earlier assumptions on borrow-ing and lending). Indeed, the optimality condition (2)simplifies to

UK= 0⇔ ƐK− cKexp−'V K= 0 (5)

Therefore, the CARA-optimal resource vector is afunction of the scalar ' only, which we will de-note by KCARA'. Similarly, let UCARA' denote theassociated utility ƐuCARAV KCARA'. Third, withCARA, the expected utility function is directly ex-pressed in terms of the characteristic function ofthe value function, which is defined as 1V Kt =ƐexpitV K. It directly follows that UCARAK =.e−'W 1V Ki' + 0. (The characteristic function hassimple closed-form expressions if has a normal,exponential, gamma, or uniform distribution.) Finally,CARA allows us to characterize the optimal resourceadjustment vector for small increases in risk aroundKn. (Recall that HKn is the Hessian matrix ofƐV Kn.)

Proposition 4. In the neighborhood of the risk-neutralcase, small CARA risk aversion sets KCARA' = Kn +'d/d'KCARA0 + o', where the optimal resourceadjustment vector is

d

d'KCARA0= 1

2H−1Kn2Kn (6)

While it is known that the value variance shouldbe decreasing with risk aversion around the risk-neutral solution (see the next subsection), the propo-sition explains how that is done. Recall that concavityimplies that the Hessian is negative definite so that

its diagonal elements are negative, but not necessar-ily its off-diagonal elements. Often, however, H−1 isdiagonally dominant (a sufficient, but not necessarycondition for concavity) so that the optimal adjust-ment vector is roughly in the negative or oppositedirection of 2Kn (it really is a linear combination).In other words, an increase in risk aversion shouldadjust resource levels in the direction that reducesvalue variance.So, should risk-averse managers set smaller re-

source levels than risk-neutral managers? While theanswer is affirmative for single-resource problems,the remainder of this paper will show that such anunambiguous result does not extend to a bundle ofresources. The gradient 2Kn captures the effectof increasing resource levels on the variance of oper-ating profits. In a newsvendor setting, increasing thecapacity or the feasible set of activity vectors x typ-ically increases the variance of operating profits v′x.A positive gradient 2Kn together with a negativeHessian imply that the expected effect of risk aversionis to decrease resource levels. The remaining sections,however, will show that in a network the benefit fromincreasing some resource levels may outweigh its costwhen correlations are sufficiently negative. In otherwords, increasing capacity in a network may decreasevalue variance.

3.3. The Mean-Variance Formulation andHedging Effectiveness

Similar to financial portfolios, the effect of risk aver-sion on the configuration of a portfolio of real assets isoften illustrated using a mean-variance (MV) formu-lation, which seeks to maximize

UMVK=K− '

22K. (7)

Let KMV' denote a maximizer of (7) and UMV'the associated MV utility UMVKMV'. Expressions(2) and (3) directly yield an expression for the gradient

UMVK = ƐK− cK

−'CovKK (8)

which is useful for gradient-based numerical opti-mization via simulation. It is well known that theMV solution exhibits smaller value variance than therisk-neutral solution.4 Proposition 4 shows that riskaversion adjusts resource levels in the direction thatreduces variance, which also follows from (8).MV formulations are hard to justify from a theoret-

ical perspective and have several undesirable proper-ties.5 They are better viewed as approximations—theremainder suggests that the MV approximation is

4 By definition, UMVKMV ≥ UMVKn and Kn ≥ KMV so that 2KMV≤ 2Kn 5 Typical justifications include normally distributed returns or quad-ratic preferences, both of which are problematic. In newsvendor


Figure 2 An Estimate of the Coefficient of Risk Aversion Sufficesto Identify the Resource Investment on the Frontier Thatthe Mean-Variance Formulation Recommends

γ /2

Mean of network value µ(K)

Variance of network values σ2(K)

UMV(γ)

Best minimalvariance K0

Mean-variancefrontier MV

Risk-neutralnewsvendor Kn

µ(KMV(γ))

σ2(KMV(γ))

quite reasonable for newsvendor networks with mod-erate risk aversion—with the two practical benefitsdiscussed in Van Mieghem (2003): (a) They are imple-mentable because only two moments are required,which can be estimated. (b) They are useful in thesense that they provide “good recommendations”even when the decision maker does not know herutility function. “Good recommendations” are thoseinvestments that are on the MV frontier MV, whichis the set of MV efficient investments that embodythe trade-off between expected return and risk. Givena frontier, all that is needed is a reasonable estimateof the decision-maker’s risk attitude ', which directlyidentifies the recommended investment on the frontieras shown in Figure 2.The standard objective of hedging suggests to mea-

sure hedging effectiveness by assessing howmuch riskcan be reduced while giving up little expected return.In MV terms: Howmuch can variance be reduced withlittle impact on mean value? The local sensitivity ofexpected return to optimal reduction of variance riskaround the optimal risk-neutral portfolio Kn is mea-sured by the curvature of the MV frontier, as shownin the appendix:

′′MV2Kn= 1

′2KnH−1Kn2Kn (9)

This quantitative theoretical measure of hedging ef-fectiveness reconfirms the importance of 2Kn:effectiveness increases if the gradient of variance islarge, in line with intuition.

networks, capacity constraints preclude normally distributed oper-ating profits. In general, quadratic preferences ux = .x − 02

require the value function V Kn to be bounded a.s. by 0 for theexpected utility to be concave increasing. MV utilities also haveincreasing absolute risk aversion and a bliss point after which util-ity is decreasing in wealth.

The remainder of this paper applies the general re-sults of this section first to the single-resource prob-lem, which will serve as our base case, and then to thethree canonical networks of Figure 1.

4. The Base Case: The Single-ResourceNewsvendor

Let us review how risk affects the single-resourcenewsvendor problem before delving into networks. Toensure a profitable investment, return should exceedcost: v1 ≥ c1.

