capacity management, investment, and hedging

Commissioned PaperCapacity Management, Investment,and Hedging: Review and Recent

Developments

Jan A. Van MieghemKellogg School of Management, Northwestern University,

Evanston, Illinois [email protected]

This paper reviews the literature on strategic capacity management concerned with deter-mining the sizes, types, and timing of capacity investments and adjustments under uncer-

tainty. Specific attention is given to recent developments to incorporate multiple decisionmakers, multiple capacity types, hedging, and risk aversion. Capacity is a measure of pro-cessing abilities and limitations and is represented as a vector of stocks of various processingresources, while investment is the change of capacity and includes expansion and contrac-tion. After discussing general issues in capacity investment problems, the paper reviewsmodels of capacity investment under uncertainty in three settings:The first reviews optimal capacity investment by single and multiple risk-neutral decision

makers in a stationary environment where capacity remains constant. Allowing for multiplecapacity types, the associated optimal capacity portfolio specifies the amounts and locationsof safety capacity in a processing network. Its key feature is that it is unbalanced; i.e., regard-less of how uncertainties are realized, one typically will never fully utilize all capacities.The second setting reviews the adjustment of capacity over time and the structure of opti-mal investment dynamics. The paper ends by reviewing how to incorporate risk aversion incapacity investment and contrasts hedging strategies involving financial versus operationalmeans.(Capacity; Investment; Expansion; Planning; Real Options; Hedging; Risk; Mean-Variance)

1. IntroductionThis paper reviews the literature on strategic capac-ity management concerned with determining thesizes, types, and timing of capacity adjustmentsunder uncertainty. Specific attention is given torecent developments to incorporate multiple deci-sion makers, multiple capacity types, hedging, andrisk aversion. Given the diverse interpretations and

application domains of capacity,1 any review must beselective. Capacity typically describes abilities and lim-itations. In operations management, it is natural to

1 There are more than 15,000 peer-reviewed articles in the ProQuestBusiness Databases with “capacity” in the title or key words, 2,632of them published during 1999–2002 alone. Restricting the searchby adding “planning,” “expansion,” or “investment” to “capacity”retained more than 5,000 articles.

1523-4614/03/0504/02691526-5498 electronic ISSN

Manufacturing & Service Operations Management © 2003 INFORMSVol. 5, No. 4, Fall 2003, pp. 269–302

VAN MIEGHEMCapacity Management, Investment, and Hedging: Review and Recent Developments

consider a network of various processing resources,also called a processing network. The types andamounts of these resources, which are the prime eco-nomic factors of production, are important determi-nants of the network’s production abilities and limi-tations. Deciding on the types of resources relates toprocessing-network design in operations research, orcharacterizing the production and operating-profitfunctions in economics. Deciding on the amountsinvolves optimization of a given network oroperating-profit function. While many other factors,such as inventory supply and energy shortages, qual-ity and yield losses, scheduling, lead times, and relia-bility, may also limit production, this paper will turnup the brightness on the direct effects of resourcescarcity and uncertainty, and turn it down on otherfactors. In this paper then, capacity is a measure ofprocessing abilities and limitations that stem from thescarcity of various processing resources and is rep-resented as a vector of stocks of various processingresources. The operational consequence is that capac-ities in this paper almost always can be interpretedas some upper bounds on processing quantities, buta more general, higher-level economics interpretationthat links capacity directly to operating profits willalso be discussed.While capacity refers to stocks of various resources,

investment refers to the change of that stock over time.Investment2 thus involves the monetary flow stem-ming from capacity expansion and contraction in theexpectation of future rewards. According to Dixit andPindyck (1994), most investment decisions share threeimportant characteristics in varying degrees. First, theinvestment is partially or completely irreversible inthat one cannot recover its full cost should one havea change of mind. Second, there is uncertainty overthe future rewards from the investment. Third, thereis some leeway about the timing or dynamics of theinvestment. In addition to these three, this paper addsa fourth characteristic: multidimensionality. Typically,a firm invests in multiple types of resources that havedifferent financial and operational properties. Deci-sions about the types and levels of investment are

2 Originally, investment was “the act of putting clothes or vestmentson.” (Shakespeare 1597, 2 Hendrick IV, iv. i. 45)

interdependent and the firm’s productive capabilitiesdepend on the complete vector of capacity levels,which we will call its capacity portfolio.3

Our outline and objectives are as follows. Section 2reviews various literatures that deal with capacityinvestment. Section 3 reviews important issues in theformulation of any capacity problem. The remain-der of the paper reviews optimal capacity investmentunder uncertainty that is partially irreversible.Section 4 reviews optimal capacity investment by

risk-neutral decision maker(s) in a stationary envi-ronment where capacity remains constant over time.The emphasis of this section is on determining theoptimal levels of various types of capacities and thetrade-offs among them. The optimal portfolio speci-fies the amounts and locations of safety capacity inthe network. Section 4.1 reviews canonical capacitymodels that adopt queuing or newsvendor networkformulations, an example of which is studied in §4.2.Section 4.3 reviews game-theoretic capacity invest-ment by multiple agents who each control a subset ofthe processing network.Section 5 reviews dynamic capacity investment

models whose emphasis is on characterizing the tim-ing of capacity adjustments. Often, optimal invest-ment dynamics follow a so-called “ISD policy,” whichis characterized by a continuation region: When thecapacity vector falls in this region, it is optimalnot to adjust capacity; otherwise, capacity shouldbe adjusted to an appropriate point on the region’sboundary. Managerially, this implies that capacity isnot adjusted continuously because of several “fric-tions,” such as irreversibility, nonconvex costs, orlumpiness, that will be discussed in §5.3. In addition,the optimal investment sequence defines an endoge-nous relative flexibility of resource capacity, meaningthat some resource capacities will be more frequentlyadjusted than others.Section 6 incorporates risk aversion in capacity

investment and reviews hedging strategies involv-

3 For the remainder, the term “capacity portfolio” is assumed toimply interdependent resource capacity decisions. Interdependencemay stem from, for example, some resource sharing among prod-ucts or dynamic routing in the processing network. Otherwise, theprocessing network may be separable and the portfolio problemmay decompose into independent, lower-dimensional problems.

270 Manufacturing & Service Operations Management/Vol. 5, No. 4, Fall 2003


Figure 1 An Example of a Capacity Portfolio Investment Problem

Final Assembly 1

Capacity K1

Product 1

Product 2

Final Test

Capacity K3

Product 1 (demand D1)

Product 2 (demand D2)Final Assembly 2

Capacity K2

ing financial versus operational means. For managers,uncertainty, quite naturally, leads to considerationsof notions of risk and methods to manage and miti-gate that risk. Section 6.1 reviews the economic the-ory whether firms should care about risk, and ifso, how to incorporate it in capacity decisions. Wereview expected utility formulations and the condi-tions under which they lead to mean-variance for-mulations. Section 6.2 reviews financial hedging anddiscusses how capacity investment is similar to sell-ing a call option. Hedging means “to protect oneselffrom losing or failing by a counterbalancing action”or “to protect oneself financially as: (a) To buy or sellcommodity futures as a protection against loss due toprice fluctuation; (b) to minimize the risk of a bet”(Merriam-Webster’s Collegiate Dictionary 1998). We referto hedging that uses counterbalancing positions infutures and financial derivative instruments as finan-cial hedging. Section 6.3 reviews operational hedging, bywhich we mean mitigating risk by counterbalancingactions in the processing network that do not involvefinancial instruments. Operational hedging, thus, mayinclude various types of processing flexibility, suchas dual-sourcing, component commonality, having theoption to run overtime, dynamic substitution, routing,transshipping, or shifting processing among differenttypes of capital, locations, or subcontractors, holdingsafety stocks, having warranty guarantees, etc. Someof these actions (e.g., multisourcing, component com-monality, product-flexible resources, and having alter-nate uses of resources) aim to pool the safety capacityof multiple resources. “Counterbalancing their capac-ities” to mitigate risk is exactly the form of opera-tional hedging that is studied in this paper. Finally, §7concludes.To motivate the study of stochastic capacity port-

folio investment and hedging, consider the followingexample.