4.1. Risk ExposureThe optimal resource level for a risk-neutral newsven-dor exposed to univariate demand risk solves (3) orv1PD1 > Kn

1 = c1. The optimal service probability forany demand distribution is 1 − c1/v1 and dependssolely on the ratio of unit cost to unit return; it isalso called the critical fractile. Thus, Kn

1 is below, at,or above the demand median if the critical fractile isbelow, equal to, or above 1/2, respectively. For nor-mal demand, the risk-neutral resource level is linearin demand standard deviation: Kn

1 = 1+ zn11, where

zn1= 1−c1/v1. Increased risk exposure (in the sense

of an increasing mean-preserving spread, i.e., increase1) then leads to a decrease, no change, or increase ofthe resource level if the critical fractile is below, equalto, or above 1/2, respectively. Figure 3 illustrates thesefacts with a numerical study with normal demandwith 1 = 1. (All graphs in this paper have ' goingfrom 0 to 2 5 in 0 25 increments, unless indicated oth-erwise. The risk-neutral case is denoted by “o,” whichis mnemonic for ' = 0.)

4.2. Risk AversionThe seminal work by Eeckhoudt et al. (1995) provesthat the optimal resource level for a single-resourcerisk-averse newsvendor is always below the risk-neutral level for any concave utility function. More-over, the optimal level always decreases in riskaversion so that the total monetary investment alwaysdecreases in risk aversion.6 (This result is “global”in that it holds whenever a concave utility functionu· is replaced by a “more concave” hu· as dis-cussed earlier.) They also show that the impact ofan increased risk exposure on the risk-averse optimal

6 The celebrated two-fund separation result of financial economicshas established that the investment in the risky asset CKudecreases in risk aversion for the standard one-safe, one-riskyfinancial asset portfolio. That result does not apply here becauseour real assets have nonlinear payoffs (nor does its key insight thatthe optimal relative configuration of risky assets is independent ofan investor’s coefficient of risk aversion hold here). Nevertheless,Eeckhoudt et al. (1995) showed that the investment in the risky realasset does always decrease in risk aversion for the single-resourcenewsvendor.


Figure 3 With Normally Distributed Demand, Risk-Neutral Single-Resource Levels Are Linear (Labeled “o”) in Demand Standard Deviation withSlope Depending on the Critical Fractile. Risk-Averse Capacities Are Roughly Quadratic and Below Risk-Neutral Levels

Risk exposure (std. dev. σ1) Risk exposure (std. dev. σ1) Risk exposure (std. dev. σ1)

(a) Critical fractile < 1/2 (b) Critical fractile = 1/2 (c) Critical fractile > 1/2

0 1/3

0.8

1.0

1.2

0

0.8

1.2

0

0.8

1.2

1/3 1/3

1.0 1.0

Risk aversion γ

γ

γ

level is ambiguous, as Figure 3 illustrates for CARAutility.This agrees with the optimal local risk-adjustment

specified in Proposition 4, which can be solvedanalytically:

Property 1. The optimal risk-adjustment (6) for thesingle-resource newsvendor problem with normally dis-tributed demand and CARA utility is

d

d'K0 = −2

1 c

(1+ zn1− c/v

zn

)≤ 0

where zn =−1(1− c

v

)

The interesting insight from this property is thatthe local optimal adjustment due to risk aversion isproportional to the demand variance. This impliesthat, for normal demand, risk-averse resource levels(at least near the risk-neutral line) are quadratic indemand standard deviation. This is in stark contrastwith the risk-neutral case, where investment levelsare only proportional to demand standard deviation.This quadratic dependence explains the ambiguity ofthe impact of increased risk exposure on risk-averseresource levels for a single-resource newsvendor andextends to the three canonical networks (and is con-jectured to be a general effect).The property also illustrates that risk-averse invest-

ment is analytically complex, even in the single-re-source case. It depends on all model parameters:demand uncertainty (mean 1 and std. dev. 1) andfinancials v and c. The independence of mean demandis unique to CARA; all other utility functions havewealth dependence and are thus also dependent onaverage demand.7 Even in this simplest of networks,

7 For example, for DARA utility ux= xq with x > 0 and 0< q < 1,the optimal risk adjustment d/dqK0 is −H−1Ɛ− c lnV +W

one cannot guarantee that the adjustment magnitudeis monotone in c or v, except if c/v > 1/2, in which casethe magnitude of the adjustment is increasing in v. It iseven harder to characterize the conditions for effectivehedging. According to (9), this amounts to identifyingthe parameters that yield a modest frontier curvature.While this curvature can be computed analytically forthe single-resource case, the expression

′2KnH−1Kn2Kn

=−431 c2vzn

(1+ 1− c/vzn

zn

)2

defies simple insights.Numerical optimization with normal demand8

allows some additional observations on the impactof risk aversion. Figure 4 compares KCARA'with KMV' and the associated utility levelsUCARAKCARA' with UCARAKMV' for a represen-tative situation (normal demand with 1 = 1, 1 = 1/3,c = 0 5, and v = 1, so that Kn = 1). The left panel ofFigure 4 shows an optimal risk adjustment that isnear linear in risk aversion and larger than what MVsuggests, a finding that extends to the three canonicalnetworks. (All our numerical work suggests thatKCARA' is convex decreasing if c/v < 1/2, linear ifc/v = 1/2, and concave decreasing otherwise, but withvery weak nonlinearity.) Near linearity implies thatthe risk-adjustment gradient is almost constant, whichbroadens the applicability of the local risk-adjustmentresult of Proposition 4. While the MV approximationunderadjusts resources, the right panel of Figure 4

and depends on wealth and mean demand. No simple closed-formexpressions are available.8 CARA and MV objective functions can be expressed analyticallywith normal demand so no simulation is needed.


Figure 4 Comparison of Optimal and Mean-Variance Capacity and Utilities for the Single-Resource Newsvendor Problem as a Function of RiskAversion (Normal Demand 1 = 1, 1 = 1/3, c= 05, v = 1)

UCARA(KMV(γ))

UCARA(KCARA(γ))

0 2 4 6 8 100

0.1

0.2

0.3

0.4Quality of MV approximation to CARA: Optimal utility

KMV(γ )

KCARA(γ)

0 2 4 6 8 100.4

0.6

0.8

µ1=1

Quality of MV approximation to CARA: Optimal resource level

Risk aversion (γ) Risk aversion (γ)

suggests that the impact on utility remains small formoderate levels of risk aversion. (Note that, as single-variable trajectories, the efficient frontier and the MVfrontier coincide for single-resource problems.)

4.3. Operational HedgingIn the absence of risk, any newsvendor invests in aresource level equal to the deterministic demand 1.The change in the resource level due to risk expo-sure is called its safety level, e.g., safety capacity orsafety inventory. For a risk-neutral newsvendor facingnormal demand, the hedge is Kn

1 − 1 = zn11 and a

change in risk 1 leads to a proportional decrease, nochange, or increase depending on the critical fractile.A risk-averse newsvendor reacts stronger to a changein normal risk exposure: her safety-level adjustmentis quadratic in 1. This is expressed more elegantly interms of the standardized safety level zix of a prod-uct i resource level x:

standardized safety level zix= x−i

i

(10)

With normal demand, the standardized safety levelis independent of risk exposure for a risk-neutralnewsvendor (equal to the constant zn

1 ), but decreaseslinearly in 1 for a risk-averse newsvendor.The remaining sections will show that several of

these single-resource insights extend to networks withthe important exceptions that some risk-averse capac-ities as well as total spending may exceed risk-neutrallevels. We start with the dedicated network before ana-lyzing the serial and parallel networks.

5. Resource Diversification: TheDedicated Network

The dedicated network (Figure 1(a)) is the simplestnetwork. While its two resources lack operational

dependence, as a portfolio the network enjoys purefinancial diversification and provides a “natural” orpassive hedge: total profit variability risk is reducedby not putting all eggs into one basket. Risk aver-sion thus induces resource investment dependenceand will favor the resource with lower-profit variance.Using a numerical study, we first discuss the insightsfrom the single-resource newsvendor that carry overto the dedicated network and highlight the differ-ences. Then, we generalize insights with analyticallyproven properties. To ensure that both investmentsare profitable, both market values should exceed theircosts: vi ≥ ci.

5.1. Risk ExposureFor a risk-neutral newsvendor, the dedicated networkdecomposes into two independent single-resourceproblems. With normal demand, capacities are linearin standard deviations with a slope determined by thecritical fractile but independent of demand correlation.The “o”-connected lines in Figure 5 illustrate thesefacts with a numerical study assuming bivariate nor-mal demand with unit means 11 and 1 = 0 752,equal unit investment costs c = 0 50 5, and net val-ues v = 21. (Market 1 thus is more profitable and hassmaller demand risk.) The critical fractiles are 3/4 and1/2 so that Kn

1 is increasing in 1, while Kn2 is constant

in 2.

5.2. Risk AversionAs for a single-resource newsvendor, the numeri-cal analysis of Figure 5 suggests that CARA utility-optimal capacities are decreasing in risk aversion androughly quadratic in standard deviation for nonnega-tively correlated normal demand. The striking differ-ence is that Ku

2 increases in risk aversion and standarddeviation for negative correlation.


Figure 5 With Normally Distributed Demand, the Risk-Neutral Dedicated Network Adjusts Capacities Linearly (Labeled “o”) in Demand StandardDeviation Independent of Correlation. Risk-Averse Levels Are Quadratic and Can Be Above Risk-Neutral Levels with Strong DiversificationBenefits at Low Correlation

0 0.30.8

1.0

1.2

0.8

1.0

1.2

0.8

1.0

1.2

Risk exposure (std. dev. σ1 = 0.75σ2) Risk exposure (std. dev. σ1 = 0.75σ2) Risk exposure (std. dev. σ1 = 0.75σ2)0 0.3 0 0.3

Risk aversion γ

K2K2

K2

K1 K1 K1

(b) Zero correlation (ρ = 0) (c) Positive correlation (ρ = +1)(a) Negative correlation (ρ = –1)

γγ

γ

γ

γ

Consider Figure 6 to further investigate and explainthe crucial role of correlation (for the same data asFigure 5). Figure 6(b) shows that Ku

2 increases inrisk aversion for any correlation coefficient belowabout −0 45. Furthermore, for below about −0 7, itsincrease outweighs the decrease of Ku

1 so that the totalspend of a risk-averse newsvendor exceeds the risk-neutral monetary investment (Figure 6(d)). Finally,Figure 6(c) shows that a risk-averse newsvendorstrictly prefers more negatively correlated demand.Put another way: correlation is another measure ofrisk exposure. (Figure 6(c) also shows that the mean-variance frontiers dominate (by definition!) yet closelyapproximate the CARA efficient frontier in the rele-vant domain of hedging—up to 50% variance reduc-tion with less than 10% value cost compared to therisk-neutral configuration.) These effects do not existin the single-resource case and can be explained byrisk diversification or hedging.

5.3. Operational HedgingThe dedicated network enjoys classic diversificationbenefits. Investing in two resources yields a diver-sified portfolio that pools risks and can be config-ured in the same manner that financial investors(re)balance portfolios to optimize their MV utilities.Increased risk aversion reduces profit variance at theexpense of some mean value. Abstracting from capac-ity constraints, market i has profit variability risk(std. dev.) vii. If we were to invest one dollar andallocate weight wi to market i (w1 + w2 = 1), theprofit risk of the portfolio would be w1v11

2 +

2w1v11w2v22+ w2v2221/2, which increases in

correlation , but never exceeds the sum of marketprofit risks. In other words, risk-pooling benefits arezero at =+1, but grow with smaller correlation andmore equal profit variances. (A zero-variance portfo-lio or perfect hedge obtains with two perfectly nega-tively correlated assets with equal v11 = v22.) Thisrisk-pooling benefit from diversification suggests thatincreased risk aversion will rebalance investment tofavor the market with lower-profit standard deviation.With capacity constraints, the diversification benefit

must be balanced with its cost in terms of investmentcosts and changes in expected profits. The numericalanalysis of Figure 6(d) shows that at sufficiently nega-tive correlation, the diversification benefits are so largethat a risk-averse newsvendor will invest more than arisk-neutral newsvendor! In our example, resource 2has lower profit risk (v22 = 2/3v11) and its fractionof total capacity (and investment) indeed increasesin risk aversion except for high positive correlation,according to Figure 7. (This demonstrates the impor-tance of profit over demand standard deviations; recallthat market 1 is more profitable and has less demandrisk: 1 = 0 752.) Note that rebalancing the capac-ity portfolio toward resource 2 is beneficial even withuncorrelated demand (although it then coincides witha reduction of both capacity levels). At =+1, diver-sification benefits disappear and capacity mix choiceis dominated by expected profit maximization, whichfavors the higher margins of market 1.The key insights are that risk-averse newsvendors

gain diversification benefits by rebalancing network


Figure 6 Capacities, Frontiers, and Monetary Investment in the Dedicated Network as a Function of Correlation and Risk Aversion1 = 03 2 = 04