Example. Consider a stylized representation of thecapacity-planning problem that disk-drive manufac-turers routinely face, as described by Van Mieghem(1998b) and illustrated in Figure 1. To support theirgrowth and frequent new product and technol-ogy introductions, such companies repeatedly makeinvestments in property and equipment.4 Assume twohigh-end disk-drive product families are scheduledto go into volume production in the first calendarquarter of next year. Product 1 boasts faster seektime and higher data-throughput rate, while Prod-uct 2 is more energy-efficient and reliable. Giventheir different product designs, each family’s “head-disk assembly” (HDA) and printed-circuit board finalassembly requires its own product-specific equipment.Both families, however, can be tested in a single,shared facility. Disk-drive testing involves connectingthe drive to intelligent drive testers (IDTs), fast com-puters that perform a set of read-write tests. IDTs canquickly switch over between testing different prod-ucts. Product life cycles of disk drives are short. Thetwo new products are planned to be in volume pro-duction for only four quarters. There are significantfixed costs associated with commissioning and start-ing up the three new facilities. In addition to the fixedcosts, the facility costs are also driven by the volumecapacity, reflecting higher labor, space requirements,and tooling costs.Capital investments at such a disk drive com-

pany are typically the result of a capacity require-ments planning (CRP) process, which links withthe production-planning and materials requirementsplanning (MRP) process. The monthly “demand-planning” cycle begins with individual marketing and

4 For example, Seagate Technologies invested $920 million in fis-cal 1997. This amount included $301 million for manufacturingfacilities and equipment related to subassembly and disc-drivefinal assembly and test (FA&T) facilities.

Manufacturing & Service Operations Management/Vol. 5, No. 4, Fall 2003 271


sales managers using their knowledge of planned pro-motions and local markets to estimate sales poten-tial for the following 12, or even 24 months. Theseestimates capture both planned purchases by majorOEMs, as well as possible orders by distributors,resellers, dealers, and retailers. Obviously, some ofthese estimates are more reliable than others andthe accuracy of these forecasts degrades sharplybeyond the immediate quarter so that significantuncertainty in total demand remains. The demandforecast represents the combined estimates of themonthly, worldwide demand for each individualproduct. Given that the two products are imper-fect substitutes, their aggregate demand is muchbetter known than the specific mix. Indeed, there issignificant uncertainty regarding the adoption of aparticular product. Conceptually, the demand fore-cast captures demand uncertainty by a probabilitydistribution. The capacity-investment problem is todecide on the size and timing of changes in the capac-ity of three resources (FA1, FA2, and Test) given ajoint probability distribution of quarterly demand forthe two product families. The capacity portfolio isdenoted by the vector K, where K1 is the capacity ofFA1, K2 of FA2, and K3 of Test.

In practice, dealing with uncertainty in a consistentmanner throughout a global corporation is a nontriv-ial task. Managers know the academic dictum that“point forecasts” in the form of a single number aretypically wrong. Yet, aggregating demand forecaststhat feature not only means but, at a minimum, alsovariances and some measure of covariances is noteasy. Incorporating such uncertainty into the com-plex procedure of capacity planning is even harder.In addition, demand is not completely exogenous:Sales-force incentives and compensation are typicallyset to enhance the likelihood of “meeting the num-bers.” (Section 4.3 will elaborate on the endogeneity ofdemand uncertainty.) Therefore, it is not uncommonfor current commercial capacity-planning software toconsider only a single scenario in the forecast. Thisis sometimes called the “sales plan” and is, typically,the input to aggregate planning, MRP, and CRP sys-tems. While such approach, which we will call deter-ministic or sales-plan driven capacity planning, virtually

ignores uncertainty, it “works,” even under decen-tralized decision making. In other words, it is easilyexplained and understood, and provides a first-orderestimate of capacity requirements. We shall see thatit typically leads to a balanced capacity configura-tion, which is attractive from a cost-perspective, as itallows for the possibility to fully utilize all resourcessimultaneously, resulting in nice accounting efficiencymetrics. Another capacity plan that may show up inpractice is a plan that minimizes lost sales. In somesettings, marketing managers may state that “a cus-tomer lost once is lost forever” and advocate amplecapacity to prevent that. We refer to such a plan thatis attractive from a revenue-perspective as a “totalcoverage” capacity plan.Incorporating uncertainty, however, typically

changes the capacity investment and can improveperformance and mitigate incentive conflicts, whichis what this paper aims to illustrate. For example,as observed in Harrison and Van Mieghem (1999),the optimal investment strategy in a stochastic modeltypically involves some degree of capacity imbalance,which can never be optimal in the deterministicversion of the model. Also, it suggests an expected-profit-maximizing compromise between the twoconflicting incentives of cost-efficiency of productionand revenue maximization of sales. Finally, a stochas-tic capacity model can capture the risk aversion ofdecision makers and can show how capacity canbe used to mitigate risk and improve performance.These conceptual distinctions between deterministicand stochastic capacity investment, as well as theirramifications for capacity planning practice, will bediscussed in detail in the context of the example in§§4.2 and 6.3.Finally, a caveat: The models in this paper are

intended to assist decision making, but seasonedresearchers know that practice is more complex.Before a firm can consider capacity decisions, it mustarticulate its business strategy, decide on its compet-itive positioning, and on which markets to enter orexit, etc. Here, we restrict attention to mathemati-cal models that provide insights into the nature andfinancial value of smart investments and that comple-ment other political and strategic considerations thatinfluence capacity decisions.



2. Capacity Research andRelated Literatures

This section surveys several major fields that studycapacity and explains their commonalities and dif-ferences in emphasis. Specific references will bediscussed throughout this paper.The capacity expansion literature in operations

research is concerned with determining the size,timing, and location of buying additional capacity,according to Luss (1982), which is the latest com-prehensive survey to our knowledge. In the begin-ning, the field’s basic concern was how to meet thegrowing demand in growing economies. The semi-nal paper by Manne (1961) studies the fundamentaltrade-off between the economies-of-scale savings oflarge expansion sizes versus the opportunity cost ofinstalling capacity before it is needed. Often, this lit-erature is deterministic and focuses on minimizingthe (discounted) cost of all expansions. Some authorsallow for uncertainty: Examples of “post-Luss (1982),”which is the focus of this paper, include Davis et al.(1987), Paraskevopoulos et al. (1991), and Bean et al.(1992). This stream of research is very much related toour topic, but is typically restricted to the expansionof capacity of one resource and cost minimization.While the capacity-expansion literature assumes

that capacity is infinitely durable and is neverreplaced (let alone, reduced), the equipment replace-ment literature focuses on replacement, while typi-cally ignoring demand changes or scale economies.Rajagopalan (1998) gives a recent review of that lit-erature and presents a unified approach to capacityexpansion and equipment replacement in a determin-istic setting.It appears that over the course of the last twenty

years, the study of plant location has focused ontransportation issues and somewhat divorced itselffrom the study of type, size, and timing of capacityinvestment.The technology management, new product development,

and operations strategy literature deals with decid-ing on the choice of technology, among many otherthings. For example, should we invest in flexible orspecialized technology? When to switch to a newtechnology? When should new product designs sharecommon components? How to allocate investment

among a set of new product projects? When technol-ogy is defined in terms of the capabilities of a net-work of different resources, then such questions canbe addressed with stochastic capacity portfolio invest-ment models. Indeed, a multiresource framework canbe used to select the technology (defined by resourceswith optimal positive capacity levels), along with itscapacity plan. For example, capacity papers that studyinvestment in flexible technology include Fine andFreund (1990), He and Pindyck (1992), Jordan andGraves (1995), Van Mieghem (1998a), Netessine et al.(2002), and Bish and Wang (2002). Capacity papersthat study timing of new technology adoption includeLi and Tirupati (1994) and Rajagopalan et al. (1998).Component commonality is studied with a capacityportfolio investment model in Van Mieghem (2003a).The production or aggregate planning literature stud-

ies the problem of the “acquisition and allocationof limited resources to production activities so as tosatisfy customer demand over a specified time hori-zon” (Graves 2002, p. 726). The answer typically isderived via an optimization problem, often a lin-ear programming model, and yields a mixed strat-egy of “chase demand” by having excess capacityor time flexibility, and “level production” by hav-ing inventories. Clearly, aggregate planning is con-cerned with the determination of the level of process-ing resources over time, but there are two significantdifferences with stochastic capacity investment. First,the resources under consideration are often restrictedto workforce size, inventory planning, subcontracting,and overtime scheduling. Second, virtually all aggre-gate planning considers a deterministic future, verysimilar to the deterministic planning described in theIntroduction.While nothing precludes the inclusion of capital

equipment adjustments in aggregate planning mod-els, the planning horizon typically is short-to-mediumterm, such that capital equipment is fixed, but itsutilization and allocation to products over time isvariable. As such, aggregate planning takes a first-order approach to endogenizing the fundamentaltrade-offs in production planning among capacityutilization (regular time, overtime, subcontracted),inventory, and service and responsiveness (e.g., back-logging or lost sales). Exceptions that consider capac-ity expansion and inventory management jointly in



an aggregate planning model include Bradley andArntzen (1999), Atamtürk and Hochbaum (2001),and Rajagopalan and Swaminathan (2001). Bradleyand Arntzen (1999) present a mixed-integer pro-gram to maximize return on assets and apply it totwo firms to illustrate the capacity-inventory trade-off. Atamtürk and Hochbaum (2001) investigate thetrade-offs between acquiring capacity, subcontracting,production, and holding inventory to satisfy non-stationary deterministic demand for a single periodover a finite horizon. Rajagopalan and Swaminathan(2001) consider a multiproduct environment wheredemand for items is known and growing graduallywhile capacity additions are discrete. Therefore, peri-ods immediately following a capacity increase arecharacterized by excess capacity. Their model stud-ies the following trade-off: Should excess capacity beused to do more equipment changeovers, and thus,reduce inventories, or should more inventory be builtin order to delay future capacity expansions?The input to aggregate planning includes a deter-

ministic demand forecast for each period in the plan-ning horizon. To consider the impact of uncertainty,aggregate planning typically makes two suggestions:Perform sensitivity analysis on the inputs of theaggregate plan and use safety inventory or safetycapacity to satisfy demand higher than forecasted.While such approaches may work well in some envi-ronments, it may be desirable to incorporate theeffects of uncertainty directly in the model to ascertainhow the optimal capacity plan is affected. Stochasticcapacity investment turns up the brightness on thedirect effect of uncertainty and turns it down on sometactical activities. As such, it will automatically andoptimally (as opposed to manually and heuristically)incorporate sensitivity analysis of input uncertaintyand endogenously define safety factors.From a hierarchical5 perspective, aggregate plan-

ning operates at a lower level, and with shorterplanning horizons, than resource planning, which isclosely related to stochastic capacity-portfolio invest-ment. Resource planning often neglects changes ininventory and overtime scheduling. Such a view is

5 Cf. Sethi et al. (2002), for a survey on hierarchical control, includ-ing applications on capacity expansion and equipment replacementproblems.

similar to the view in economics that “current out-put flow depends on installed capital stock, and per-haps on flows of instantaneously variable inputs likelabor and raw materials, through a production func-tion We can regard profit flow as the outcome ofan instantaneous optimization problem where vari-able inputs such as labor or raw materials are cho-sen holding the level of capital fixed” (Dixit andPindyck 1994, pp. 357–359). CRP often shows up inthe context of production planning and planning soft-ware like MRP and MRPII (Hopp and Spearman 1996,Nahmias 1993). Despite its name, however, CRP typi-cally verifies whether the MRP-generated productionplans are feasible given the capacity in place; that is,it checks feasibility of resource allocation rather thanplan resource investment or adjustments.The inventory and supply chain management litera-

ture is concerned with the flow of material througha multiechelon inventory system. In contrast to theaggregate-planning literature, this field often explicitlyconsiders uncertainty, but rarely capacity. (In single-period models, however, there is no essential differ-ence between capacity or inventory, as shall be illus-trated in §4.) Kapuscinski and Tayur (1998) provide arecent review of articles that consider capacitated sup-ply chains. Most of those deal with given, fixed capac-ities. Capacity, here, is the upper bound on produc-tion quantities, which is typically deterministic. Anexception is Hu et al. (2002), who consider stochasticupper-bounds, i.e., capacity uncertainty, in addition todemand uncertainty. Inventory papers that also con-sider capacity investment decisions include Angelusand Porteus (2002), Bradley and Glynn (2002), VanMieghem and Rudi (2002), and will be discussed in theremaining sections. Networks where multiple agentscontrol their own productive capacity require game-theoretic models as §4.3 will discuss.Investment in capital and labor has been at the core

of the economics literature since its inception. Invest-ment contributes to future output, economic growth,current demand, and employment. Given that invest-ment empirically is lumpy (versus continuous overtime), linear theories had to be expanded. When itis costly to reverse investment in capital or labor,a firm’s investment decisions exhibit the empirically



observed dynamics. This occurs, for example, when afirm faces labor firing costs or when it cannot recoupthe acquisition price of capital when it is resold.While the possibility that investment may be costly toreverse has been recognized in the literature at leastas far back as Arrow (1968), irreversible investmentstarted receiving more attention in the late 1980s andearly 1990s in a stochastic framework. It exploits ananalogy with the theory of options in financial mar-kets and, therefore, came to be known as the “realoptions approach to investment,” according to pio-neers Dixit and Pindyck (1994). The resources thatwe consider are real assets and our discussion mayvery well be labeled as “a real options approach.”Section 5 will review why capacity is typically notadjusted continuously over time.The specialization of economics called corporate

finance considers methods to value and finance invest-ments in projects and property, plant, and equipment.The traditional valuation methods for projects includenet present value, payback years, and various typesof hurdles on financial ratios. A central question inthis field is: What is the objective of the firm, and howdo we value uncertain future cash flows? Clearly, thisrelates to risk sensitivity, which will be summarized in§6.1. Corporate finance also focuses on the means offinancing the investment and the impact of debt ver-sus equity on the capital structure of the firm, whichis beyond the scope of this paper. As we shall discusslater, stochastic capacity models assume (often implic-itly) either perfect capital markets, so that frictionlessborrowing is possible, or that the investment size isrelatively small, so that it can be internally financedwithout material impact on the overall valuation ofthe firm.

3. General Issues in CapacityInvestment Problems

This section discusses important issues in typicalcapacity investment problems. In the course of thatdiscussion, some definitions and notations will beintroduced. Consider a firm that has n different“means of processing,” which we will call resources.Its capacity portfolio at time t is denoted by the nonneg-ative capacity vector Kt ∈ n

+ whose ith component

represents the level of resource i that is available forprocessing at time t. The capacity problem is to char-acterize the desired capacity portfolio over time.

3.1. Capacity Constraint FormulationGeneral higher-level models, often used in economics,capture the impact of capacity by a direct functionaldependence of operating profits on the capacity stock.The operating-profit functions tKt denote theoperating profit (excluding capacity investment costs)as a function of time t, the capacity vector Kt availableat that time, and uncertainty represented by , whichrefers to “state of the world,” or in more general mod-els, a sample path. (Incorporating emphasizes thatthe entity is random.) The typical assumption is thatoperating profits are concave in the capacity vectorK, which captures decreasing marginal returns frominvestment. This higher-level formulation is remark-ably general and flexible. No further specific assump-tions on exactly how capacity constrains processingquantities are needed to characterize general capac-ity dynamics, as will be reviewed in §5. In gen-eral, K need not even be interpretable in terms ofmaximal product quantities. Stochastic dependencecaptures broad capacity formulations, including “ran-dom capacity,” where profit is a stochastic function ofcapacity.In operations management, one often extends and

details the formulation so that the operating-profitfunction tKt becomes endogenous to the model.Typically, the model explicitly specifies the input-output relationship and its dependence on capacity,process structure, and management. The operating-profit functions tKt then become the outcomeof tactical optimization problems that depend onthe state of the process and its capacity vector. Forexample, this formulation could capture a compli-cated setting with product-sequence-dependent setuptimes. The outputs and operating profit during aperiod, then, depend not only on the available capac-ity vector K, but also on the ex-post optimal produc-tion schedule, which can be a function of the actualoutput demand and input supply during that period.This captures a setting where maximal output quanti-ties of various products are state-dependent and non-separable among products.