–1.0 –0.5 0 0.5 1.00.6

0.8

1.0

1.2

0.6

0.8

1.0

1.2

Correlation coefficient ρ–1.0 –0.5 0 0.5 1.0

–1.0 –0.5 0 0.5 1.0

Correlation coefficient ρ

Correlation coefficient ρ

(a) Capacity K1

$0.8

$0.9

$1.0

$1.1

$1.2

0 0.5$1.50

$1.66

(b) Capacity K2

(d) Total financial investment c′K(c) Efficient vs. mean-variance frontiers

Risk aversion γ

γ

γ

γ

ρ = +1

ρ = –1

γ

Correlation

Variance of network value, σ2(K)

Mea

n of

net

wor

k va

lue

µ(K

)

MVfrontier

Efficientfrontier

capacity toward resources that supply the lower-profitvariance market. Total spending may even exceed therisk-neutral monetary investment.

5.4. GeneralizationsThese insights from the numerical analysis can begeneralized to general demand distributions and util-ity functions under certain conditions. (Precise ana-lytic conditions of all properties are relegated to theirproofs in the appendix.)

Property 2 (Dedicated Network). Ku ≤ Kn andresource levels always decrease in risk aversion if correla-tion = 1. However, Ku

2 ≥Kn2 and increases in risk aversion

with correlation =−1, v22 < v11, moderate risk aver-sion, and standardized safety levels ziK

ni > 0.

Remark. For symmetric demand distributions, itsuffices that ci < 2vi to yield positive standardizedsafety levels ziK

ni . The proof specifies (weaker) con-

ditions analytically and extends the ingenious tech-nique of Eeckhoudt et al. (1995) of bounding themarginal utility to a newsvendor network.

The risk-averse investor rebalances the capacity mixtoward the lower-profit variance market, and the con-dition v22 < v11 suggests that this lowers the vari-ance of the network profit. Indeed, profit variance isdecreasing in K2 so that risk aversion increases thelevel of K2 beyond the risk-neutral level:

Property 3 (Dedicated Network). 22Kn ≤ 0

and d/d'K20 ≥ 0 with correlation = −1, v22 <v11, moderate risk aversion, and standardized safety levelsziK

ni > 0.

These theoretical results formalize and generalizethe intuitive explanation in terms of the diversificationbenefits and costs. The result of Eeckhoudt et al. (1995)extends to dedicated networks if correlation = 1, butfails with negative correlation and moderate costs andrisk aversion; diversification benefits are the culprit.A continuity argument directly extends these results toa neighborhood of =±1: At high correlation, diver-sification benefits become negligible and both resourcelevels always decrease in risk aversion. It is easy toshow that this also happens with CARA utility if


Figure 7 Operational Hedging in the Dedicated Network Is Manifested by Capacity Rebalancing Toward the Minimal-Profit Variance Market ifDiversification Benefits Outweigh Costs (Panels (a) and (b))

0 0.3

(a) Negative correlation (ρ = –1) (b) Zero correlation (ρ = 0) (c) Positive correlation (ρ = +1)

0 0.30 0.3Risk exposure (std. dev. σ1 = 0.75σ2) Risk exposure (std. dev. σ1 = 0.75σ2) Risk exposure (std. dev. σ1 = 0.75σ2)

42%

50%

52%

42% 42%

50%

52%

50%

52%

Risk aversion γ

Capacity mix K2/(K1+ K2)

K2/(K1+ K2)

K2/(K1+ K2)γ

γ

demands are independent (and thus = 0).9 At suf-ficiently low correlations, however, the diversificationbenefit is sufficient for a risk-averse newsvendor toincrease the resource supplying the lower-profit vari-ance market provided capacity is not too expensive.(This suggests that a negative correlation threshold—like −0 45 in Figure 6(b)—may exist for CARAand normal demand below which Ku

2 ≥ Kn2 and above

which Ku2 ≤Kn

2 .)

6. Resource Sharing andComplementarity: The SerialNetwork

To see how the insights from the dedicated networkcarry over to networks whose resources feature oper-ational dependence, consider the serial network ofFigure 1. Besides financial diversification, this net-work also benefits from operational flexibility throughthe shared resource. We will see that allocation flex-ibility adds two additional benefits: demand poolingand ex post revenue maximization, which provides anactive hedge.The question in the two-stage serial network is

what the three resource levels should be, and thetension is about complementarity (between dedicatedand shared resources) and bottlenecks. Expected profitoptimization (3) sets the probability that a resource isa bottleneck proportional to its cost. Thus, any costvector c is admissible in the serial network as long

9 The optimality equations then decouple and both resourcesbehave as single-resource systems. CARA is necessary because ingeneral, the wealth effect couples investments even with indepen-dent demand.

as both products or markets are economically viable:ci + c3 < vi.Again, we first discuss insights from a numerical

study and then generalize. The numerical data is asearlier except that marginal investment costs are c =0 10 10 4. This choice allows meaningful compar-isons because, abstracting from resource sharing, ityields the same critical fractile as for the dedicated net-work.