A simpler and fairly typical capacity constraintformulation is via a “recourse” linear-programmingproblem, as illustrated by the following example: Afirm sells m products in a competitive market whereprices are uncertain. Let pt represent the unit pricevector for period t, which is observed at the begin-ning of the period before Kt is chosen. According tothe philosophy of continuous improvement, the firmis improving its manufacturing technologies, but notin a deterministic fashion. The firm’s capacity con-sumption matrix At and marginal processing costs ctfor period t are also observed at the beginning of theperiod, before Kt is chosen. Assuming that the firm’sprocessing quantities, xt , during that period are lin-early constrained, the firm will set period t process-ing according to the linear program, to maximize theoperating profit:

tKt=maxxt∈m+

pt− ct′xt (1)

s.t. Atxt ≤Kt (2)

where ′ denotes transpose and vector inequalitiesshould be interpreted componentwise. The resultingoperating-profit function t· is concave for each and t.The operational consequence is that capacities in

operations-management models (and in this paper)can almost always be interpreted as some upperbounds on processing rates. Often, the capacity vec-tor is the right-hand side of a linear constraint onprocessing quantities during a period (i.e., process-ing rates) similar to (2). Clearly, appropriately defin-ing or adding a “period” corresponds to determin-ing or increasing the total number of hours (or shifts)worked, which can modify the total capacity over agiven horizon. Newsvendor networks, which will bediscussed in the next section, fall into this formula-tion where the impact of capacity is modeled explic-itly via “hard,” linear constraints. Another typicalcapacity constraint formulation is via queuing model,which highlights the uncertainty of processing andthe impact of tactical scheduling and resource starva-tion on realized capacity, as will be discussed in §4.1.In reality, capacity constraints may be “soft,” in

the sense that the output of resource i is not rigidlybounded, but can be increased, albeit at an increas-

ing cost. The increasing marginal costs may reflectextraordinary charges including expediting, overtime,etc. Such capacity constraints can be captured implic-itly by a concave operating-profit function, or explic-itly by processing costs that are piecewise linear orconvex increasing when quantity exceeds a criticalnumber, say Ki.

3.2. Capacity Adjustment CostsCapacity adjustment cost are the investment costsincurred when changing capacity. When adjustingcapacity vector Kt−1 to Kt at time t, the associatedcost is, in general, a bivariate function of Kt−1Kt.In addition, when evaluated at an earlier time, it maybe uncertain. Typically, however, the adjustment costat time t is assumed to only depend on the capacitychange and is denoted by CtKt−Kt−1. In addition, Ct

is assumed to be convex to guarantee a well-behavedconcave capacity investment optimization problem.Let x+ and x− denote the vectors with componentsmax0xi and max0−xi, respectively. The typicaleconomic assumption is that Ct is a kinked piecewiselinear convex function Ct :

Ctx= c′K tx+− r ′K tx

− (3)

where the marginal investment costs cK t and dis-investment revenues rK t are usually, but not neces-sarily, positive. Very seldom can capital investmentbe reversed at no cost; that unique setting wherecK t = rK t is called reversible or frictionless investment.Typically, the focus is on resources that are costly toreverse,6 meaning that cK t > rK t , so that only a frac-tion of the investment cost is recovered when sellingreal assets. If rK t = 0, nothing is recovered, which iscalled irreversible investment.The nature of the adjustment cost function depends

on the type of adjustment that is considered and,hence, the convexity assumption of adjustment costsis not always appropriate. It may be appropri-ate when capacity is divisible or when considering

6 Optimal capacities of costlessly reversible resources are foundby optimizing the (extended) operating-profit function. Thoseresources are assumed to have been “maximized out” and, there-fore, don’t appear in the operating-profit function.



investment at the macro-economic level.7 Also, ongo-ing or gradual adjustments, such as maintenance,training, etc., might be labeled “frictionless” invest-ments, and hence, modeled by a linear adjustmentcost. In contrast, however, infrequent and majorcapacity adjustments typically stem from some fric-tion or nonlinearity in the adjustment costs and oftenenjoy economies of scale, implying that Ct· is con-cave. For example, at the firm level, there is often afixed cost associated with making any capacity adjust-ment, creating a discontinuity in Ct at 0 Ct0 = 0,while limKt−Kt−1→0CtKt−Kt−1> 0. Luss (1982) givestwo often-used adjustment cost functions that exhibiteconomies of scale. The first one, called the “fixedcharge” cost function, is the affine version of (3), butwith a discontinuity at 0. Its single-resource versionfor capacity additions is Cx= c0+c1x for x > 0 withC0 = 0. Such affine capacity adjustment costs canbe handled fairly well as discussed in Van Mieghemand Rudi (2002); a pure fixed cost only affects theboundary invest-or-not decisions, while the size of theadjustment remains given by interior optimality con-ditions that are independent of the fixed cost compo-nent. The second often-used adjustment cost functionwith economies of scale is the “power” cost function,which is strictly concave. Its single resource version is:Cx= kx, where k > 0 and 0< < 1. Frictions fromirreversibility and/or economies of scale in adjust-ment cost lead to a firm’s optimal capacity dynam-ics involving occasional large changes, as will bediscussed in §5.

3.3. Capacity Investment Objective, IncludingPlanning Horizon, Discounting, and DecisionMakers

Traditionally, the objective is to maximize expectednet present value of the firm. (Section 6 will dis-cuss more recent work that moves beyond maximiz-ing expected present values and incorporates riskaversion.) Net present value calculations require aplanning horizon T and discounting, which typi-cally assumes a constant per-period discount factor

7 In a macro-economic equilibrium setting, the marginal cost ofinvestment should eventually increase, because the shadow priceof investment should eventually rise as resources are diverted fromother uses, which tends to convexify the cost of investing.

> 0. Obviously, the values of the planning horizonT and discount factor influence dynamic capacitydecisions. Capacity models typically use longer plan-ning horizons (several years and sometimes decades)than aggregate planning models (several monthsand sometimes years). This longer planning hori-zon reduces forecasting accuracy (see below) andincreases uncertainty, making stochastic models moredesirable. On the other hand, higher discount factorsmitigate the impact of longer horizons. (The appro-priate choice of is a central problem in finance thatgoes beyond the scope of this paper.) Discounting alsoputs restrictions on the adjustment cost functions. Forexample, one typically assumes that the present valueof a unit of used capacity cannot be higher than anew unit, i.e., cK t ≥ −trK for > t, to exclude unre-alistic capacity dynamics. In addition, finite horizonmodels (T <) require an additional element in theirformulation: A salvage function f K, which is thefinal (salvage) value for capacity portfolio K giventhat state obtains.The traditional objective of capacity-investment

problems is either processing-network optimization ornetwork design by a single decision maker. Decidingon the amounts of capacity involves optimization ofa given processing network or operating-profit func-tion. By necessity, all single-resource capacity modelsfall into this class. By considering a portfolio of differ-ent types of resources, however, the capacity problemcan amount to network design, or selecting the mostappropriate mix of types of resources and their config-uration (cf. the discussion on technology managementin §2). Often, however, capacity decisions are strategicand may depend on multiple decision makers, includ-ing other firm’s decisions, as §4.3 will review.

3.4. Continuous Versus Discrete CapacityMany models assume that the capacity vector is anonnegative real variable, so that capacity is divis-ible and, thus, its investment can be continuous or“incremental.” While this is a valid assumption in set-tings where the number of possible capacity sizes islarge, more detailed and precise capacity-investmentmodels may treat capacity as a discrete variable. Such“lumpy” investment is appropriate when capacityis indivisible and can only be installed in a small



number of possible sizes. Integer restrictions typicallymake these models less amenable to analysis.

3.5. LeadtimesLeadtimes refer to the time between the purchase andavailability of new capacity. With growing stochas-tic demand, capacity leadtimes increase the risk ofcapacity shortage caused by demand uncertainty. Thebasic approaches to protect against such risk are toeither expand capacity earlier or to adopt larger capac-ity increments. Only a handful of papers, reviewedin Ryan (2002), consider leadtimes in a stochastic set-ting. Erlenkotter et al. (1989) consider one capacityexpansion of a single resource where expansion com-pletion timing is held constant but the leadtime isa decision variable. They show that the presence ofuncertainty has the effect of reducing the optimal lead-time compared to its optimal value in a deterministicmodel. When demand follows a geometric Brownianmotion, Ryan (2002) shows that leadtimes influencetiming, cost parameters determine expansion size, anddemand characteristics affect both timing and size.