6.1. Risk ExposureRisk-neutral capacities in newsvendor networks withnormal demand are linear in (marginal) demand stan-dard deviations. Indeed, the multivariate normal dis-tribution scales linearly in standard deviations andthe risk-neutral conditions (3) are expressed purely inprobability fractiles. In contrast to the dedicated net-work, however, the slopes now depend on demandcorrelation as well as on the value-cost ratios c/v.The “o”-connected lines in Figure 8 illustrate thesefacts through a numerical study where both risk-neutral dedicated resource levels decrease in correla-tion, while the risk-neutral shared resource level Kn

3increases in correlation. The shared resource providestwo benefits that were absent in the dedicated networkand explains the impact of correlation on risk-neutralcapacities:(1) Demand pooling refers to serving multiple mar-

kets by one resource, e.g., through inventory central-ization as first studied by Eppen (1979). Diversificationrefers to serving multiple markets from one portfolioor network. Demand pooling thus is a special formof diversification and risk pooling. It provides therisk mitigation benefits of §4 that are valued by risk-averse investors. In addition, demand pooling reduces


Figure 8 With Normally Distributed Demand, the Risk-Neutral Serial Network Adjusts Capacities Linearly (Labeled “o”) in Demand StandardDeviation with Slope Dependent on Correlation. Risk-Averse Levels Are Quadratic and Can Be Above Risk-Neutral Levels at LowCorrelation Reflecting Strong Risk-Pooling Benefits and Complementarity

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Risk exposure (std. dev. σ1 = 0.75 σ2) Risk exposure (std. dev. σ1 = 0.75σ2) Risk exposure (std. dev. σ1 = 0.75σ2)0 0.3 0 0.3 0 0.3

Dedicated K1

Risk aversion γ

Shared K3

Dedicated K2

K3

K2

K1

K3

K1

K2


γγ

γ

γ

γ

γ

γ

γ

mismatch costs and affects the sizing of the sharedresource10 which also benefits a risk-neutral newsven-dor (but less so as correlation increases).11

(2) Ex post revenue profit maximization refers to theex post allocation option to steer or switch the out-put mix toward the more profitable market when theshared resource is capacity constrained. This bene-fit was first identified in Van Mieghem (1998, Propo-sition 3) and requires a market profit differential(v1 = v2). It is distinct from risk pooling as it surviveseven with perfect positive correlation (as the remain-der will demonstrate). Indeed, by exploiting upsidevariations, it increases profits and thus also benefitsrisk-neutral newsvendors.

6.2. Risk AversionThe numerical study suggests that the CARA util-ity dedicated capacities with normal demand mir-ror those in the dedicated network: In Figure 8, bothappear roughly quadratic in standard deviation and

10 The shared capacity is driven by a critical fractile of the sum ofmarket profit distributions whose standard deviation is less thanindividual distributions with strong negative correlations. Eppen(1979) called this effect statistical economies of scale.11 This agrees with Proposition 3 in Van Mieghem and Rudi (2002),which states that the mean value of any newsvendor networkwhose operating profit is submodular in D (which is the casefor the serial network) is decreasing in any correlation coefficient.Figure 9(a) also illustrates that even a risk-neutral newsvendorstrictly prefers more negatively correlated demand.

Ku2 as well as the shared resource level Ku

3 increasein risk aversion for strongly negatively correlateddemand. In this example, the combined benefits fromrisk pooling and revenue maximization are so largewith strongly negative correlations that a risk-aversenewsvendor invests more money in the serial networkthan a risk-neutral investor (Figure 9(b)).

6.3. Operational HedgingSerial network newsvendors purposely imbalancecapacity to mitigate profit risk. A capacity portfo-lio K is balanced if all resources can be fully uti-lized simultaneously. In a newsvendor network, thisrequires the existence of an activity vector x ≥ 0 suchthat Ax = K (Van Mieghem 2003, Definition 1). Inthe serial network, this condition simplifies to K1 +K2 =K3, which is optimal with deterministic demand.With demand uncertainty, however, Harrison and VanMieghem (1999) first showed that it is optimal for arisk-neutral newsvendor to under-invest in the sharedresource (K3 < K1+K2). Three effects now explain suchcapacity imbalance. First, demand pooling is an obvi-ous driver and suggests that relative capacity imbal-ance would decrease in correlation. Figure 10 confirmsthat suggestion in our numerical study, but also showsthat capacity imbalance may remain optimal with per-fect positive correlation. This highlights the revenuemaximization option as the second driver (given thatthere is no risk-pooling benefit at = 1). Finally, to


Figure 9 Serial Network Newsvendors Prefer Smaller Correlations (Frontiers Move Northwest). Risk Aversion May Increase Spending AboveRisk-Neutral Levels with Low Correlation 1 = 03 2 = 04

–1.0 –0.5 0 0.5 1.0$0.8

$0.9

$1.0

$1.1

$1.2

0 0.1 0.2 0.3 0.4 0.5$1.3

$1.4

$1.5

$1.6

$1.7

$1.8

$1.9(b) Total financial investment c′K(a) Efficient vs. mean-variance frontiers

Correlation coefficient ρValue variance σ2(K)

Mea

n va

lue

µ(K

) Correlation

MVfrontier

γ

γEfficientfrontier

ρ = –1

ρ = +1

exercise its switching option, the shared resource musthave upstream capacity leeway.Relative capacity imbalance decreases in correlation,

but is amplified by risk aversion, especially at corre-lation = 1 according to the numerical study in Fig-ure 10. In the absence of risk pooling, this reflects therevenue maximization option. While all three resourcelevels decrease in risk for = 1, the reduction inthe shared resource exceeds that in the dedicatedresources, thereby increasing the relative potential ofthe switching option. (The smaller curvature of thefrontiers in Figure 9(a) suggests that capacity imbal-ance may be more effective at higher correlations.Numerical analysis also shows that capacity imbal-ance increases in the profit differential v1−v2, but dis-appears at = 1 if v1 = v2.)Given that the optimal risk-averse resource vector

seeks to reduce profit variance, this means that more

Figure 10 Operational Hedging in the Serial Network Rebalances Capacity Away from the Shared Resource. Imbalance Decreases in Correlationbut Increases in Risk Aversion

0%

10%

20%

30%

40%

50%

0%

10%

20%

30%

40%

50%

0%

10%

20%

30%

40%

50%

Risk exposure (std. dev. σ1 = 0.75σ2)0 0.3




Risk aversion γ

Capacityimbalance

(K1+ 2–K3)/K1+ 2 + 3 Capacityimbalance

(K1+ 2–K3)/K1+ 2 + 3

Capacityimbalance

(K1+ 2–K3)/K1+ 2 + 3

γ

γ

resource imbalance reduces more profit risk. Resourceimbalance thus truly is an operational hedge.