3.6. Physical Depreciation and DegradationPhysical depreciation and degradation of capacityrefers to the diminishing financial and operatingvalue of a resource over time. Depreciation is typi-cally modeled by a financial loss to the firm’s value.(The depreciation on the marginal capacity unit ofresource i is proportional to its marginal value andis typically captured by increasing the discount rate.)Similarly, physical degradation can be modeled by adecrease in the capacity vector, typically proportionalto the installed capacity K, or as in equipment replace-ment models as reviewed by Rajagopalan (1998).

3.7. Tactical Activities, Starvation, and InventoryGeneral capacity-investment models take operating-profit functions tK as primitives. While theobserved dynamics of inventories and other tacticalflows could be incorporated in state- or sample-path, many capacity models simply ignore tactical flows,as discussed in §2.Clearly, modeling must strike a balance between

complexity and realism: The appropriateness of ignor-ing tactical flows may depend on the time-scale and

planning horizon under study. For example, in somesettings, capacity adjustments are only allowed infre-quently. When adjustments of tactical flows occuron a much smaller time-scale than capacity adjust-ments, tactical flows can be optimized, given the cap-ital stock that only varies on a larger time-scale andthus can be taken as fixed for the tactical flow opti-mization. Such “time-scale separation” would makeinventory changes, scheduling, and other tactical deci-sions “invisible” at the higher level of capacity plan-ning. The justification in Eppen et al. (1989) for ignor-ing inventory-carryover between periods can be inter-preted as a time-scale separation argument: “[ignor-ing inventory-carryover is] consistent with the factthat each time period is of sufficient length (one year)so that production levels can be altered within thetime period in order to satisfy as closely as possiblethe demand that is actually experienced.” In manysettings, however, actual realized output quantitiesoften depend on tactical activities and flows, such asproduct-sequence-dependent setups and schedulingthat determines availability of raw material and work-in-progress or resource starvation. For example, in sea-sonal environments, it is not unusual to set capacityfor constant processing and buffer seasonalities withinventory build-up and depletion. To capture thesedetailed operational effects and trade-offs, it is oftennecessary to incorporate tactical activities and inven-tory carryover, especially in capacity portfolio models.When activities or products are not divisible, schedul-ing conflicts may lead to resource blocking and starv-ing, which may reduce actual process capacity belowindividual bottleneck resource capacities. (See §4.1.)Interactivity inventory buffers those problems.Mathematically, capacity investment is similar to

inventory management in that it deals with stochas-tic optimization of very similar functions. In single-period models, there is no essential differencebetween inventory and capacity. There are some dif-ferences in multiperiod (i.e., dynamic) models. First,capacity is “utilized,” but not “consumed:” Unlikeinventory, the capacity vector is not depleted bydemand. It may, however, degrade faster with higherusages, requiring faster depreciation. In inventorymodels, different periods are linked via inventory-carryover, which is often neglected in capacity mod-els. On the other hand, inventory models mostly



focus on “moving the goods” from stage to stage,whereas capacity portfolio models focus more onprocessing and the resulting interproduct couplings.Second, inventory models typically assume thatunused inventory and all capacity are not perish-able and that they are available for use in thenext period. Third, capacity models typically havelonger time horizons than inventory models so thatdiscounting and forecasting gain in importance incapacity models. Last, and least, stationary capac-ity models typically collapse to an essentially staticcapacity problem: Given that no new informationis ever gained, the optimal capacity investment pol-icy typically makes only one initial investment ever(see next section). Stationary inventory models, onthe other hand, retain dynamic reordering, while theordering policy itself also is described by one or twocritical numbers which also can be found in an essen-tially static problem. Dynamic capacity models, wheretiming becomes nontrivial, require a more sophisti-cated stochastic formulation: Typically the problem isnonstationary, or capacity degrades over time.

3.8. Unsatisfied Demand and Capacity ShortagesWhen demand is uncertain, capacity is costly, orcapacity adjustments are not instantaneous, there willbe instances of insufficient capacity to meet demand.Depending on the view one takes, these are calledcapacity shortages or excess demand. In practice, firmsemploy tactical countermeasures, such as allocationschemes, increased pricing, backlogging, or advanceinventory build-up, to manage shortages. When suchtactical countermeasures are not incorporated in themodel, assumptions must be made on what happenswith excess demand. There is a strand in the capacityliterature that “assumes that available capacity mustmeet or exceed demand” (Bean et al. 1992, p. S210),which is also called the “no backlogs in demand”assumption by Manne (1961). Obviously, with uncer-tain demand, a no-capacity-shortage assumption mustbe accompanied by a zero-capacity-leadtime assump-tion. The alternative is to allow for excess demand,which either is backlogged, or lost, or some combina-tion of both. In either case, a demand-shortage penaltyis typically included, which may represent the lossof goodwill that will manifest itself in a reduction offuture demand and is very hard to quantify, or the

cost to fill the demand through an alternate process,as discussed in Manne (1961). The lost-sales case is rel-atively easily incorporated in single-stage processes,while backlogging requires one to incorporate nega-tive inventory carry-over. In a capacity-expansion set-ting where no contraction is allowed, backlogging caneasily be incorporated, as shown in Manne (1961), fora single resource and growing demand, and in VanMieghem and Rudi (2002) for a capacity portfolio andstationary demand. The “no backlogging” assump-tion is typically easier to analyze because it corre-sponds to the limiting case where the shortage penaltybecomes infinite and the peak process, sup≤tD , con-tains all relevant information. In multiproduct sys-tems, as studied in capacity-portfolio problems, analternative option to backlogging or lost sales is thatunsatisfied demand for one product “spills over” toanother product. In other words, ex-post substitutionprovides another mechanism for matching capacitywith demand. Substitution has started to be exploredin an inventory context, and its effects are likely tobe similar in capacity problems. An important issueis whether the substitution is executed by the firm orby customers, as discussed in Bassok et al. (1999) andNetessine and Rudi (2003).

3.9. Uncertainty, Information Structure, Learning,and Forecasting

Arguably, the most important factor in a dynamiccapacity-investment model is the description of uncer-tainty and its resolution over time. For the theorist,the tractability of the model is directly related to themathematical properties of the stochastic process thatrepresents uncertainty and information resolution, aswill be illustrated in §5. In theory, that stochastic pro-cess should depend on the employed forecasting pro-cedures. While forecasting is extremely relevant to thepractitioner, however, it is rarely discussed in capacity-investment research, Ryan (2003) being a notableexception. (Chand et al. 2002, review the literatureon forecast and rolling horizons.) Paraskevopouloset al. (1991) distinguish three sources of uncertainty:The uncertainty (“forecast error”) in econometricallyestimated demand equations and exogenous assump-tions, uncertainty in structural shifts in demand overtime, and uncertainty in system parameters, such as



the rate of learning. (See Hiller and Shapiro 1986, for acapacity-investment model that incorporates learningeffects.) Often one can “learn” and update the demandforecast as information is revealed and observed overtime. In such setting, Burnetas and Gilbert (2001) ana-lyze the trade-off between better demand informationby waiting to procure capacity versus the increasedcapacity cost when capacity acquisition cost increasesover time.General capacity models assume an operating-

profit function and a stochastic environment thatdoes not specify the origin of uncertainty. They cancapture uncertainty in supply, internal processing,demand, costs, prices, and environmental factors. Inoperations-management models, demand is often thegenerator of uncertainty and typically is assumedto be exogenous. The other extreme would be toassume that demand is endogenous, reflecting prac-tices where marketing and sales would agree on asales plan and then each function would be respon-sible for making it happen. Manufacturing producesaccording to the plan and marketing and sales dowhatever is necessary to realize sales according to theplan. With an endogeneity assumption, deterministicor sales-plan driven capacity planning, as describedin the Introduction, may be appropriate to assess howto deploy capacity to meet the sales plan, as well asthe cost of doing so. In that case, however, the capac-ity problem has just been translated into the problemof determining the best sales plan. Of course, reality issomewhere in between: Demand is neither completelyexogenous nor completely endogenous; a propertythat has not received much attention in the opera-tions literature. A notable exception is Cachon andLariviere (1999) where demand is influenced by thescarcity of capacity via capacity allocation schemes.Then, demand can be decreasing in capacity if thefirm’s customers become convinced that their require-ments will surely be met.