6.4. GeneralizationsThe insights from the numerical study hold for gen-eral demand distributions and utility functions undercertain conditions:

Property 4 (Serial Network). Ku2 ≥ Kn

2 and in-creases in risk aversion with correlation = −1, v22 <v11, and moderate costs and risk aversion.

(The proof details the cost conditions which can beexpressed in terms of the safety resource levels andthus c/v fractions.) The driving force behind the in-crease of K2 again is the reduction of profit variance:

Property 5 (Serial Network). Assume correla-tion = −1 and moderate costs and risk aversion. If


Figure 11 With Normally Distributed Demand, the Risk-Neutral Parallel Network also Adjusts Capacities Linearly (Labeled “o”) in DemandStandard Deviation with Slope Dependent on Correlation. Risk-Averse Levels Are Roughly Quadratic. Flexibility Always Increases inRisk Aversion Reflecting Strong Risk Pooling and Revenue Maximization Benefits and Substitution

Risk exposure (std. dev. σ1 = 0.75σ2)0

0

1.0

1.2

0

1.0

1.2

0

1.0

1.2

0.3Risk exposure (std. dev. σ1 = 0.75σ2)

0 0.3Risk exposure (std. dev. σ1 = 0.75σ2)

0 0.3


Dedicated K1

Risk aversion γ

K1

K1

K2

K3K3

K2

Flexible K3

Dedicated K2

γ

γ

γ

γγ

γ

γ

γ

v22 < v11, then 22Kn ≤ 0 and d/d'K20 ≥ 0. If

v1 = v2, then 32Kn≥ 0.

The impact of risk aversion on the shared resourcelevel is parameter dependent. Figure 8 shows thatthe shared resource can increase in risk aversion withstrongly negative correlations, reflecting the comple-mentarity in the network. The diversification benefitdrives more risk-averse agents to increase K2, and asufficiently strong increase induces an increase in theshared resource to alleviate its potential of being a bot-tleneck. With higher correlations, the diversificationbenefit weakens as does its complementarity on K3,which becomes decreasing in risk aversion. Asidefrom this complementarity, the shared resource levelis also driven by the revenue maximization option:With v1 = 2v2, its value at =−1 was sufficient for K3to increase in risk aversion in our numerical results,while K3 decreases in risk aversion without revenuemaximization (i.e., if v1 = v2) according to Property 5.

7. Resource Flexibility andSubstitution: The Parallel Network

The parallel network can be viewed as a dedicated net-work augmented with the option to use a third flexi-ble resource which can only improve upon the dedi-cated network’s performance. Allocation flexibility inthe parallel network again provides an active hedge,but its dependence differs from that in the serial net-work. In the latter, the shared resource was a potentialbottleneck and a complement of the upstream dedi-cated resources, but its capacity was underweight to

enable its switching option. Here, the flexible resourceadds redundancy and is a substitute of the dedicatedresources. The question in the single-stage parallel net-work is whether the substitutable resource 3 is a viablealternative and what its resource level should be. Thesubstitution tension depends on the relative cost ofresource 3, and the natural and simplest assumptionsfor the parallel network are maxc1 c2 < c3 < c1 + c2and ci ≤ vi.Again, we first make observations from a numeri-

cal study and then generalize. The numerical data isthe same as before, but with more expensive flexiblecapacity costs c3 = 0 7> c1 = c2 = 0 5.

7.1. Risk ExposureAgain, risk-neutral capacities with normal demandremain linear in (marginal) demand standard devia-tions with slopes depending on demand correlation aswell as on the value-cost ratios c/v. The “o”-connectedlines in Figure 11 illustrate these facts through anumerical study where both risk-neutral dedicatedresource levels increase in correlation, while the risk-neutral flexible resource level decreases. The latterstems from decreasing risk-pooling benefits, and thesubstitution effect explains the former. Interestingly,flexibility is used even at perfect positive correlation;while there is no risk pooling, the revenue maximiza-tion option of flexibility remains.

7.2. Risk AversionThe numerical study suggests that the CARA utilitydedicated capacities with normal demand again mir-ror those in the dedicated network. In Figure 11, both


Figure 12 Parallel Network Newsvendors Prefer Smaller Correlations (Frontiers Move Northwest). Risk Aversion May Increase Spending AboveRisk-Neutral Levels with Low Correlation 1 = 03 2 = 04

–1.0 –0.5 0 0.5 1.0$0.8

$0.9

$1.0

$1.1

$1.2

0 0.5$1.5

$1.6

$1.7

$1.8

(b) Total financial investment c′K(a) Efficient vs. mean-variance frontiers

Correlation coefficient ρValue variance σ2(K)

Mea

n va

lue

µ(K

) Correlation

MVfrontier

Efficientfrontier

γ

γρ = –1

ρ = +1

appear roughly quadratic in standard deviation andKu2 increases in risk aversion for strongly negatively

correlated demand. Interestingly, more risk-aversenewsvendors use more flexibility for any correlationin this example, i.e., the result of Eeckhoudt et al.(1995) does not extend to serial networks (even notat = 1). With strongly negative correlations, its ben-efit is so large that it is optimal to “oversubstitute”dedicated capacity with flexible at higher total spend-ing than under risk neutrality (Figure 12(b)). Figure12(a) shows that even a risk-neutral newsvendor againstrictly prefers more negatively correlated demandand that the parallel network dominates the dedicatednetwork (compare with Figure 6(c)).