4. Optimal Capacity Investment:Types and Amounts

This section reviews optimal capacity levels for vari-ous types under risk-neutral investment. (Timing andrisk aversion will be discussed in §§5 and 6, respec-

tively.) Here, we consider a stationary environmentwhere it is optimal to keep capacity constant overtime. The simplest such environment is an i.i.d. struc-ture, which Eberly and Van Mieghem (1997) defineas capacity portfolio models with (1) stationary oper-ating profit and kinked, piecewise linear adjustmentcost functions (i.e., t = , rt = r, ct = c), (2) sta-tionary stochastic structure (i.e., probability measuresfor t and 1, where = tt are identical), and(3) independent periods (i.e., probability measuresfor any pair, ij , is equal to the product of theirmeasures). They show that an i.i.d. setting in infi-nite horizon8 reduces the general dynamic capacityproblem to a single, initial capacity investment, effec-tively collapsing the problem to a single-period prob-lem. While losing dynamics, these essentially staticcapacity-investment models are able to capture richmodeling detail in the processing network and thenature of uncertainty.Section 4.1 reviews canonical stochastic capacity

models that adopt either queuing or newsvendor net-work formulations. Section 4.2 illustrates newsven-dor network analysis in the context of the example inthe Introduction. Section 4.3 reviews game-theoreticcapacity investment by multiple agents, who eachcontrol a subset of the processing network.

4.1. Canonical Capacity Models in an i.i.d.Setting with Multivariate Uncertainty:Queuing Versus Newsvendor Models

Queuing Models. As discussed above, operations-management models often use detailed formulationswhere the operating-profit function K becomesendogenous. One formulation involves queuing mod-els, which are typically set in continuous-time andfocus on flow times and responsiveness in the pres-ence of stochastic processing and stochastic demand.Multiple resources lead to queuing networks in theobvious way and product-dependent scheduling, pro-cessing, and routing leads to multiclass queuing net-works. Such systems allow for multivariate (oftenproduct-dependent) uncertainty, although differentclasses typically are assumed to be independent

8 That property retains in finite horizon settings if the final valuefunction f is identical to the disinvestment cost: f K = r′K.Otherwise, end-of-horizon investment effects may appear.



(queuing theory is not very successful in handlingcorrelated arrivals). Queuing models can study tacti-cal activities, such as dynamic sequencing and routingin a given network, but are vastly underutilized inthe study of capacity investment (perhaps because oftheir increased complexity and tactical detail).In queuing models, the “maximal average process-

ing rate” at a node (location) in the processing net-work represents the capacity of that node. In thesimplest setting, a node contains a single resource, or“server,” that processes a single product, which mayrepresent a good, a customer, or both. Traditionally,the maximal average processing rate at resource i isdenoted by i, which is the reciprocal of the aver-age time mi to process the product at that resource.Capacity then constrains the average processing rateat resource i, which is typically denoted by i, asi ≤ i. This is often scaled into an equivalentcapacity utilization constraint, which is traditionallydenoted by i = mii ≤ 1. The latter representationdirectly allows extension to a multiproduct setting,or a multiclass queuing network. Letting mij andij denote the average processing time and rate ofproduct j at resource i, the capacity constraint forresource i becomes

∑j mijij ≤ 1.

When comparing this to the linear programmingconstraint (2), processing times m and processingrates are the obvious counterparts of capacity con-sumption rates A and activity rates x, but what aboutthe level Ki of investment of resource i? There are sev-eral interpretations of capacity investment in queuing.Capacity investment can refer to increasing the servicerate i of the server at node i, typically by reducingthe mean processing time of a particular product,or of all products, served. Capacity investment canalso refer to increasing the number of servers atnode i. While capacity can be changed continuouslyby increasing processing speed, it can only assumediscrete values when increasing number of processorsin the latter case. Obviously, both approaches can beused simultaneously to adjust capacity. Multiservernodes quickly become difficult to handle analyti-cally. In some formulation that adopt, for example,fluid or Brownian approximations, one can introducean additional “capacity-scaling factor” for processingnode i, say ∗

i . The multiproduct constraint,∑

j mijij

≤ 1, then becomes∑

j mijij ≤ ∗i , which is the coun-

terpart of the linear-programming constraint Ax ≤K. Queuing models, however, typically offer a moredetailed representation of processing networks thanlinear programming models in that they incorporatethe impact of tactical scheduling conflicts on total net-work capacity. As discussed earlier, when activities orproducts are not divisible, processing cannot be inter-rupted or preempted. Resulting “bang-bang” process-ing and poor scheduling may then lead to down-stream resource starving, which may reduce actualprocess capacity below individual bottleneck resourcecapacities. For example, in multiclass queuing net-works, naive scheduling rules can induce product-specific starvation at different servers at differenttimes. This reduces the aggregate network’s effectivecapacity in the sense that the network can becomeinstable, even though each server is less than 100%utilized. (Dai and Vande Vate 2000 review the recentsurge in the study of stability conditions for multi-class queueing networks.)Queuing models become capacity-investment mod-

els when superposed with optimization and a capac-ity adjustment cost function. Some examples ofqueuing capacity models include Mendelson (1985),Loch (1991), Lederer and Li (1997), and Cachon andHarker (2002). The continental divide between inven-tory and queuing models also applies to the subfieldof capacity investment. Production-inventory modelsattempt to bridge the two worlds with representativeexamples (Caldentey and Wein 2003, Armony andPlambeck 2002). (Section 4.3 reviews these multiagentqueuing models.) Boyaci and Ray (2003) study a firmwith two substitutable products that differ only intheir prices and delivery times and are produced bydedicated capacities. They analyze how capacity costsimpact capacity-investment levels and market posi-tioning of the two products.

Newsvendor Models. Besides queuing formula-tions, another, more popular capacity-investment for-mulation is via a “recourse” linear-programmingproblem. The newsvendor network formulation inVan Mieghem and Rudi (2002) is an example of suchi.i.d. recourse-based models. Newsvendor networkproblems are typically set in discrete-time and focuson the impact of multivariate demand uncertainty,



while assuming deterministic processing. Often asequence of i.i.d. demand vectors Dt t ∈ is thegenerator of uncertainty in these models, althoughsupply, yields, and other uncertainty also can be cap-tured. At the beginning of the period (called Stage 1),the capacity vector K and inventory vector S are cho-sen. Then demand D is revealed. At the end of theperiod (Stage 2), an activity vector x is chosen to max-imize operating profit by transforming RSx of inputstock into RDx units of output and sales for givensupply and demand routing matrices RS and RD. Thekernel of a newsvendor network generalizes (1)–(2) toa network structure (but typically assumes determin-istic processing):

KSD

=maxx∈m+

p′x− c′x− c′PD−RDx− c′HS−RSx (4)

s.t. RSx ≤ S RDx ≤D Ax ≤K (5)

In addition to prices p, processing and trans-portation costs c, routing matrices RS and RD, andcapacity consumption matrix A, a newsvendor net-work’s operating-profit function, thus, incorporatesunit demand (output) shortage cost cP, and unitinventory procurement and holding costs cS and cH.The objective is to maximize the expected net presentfirm value, denoted by KS, by choosing capac-ity K and inventory S before demand is known, andchoosing activity x afterwards:

KS= ƐKSD− c′SS− c′KK (6)

where Ɛ denotes expectation and is the per-period,risk-neutral discount factor. The expected value func-tion is jointly concave in K and S so that the opti-mization problem is well behaved.The single-period optimality equations can be

expressed in terms of the expected dual prices ofthe capacity and inventory constraints. As the exam-ple in the next section will illustrate, these suf-ficient first-order conditions can be interpreted ascoupled, generalized critical-fractile conditions, thatspecify the optimal trade-off between the marginalvalue of capacity, inventory, and their correspond-ing marginal costs. In other words, newsvendor net-works construct the optimal capacity portfolio bytrading off the opportunity cost of capacity-underages

with the cost of excess-capacity. With lost sales orwith backlogging for certain networks, Van Mieghemand Rudi (2002) show that the single-period solu-tion extends to a dynamic i.i.d. structure: The optimalcapacity strategy collapses to a single initial capacityinvestment and all further dynamics involve inven-tory, but no capacity adjustments. For small networks,the model can be solved analytically; otherwise, it isalso easily solved using stochastic optimization viasimulation.By appropriately structuring the three network

matrices, newsvendor networks can feature com-monality, flexibility, substitution, or transshipment, inaddition to assembly and distribution. Multivariateuncertainty allows the study of the important impactof correlation on capacity investment. Van Mieghemand Rudi (2002) show that, if demand is normally dis-tributed, the optimal expected firm value is increas-ing in the mean demand vector and decreasing in anyvariance term. In addition, if KSD is submodu-lar in D, then the expected value is decreasing in anycovariance term (and, thus, pairwise demand correla-tion). (There is, however, no general theory yet on theimpact of covariances on the optimal capacity portfo-lio K.) By allowing for alternate or “nonbasic” activ-ities that can redeploy inputs and resources to bestrespond to resolved uncertain events, newsvendornetwork analysis can be used for network design. Forexample, newsvendor networks that study the choicebetween investing in product-flexible or dedicatedcapacity include Netessine et al. (2002), Van Mieghem(1998a), and Bish and Wang (2002), who add ex-postpricing to the capacity decisions. Kulkarni et al. (2002)study the choice between product- or process-focusedplant network configuration. Newsvendor networksanalyzing when it is worthwhile to allow for inputcommonality or substitution include Van Mieghem(2003a) and others reviewed in Van Mieghem andRudi (2002). Similar network design questions havebeen addressed by Graves and coauthors using dif-ferent, but related models; e.g., Jordan and Graves(1995) present an original and insightful analysis onthe impact of “chaining” capacities in a network, andGraves and Willems (2000) study the strategic place-ment of safety stock in a supply network.



4.2. Solution, Discussion, and Ramifications ofthe Example

Solution of the Example (Risk-Neutral). To sim-plify the exposition, consider the example from theintroduction in a single-period setting, thereby dis-pensing with inventory dynamics and discounting(i.e., = 1). For concreteness, assume the followingdata: Demand (in thousands) for the year is esti-mated for three scenarios and shows high mix uncer-tainty: a “pessimistic scenario” D = D1 = 150350with probability 1/4, the “expected scenario” D =D2 = 300300 with probability 1/2, and an “opti-mistic scenario” of D=D3 = 450250 with probabil-ity 1/4. The capacity adjustment cost function CK=cK0 + c′KK includes the fixed cost cK0 of $40 mil-lion and unit-adjustment costs of cK1 = $30 for final-assembly facility 1 (FA1), whereas the more cost-efficient FA2 facility requires a modest cK2 = $20. Thetesting facility (resource 3), however, is more expen-sive with cK3 = $80, reflecting the cost of many expen-sive IDTs. The two high-end drives are estimated tohave unit contribution margins p− c = $400$300,respectively,9 which will be denoted by net value v.This is a newsvendor network model with two out-

puts or products, three resources, and four processingactivities: activity 1 2= FA of Product 1 (2), 3= testProduct 1, 4 = test Product 2. In the static problem,we do not consider inventories. The general activityvector x is a nonnegative four-vector for any state D.Given the process structure and the assumption thatwe do not allow for intraresource buffers, we clearlyhave that x1 = x3 and x2 = x4. Thus, a reduced two-vector, which we also will call x, is a sufficient activ-ity descriptor. Without loss of generality, we assumethat each product requires approximately the sameamount of tester time. Therefore, the relevant networkmatrices in (4)–(5) are demand routing matrix RD and

9 Strategic objectives and “competitive intelligence” fixed marginsin advance so that uncertainty is manifested mainly through quan-tities demanded. While our example focuses on demand uncer-tainty, a complete analysis would also consider uncertainty inmargins, or price and variable costs.

Table 1 The Optimal Activity Vector and Marginal Values of Capacitiesin Each Demand Domain for the Example

Demand Domain xKD KD= KKD

0K=y ∈ 2

+ y1 ≤ K1 y2 ≤ K2 D 0 = 000y1+y2 ≤ K3

1K=

y ∈ 2

+ y1 < K3−K2 D1 K2 1 = 0 v20K2 < y2

2K=

y ∈ 2

+ K3−K2 < y1 < K1 D1 K3−D1 2 = 00 v2K3 < y1+y2

3K=

y ∈ 2

+ K1 < y1 K1 K3−K1 3 = v1− v20 v2K3−K1 < y2

4K=

y ∈ 2

+ K1 < y1, K1 D2 4 = v100y2 < K3−K1

capacity consumption matrix A:

RD =[1 0

0 1

]and A=

1 0

0 1

1 1

To determine the newsvendor-network solution K∗,first observe that it is suboptimal to have K3 > K1+K2

or maxK1K2 > K3. The solution domain for theoptimal capacity vector K∗, thus, becomes K ∈ 3

+maxK1K2 ≤ K3 ≤ K1 +K2. Given that v1 = $400 >v2 = $300 and resource consumption rates A31 andA32 are equal, the optimal contingent activity vectorxKD is the greedy solution to (4)–(5):

x1KD = minD1K1K3

x2KD = minD2K2K3−x1

= minD2K2K3−minD1K1

The demand space can be partitioned into fivedomains #iK, as defined in Table 1 and displayedin Figure 2, in which xKD is linear in K, andits optimal shadow value or dual price KD

def=KKD, thus, constant. The first-stage capacitydecision maximizes expected value (6) with sufficientcondition:10

K∗def= ƐK∗D− cK = 0

For newsvendor networks, gradient and expectationinterchange: ƐKD = ƐKKD, so that the

10 With a discrete demand distribution, the expected operatingprofit is piecewise linear and, hence, not-differentiable at the break-points. At those points, should be interpreted as a subgradient.



Figure 2 The Optimal Ex-Post Activity Vector x for the ExampleDepends on the Capacity K and Demand D

K3

D

x

x= D

Produce

Ω1 Ω2 Ω3

Ω4Ω0

K2

K1 K3 D1

D2

marginal operating-profit is the expected shadow vec-tor K of the linear program:

Kdef= ƐKKD= ƐKD

= ∑domains i

iP #iK (7)

where P is the demand forecast and i is the con-stant KD for D ∈ #iK. Thus, the newsvendor-network solution solves:

K∗= 0 where K=K− cK (8)

Condition (8) for the optimal capacity portfo-lio is a generalization of the critical-fractile con-dition in single-dimensional newsvendor models.The marginal value, K, generalizes the expected“underage cost,” or the expected opportunity cost ofhaving insufficient capacity. Incorporating an addi-tional demand-shortage penalty, cP, increases netvalue, v, by R′

DcP, thereby increasing the underagecost. In a multiresource, multiproduct setting, theunderage cost depends not only on the resource, butalso on the demand-vector scenario. Both resourcesharing (of the test resource) and demand dependence(both drives are imperfect substitutes) introduce cou-pling between different resources’ marginal value, aneffect that is absent in the single-dimensional model.Underage cost is balanced with overage cost, as mea-sured by the marginal cost of excess capacity.

The newsvendor solution K∗ is now found easilyby a marginal or steepest-ascent argument. Start fromthe capacity vector Kb = 300300600, which is thelowest-cost portfolio that enables meeting the most-likely demand D2, which equals ƐD. (In other words,Kb would be optimal if there were no uncertainty.)Now evaluate the marginal value of an incrementK > 0, using (8):

Kb=

0

300

0

025+

400

0

0

025−

30

20

80

=

70

55

−80

Thus, increase K1 as long as &K1 > 0, or untilK1 = 350, beyond which P4 (let Pi denote P#iKbecomes 0, and P3 = 1/4, and &K1 = 100/4− 30 =−5< 0. Second, increase K2, as long as &K2 > 0,or until K2 = 350, beyond which point P1 becomes0, and &K2 = −20 < 0. Third, increasing K3 beyond600 is suboptimal, as that would yield P2 = P3 = 0,and &K3=−80< 0. Similarly, decreasing K3 below600 is suboptimal, as that would yield P2 = 1/2, and&K3 = 4001/2+P3− 80 > 0. Thus, we have arrivedat the unique newsvendor-network solution: K∗ =350350600.The corresponding contingent activity vectors are

xK∗D1 = 150350, xK∗ D2 = 300300, andxK∗ D3 = 350250. Associated state-dependentoperating profits are K∗ D1= $165000, K∗D2= $210000, and K∗D3= $215000. Expected oper-ating profit is $200,000 while capacity investmentcosts are CK∗ = $40000+ $65500 = $105500, sothat maximal expected value is K∗= $94500. It iseasy to verify that K S D is submodular,11 whichgives insight directly into the sensitivity to demandforecast parameters: The maximal expected valueincreases in the mean demand vector, but decreases inany demand variance or covariance terms (includingany pairwise demand correlation). Thus, if productsare less substitutable, expected value will suffer.