7.3. Operational HedgingFirst, compared to the dedicated network, the flexibleresource provides additional risk mitigation and value

Figure 13 Operational Hedging in the Parallel Network Rebalances Capacity Toward the Flexible Resource. The Flexible Share Increases in RiskAversion, but Decreases in Correlation

0 0.3

(a) Negative correlation (ρ = –1)



Risk exposure (std. dev. σ1 = 0.75σ2)

(b) Zero correlation (ρ = 0) (c) Positive correlation (ρ = +1)

Risk aversion γ

Flexiblecapacity mix

K3/K1+ 2 + 3


K3/K1+ 2 + 3


K3/K1+ 2 + 3

0%

10%

20%

30%

40%

0%

10%

20%

30%

40%

0%

10%

20%

30%

40%

γ

γ

creation: the frontiers of Figure 12 lie northwest rela-tive to those of Figure 6(c). Second, parallel networknewsvendors move the investment mix toward theflexible asset to mitigate profit risk. The flexible capac-ity share decreases in correlation (reflecting decreasedrisk pooling benefits), but does not disappear at =1 (reflecting revenue maximization benefits) in thenumerical study of Figure 13. More importantly, theflexible share increases in risk aversion, especially atcorrelation = 1. (The dedicated resources K1 and K2are strongly decreasing in risk at = 1, thereby free-ing up funds that are partially invested to increase theflexible resource level. Numerical analysis shows thatflexibility increases in the profit differential v1−v2, butdisappears at = 1 if v1 = v2.)Flexibility has been interpreted as an operational

hedge in risk-neutral models. This paper shows thatthe capacity mix moves even more toward flexibility


as risk aversion increases. Given that the optimal risk-averse resource vector seeks to reduce (variance) risk,this means that more flexibility reduces more profitvariability, reinforcing its interpretation as a hedge.

7.4. GeneralizationsThe insights from the numerical study can again begeneralized to other demand distributions and util-ity functions (but somewhat less so than earlier).Assuming that a rational risk-neutral agent investsin flexibility (see Van Mieghem 1998, Proposition 7for conditions), risk aversion moves the capacity mixtoward the lower-profit variance resource 2 and theflexible resource 3:

Property 6 (Parallel Network). Assume corre-lation = −1 and moderate risk aversion and flexiblecost so that Kn

3 > 0. If 1 = 2 and v1 > v2, thenKu2 + Ku

3 ≥ Kn2 + Kn

3 and increases in risk aversion, whileKu1 ≤Kn

1 and decreases in risk aversion.

8. Conclusion, Limitations, andExtensions

This paper has shown how resource allocation in net-works can mitigate risk exposure. Risk attitude andnetwork configuration drive the strategic placement ofoperational resources like safety capacity and inven-tory to reduce financial risk such as mismatch costsand profit variability. This joint operational and finan-cial perspective was adopted to develop theory andinsight into capacity imbalance and allocation flexibil-ity to mitigate risk and serve as operational hedges.They may even exploit risk and create value.Risk-averse newsvendors may increase network

capacity and total spending above risk-neutral lev-els because rebalancing capacity may decrease profitvariance. (Increasing capacity captures the multivari-ate demand distribution more fully and increases risk-pooling benefits, especially with negative correlations.In contrast, with access to only a single asset anda single risk, decreasing capacity is the only way todecrease profit variance.)12 With normally distributeddemand, risk-averse newsvendors change resourcelevels roughly proportionally to demand variance,while risk-neutral agents adjust only proportionally tostandard deviation. This was explained in terms of thebenefits from pure diversification, demand pooling,and ex post optimization; Table 1 provides a summary.The managerial take-away is that risk-averse news-

vendors faced with increased risk exposure shouldoveradjust their resource portfolio relative to their

12 The access conditions are necessary: Chod et al. (2006) and Dinget al. (2007) show that single-resource risk-averse newsvendorswith access to a financial (i.e., second) asset or two risks may alsoincrease resource capacity beyond the risk-neutral level.

risk-neutral counterparts. Sometimes they shouldincrease capacity, but the appropriate actions dependon market profit (co)variances and network struc-ture. The theory suggests some rules of thumb forstrategic placement of safety capacity and inven-tory in networks: Inexpensive resources supplyingthe lower-profit variance market may be increasedfor operational hedging, especially with strong nega-tive correlations. When markets differ in profitability,capacity may be rebalanced toward substitutable flex-ible resources, but away from shared resources. Giventhat both types of resources are “product-flexible,”this means that the appropriate hedging action forproduct-flexible resources depends on their networkposition: increase when in parallel with dedicatedresources, decrease when in series. (We suspect thatthe redundancy inherent in parallelism plays a role inthis difference in response, but leave that investigationto future work.)Some testable hypotheses of our theory include: the

capacity mix of firms with a natural hedge is nega-tively correlated with market profit variances. Firmswith redundant flexible capacity have higher valueand smaller profit variability than similar dedicatedfirms. Resource imbalance in firms with shared capac-ity is positively correlated with market demand stan-dard deviation, but decreases in market correlation.Our analysis and insights, however, have many

limitations and much remains to be done. Whilenewsvendor networks have several advantages (seethe introduction), their main disadvantage is that theymay be too stylized to capture details necessary forpractical decision-support systems. The single-periodmodel abstracts from real dynamics. The recent work-ing paper by Zhu and Kapuscinski (2006) providesa first and promising extension of the analysis todynamic risk-averse newsvendor networks just likeBouakiz and Sobel (1992) and Chen et al. (2004) didfor the single-resource model. Few organizations arecontrolled by a single decision maker and only allowfor input inventories (yet have general resource net-works). Multiagent newsvendor networks are a nat-ural extension (e.g., Van Mieghem 1999, Goyal andNetessine 2007) as is allowing inventory stocking atmultiple stages, but adding risk aversion will furthercomplicate analysis. von Neumann andMorgenstern’scelebrated utility approach to decision making underuncertainty has well-known limitations: it is not themost general or basic way to describe human behav-ior (e.g., see Fishburn 1982, Heyman and Sobel 1984);the axioms postulated to guarantee the existence of autility function are often violated in practice; humanbehavior is far more complex than that implied byincreasing concave utility functions (see a reviewpaper by Rabin 1998, Tomlin and Wang 2005 for con-sidering loss aversion and value-at-risk measures);


and constructing a utility function or soliciting prefer-ences in practice is a daunting task. In light of this, itwould be interesting to analyze whether certain net-works are more robust to parameter estimation errorsthan others.This paper provides some general mathematical

expressions, but only applies them in the limited set-ting of three networks whose properties mostly focuson the local neighborhood of the risk-neutral solu-tion. The future task is to expand the set of networksand the generality of the analysis to the extent possi-ble. A first and natural approach is to use the modelhere as a numeric optimization tool to quickly iden-tify and compare a number of promising networkconfigurations, following the approach of Gravesand Willems (2000) and Graves and Tomlin (2003)in response to industry practitioners’ need. From aresearch perspective, the next task is to increase struc-tural insight for general networks, like Jordan andGraves (1995) successfully did for risk-neutral net-work configuration for flexibility. One would like tocharacterize which network structures and parametersyield effective hedging and how each resource shouldbe adjusted as risk exposure changes. This surely isa difficult assignment that probably is best addressedpiecemeal wise. Eventually, one would like to formu-late rules that specify which network modules, andeven complete designs, are appropriate for given envi-ronments.

9. Electronic Companion and ProofsAn electronic companion to this paper is available aspart of the online version that can be found at http://mansci.journal.informs.org/.

AcknowledgmentsThe author is grateful for stimulating discussions withPhilipp Afeche (who suggested analyzing the dedicated net-work as the simplest network), Jiri Chod, Marty Lariviere,Lauren Lu, Panos Kouvelis, Nils Rudi, and Costis Skiadas.The author greatly benefited from the excellent commentsand questions from three peer reviewers and the associateeditor, which substantially improved this paper.

Appendix. Proof of Property 2Optimal activities are simple: xiKD=minKiDi so that8iKD = Ki

KD = vi1!Di≥Ki". The demand space can

be partitioned accordingly, as shown in Figure A1. Abbre-viate i ∪ j by ij and Pij K by Pij K. According toProposition 2, the optimality conditions for the risk-averseresource vector Ku are

0= v1− c1Ɛ34u′V KuD+W

− c1Ɛ01u′V KuD+W (11)

0= v2− c2Ɛ13u′V KuD+W

− c2Ɛ04u′V KuD+W (12)

where Ɛif = ∫i

f dP denotes partial expectation over i.

Part 1. =+1. Any perfectly positively correlated distri-bution on 2

+ has an upward-sloping line as support:

D1−1

1= D2−2

2⇔ z1D1= z2D2⇔D1 = aD2+ b+

with probability 1 (13)

using the standardized safety level notation (10) and wherea = 1/2 > 0 and b+ = 1 − a2. There are two possiblecases.

Case 1. Ku falls below the demand line so that P4Ku= 0,as in Figure A1(b). The proof uses the point x1K = aK2+b+K2 to partition and bound marginal utilities as follows.Recall that V KD= v′xKD−CK, where xKD fallson the the bold line in Figure A1, which is the smallerof capacity (boundary of 0) and demand. Define V1K =v′x1K−CK. Clearly, v′xKD, and thus V is increasingin the direction of v so that V KD1 ≤ V1K ≤ V KD2for any D1 ∈ 0K and D2 ∈ 13K. Because u′ is strictlydecreasing, u′V KD1≥ u′V1K≥ u′V KD2, so that(12) yields

0 ≤ u′V1Ku+W v2− c2P13K

u− c2P0Ku

P0=1−P13⇒ P13Ku= PD2 > Ku

2 ≥ c2v2

Similarly, use the point Ku and V0Ku = v′Ku − CKu toestablish that (11) yields

0 ≤ u′V0Ku+W v1− c1P3K

u− c1P01Ku

⇒ P3Ku= PD1 > Ku

1 ≥ c1v1

Kn satisfies PDi > Kni = ci/vi so that Ku

i ≤ Kni . An increase

in risk aversion is equivalent to a concave increasing trans-formation h of the utility function u, where h′ > 0 andh′′ < 0. The gradient of UhuKW= ƐhuV K+W atKu can also be signed:

1UhuKuW

=Ɛ81Ku−c1h

′uV Ku+W u′V Ku+W

= v1− c1Ɛ34h′uV KuD+W u′V KuD+W

− c1Ɛ01h′uV KuD+W u′V KuD+W

≤ h′uV0Ku+W v1− c1Ɛ3u

′V KuD+W

− c1Ɛ01u′V KuD+W = 0

where we used (11) and the fact that h′ is decreasing. Asimilar argument shows that also 2UhuKuW ≤ 0. Thisholds for arbitrary h and u, which together with concavityof Uhu means that Khu ≤Ku.

Case 2. Ku falls on or above the demand line so thatP1Ku = 0. Partition using the points x2K = K1 a−1K1 −b+ and Ku to establish that P34Ku = PD1 > Ku

1 ≥ c1/v1and P3Ku= PD2 > Ku

2 ≥ c2/v2, respectively.Part 2. = −1. The demand support is now downward

sloping: z1D1 + z2D2 = 0 ⇔ D1 + aD2 = b, where b =1 + a2 > 0. In the setting of Figure A1(a), use the pointx1K= b−aK2K2 and scalar k1K= v′x1/v1 (the horizon-tal intercept of the normal to v through x1) to again parti-tion and bound marginal utilities to establish that P1Ku=


Figure A1 Dedicated Network Activities x Depend on Resource Levels K and Demand D

Ω0

Ω1 Ω3

Ω4

Ω1 Ω3σ2/σ1

–σ2/σ1

Ω4

D2 D2

D1 D1K1K1k1

K2

K2

v

x1

x1

Dx

D = x

K

v

(a) Negative correlation ρ = –1 (b) Positive correlation ρ = 1

K

Ω0

Note. Demand is on the downward-sloping line if =−1 (a) and on the upward-sloping line if = 1 (b).

PD2 > Ku2 ≤ c2/v2. In contrast to Part 1, Part 2 requires con-

ditions: (1) v is below the demand normal or v2/v1 < 1/2;(2) k1Ku < Ku

1 or z1Ku1 + 1 − v22/v11z2K

u2 > 0; and

(3) Ku falls above the demand line or z1Ku1 + z2K

u2 > 0.

If risk aversion is moderate so that Ku falls in a Kn neigh-borhood, then conditions (2) and (3) evaluated at Kn suffice.Similarly, establish that 2UhuKuW ≥ 0. Given concav-ity, this means that an infinitesimal increase h in risk aver-sion relative to any u that satisfies the conditions leads toKhu2 > Ku

2 .

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