11 For example, to show that the (sub)gradient '/'D1 = v′'x/'D1

is decreasing in D2, one must consider three scenarios: (1) If D1

is small, then as D2 increases from 0, v′'x/'D1 remains constantat v1 throughout #0 and #1; (2) if D1 is intermediate, then as D2

increases from 0, v′'x/'D1 remains constant at v1 throughout #0

and decreases to v1 − v2 in #2; (3) finally, if D1 is large, v′'x/'D1

remains constant at 0 throughout #4 and #3.



Discussion of the Example (Risk-Neutral). Thisexample highlights an important conceptual distinc-tion between deterministic and stochastic capacityplanning mentioned in the Introduction: The opti-mal investment strategy in a stochastic model typ-ically involves some degree of capacity imbalancewhich can never be optimal in a deterministic model.Capacity balance means that it is possible to fully uti-lize all resources simultaneously, which means in anewsvendor network:Definition 1. A capacity portfolio K in a newsven-

dor network is balanced if there exists an activityvector x ≥ 0, such that Ax =K.In the example, the condition for a balanced capac-

ity portfolio simplifies to K1+K2 = K3. In the graphicrepresentation of Figure 2, a balanced capacity portfo-lio yields a rectangular feasible region, while capacityimbalance yields a rectangle with a “cut-off corner.”Clearly, the capacity vector Kb = 300300600 is bal-anced, while the newsvendor network solution K∗

exhibits a relative “capacity imbalance” ( = K1+K2−K3/K3 = 1/6 = 13%. Capacity balance is attractivefrom a cost perspective, as it allows for the possibil-ity of simultaneous full utilization of all capacities,resulting in nice accounting efficiency metrics.Another capacity plan that may show up in prac-

tice is a plan that minimizes lost sales. In some set-tings, marketing managers may state that “a customerlost once is lost forever,” and advocate ample capac-ity to prevent that. We refer to such a plan that isattractive from a revenue perspective as a “total cover-age” capacity plan Kc. In the example, the “best” totalcoverage is Kc = 450350700. The newsvendor-network solution K∗ provides the expected-profit-maximizing compromise between these two conflictingincentives.This feature of compromising two conflicting incen-

tives is just another interpretation of the well-knownproperty of overage-underage balance in single-dimensional newsvendor models. The reason forunbalancing capacities, however, follows from extend-ing this purely financial argument involving uncer-tainty to multiple dimensions and, thus, is a fea-ture unique to capacity portfolios. In the example,the marginal value of increasing investment in testers

beyond 600 does not outweigh its cost of $80. There-fore, there is an optimal 25% probability of not beingable to meet all demand, and the optimal aggregateservice level is 75%. In a network, many capacityconfigurations can yield the same aggregate servicelevel. Safety capacity is the excess over the capacitythat would be optimal if there were no uncertainty.In the example, optimal safety capacity is K∗ −Kb =50500. Hence, another interpretation of the opti-mal capacity portfolio is that it specifies the optimalamounts and locations of safety capacity in the net-work, thereby also specifying optimal product servicelevels. Thus, optimally trading off the expected valueof safety capacity at different resources with its cost,results in a 100% service level for Product 2, whilethe higher-margin Product 1 receives a service levelof 75%. Risk-neutral financial optimization, thus, spec-ifies optimal safety-capacity amounts and locationsthat typically result in an unbalanced capacity portfo-lio because it “hedges” optimally against uncertainty.

Ramifications to Practice (Risk-Neutral). Whilethe newsvendor-network solution, by definition,yields the highest expected profit among all capacityportfolios, it has three properties that may impact thelikelihood of its implementation. First, the managerrecommending a newsvendor investment plan mustexplain to top management why they should autho-rize cash to be invested in a capacity portfolio that isknown in advance to be never fully utilized. The rea-son is that counterbalancing capacities provides thebest operational hedge: This intentionally unbalancedportfolio yields a network capacity configuration thatmaximizes expected value. Obviously, if the managercould observe the actual demand and all other uncer-tainty in advance, she could identify precise capacityneeds and would invest in a balanced capacity port-folio. Under uncertainty, however, she should “hedgeher bets” and invest in excess or “safety” capacityin some resources precisely because this yields betteraverage performance. Thus, one key driver for capac-ity imbalance is uncertainty (as we shall see in §6.3,risk aversion is another).Second, given that the newsvendor solution is typ-

ically not optimal for any ex-ante known demand,it cannot be the outcome of the ”typical” sales-plan driven capacity requirement planning described



in the Introduction. To put this in perspective, thecapacity literature has typically focused on single-resource investment for which a famous result wasfirst shown in the seminal paper by Manne (1961)(which is reviewed in §5.4): Given stochastic demandforecasts, the optimal capacity can be found by an“equivalent deterministic problem” that considers thesame capacity-investment problem but with a per-haps modified, but always deterministic, demandand with some parameters (typically the discountrate) modified to incorporate the effect of uncertainty.This result justifies the practice of sales-plan drivencapacity planning or deterministic planning basedon one scenario. With a single resource, it is clearthat one can always find one modified demand forwhich deterministic planning would yield the optimalnewsvendor solution. In contrast, such an equivalentdeterministic problem—and, hence, related justifica-tion of practice—typically does not exist for a truemultiresource capacity-portfolio problem (e.g., wheresome resources are shared among different products).Indeed, the unbalanced optimal capacity portfolio canonly be obtained by carefully weighing upside versusdownside for each capacity, which are expressed byintricate and coupled optimality conditions.Deterministic planning, on the other hand, typically

yields a balanced capacity portfolio. While there is lit-tle doubt that capacity-planning software capabilitieswill advance in the future to incorporate these intri-cate conditions, just like financial asset pricing soft-ware has, the fact that the conditions are coupled hasbroad implications for capacity-planning methods inpractice. Good capacity-portfolio planning cannot beperformed independently at separate locations withonly a corporate sales plan as input, but must be coor-dinated throughout the organization.12 Obviously, topmanagement knows that traditional sales-plan drivencapacity planning misses uncertainty and, therefore,heuristically incorporates uncertainty by perturbingits capacity proposals before implementing them.Adopting optimal stochastic capacity-portfolio plan-

12 While the process could be decentralized using incentives suchas transfer prices, determining the appropriate incentives seemsto require a central planner to first solve the organization-widecapacity problem.

ning, however, eliminates these heuristic perturba-tions and automates the incorporation of uncertainty.It seems logical to conjecture that the financial

and strategic value of adopting stochastic portfolioplanning should be highest for firms in volatile indus-tries that have some resource-sharing among differentproducts.13 (The third property that should be consid-ered when adopting a newsvendor capacity solutionis that it defines the maximal-risk capacity portfolio,which will be discussed in §6.3.)

4.3. Game-Theoretic Capacity Investment byMultiple Agents

In many settings, capacity-investment decisions arenot made in a vacuum. At a minimum, those deci-sions interact with external customer demand andmay depend on external supply markets. These cus-tomers may have access to other firms, and these sup-pliers may also supply those other firms. In short,a firm’s capacity decisions typically depend on, orinteract with, other economic agents’ decisions. Thus,it seems natural for capacity investment models toincorporate the strategic behavior of self-interestedagents. A classic textbook example is capacity pre-emption, where an incumbent overbuilds capacity todeter entry by signaling to potential competitors thatit has a small marginal cost. Thus, information asym-metry enters the picture, as well as many other game-theoretic factors, when a processing network is parti-tioned such that each subnetwork is controlled by adifferent agent.The capacity-portfolio solution methods discussed

earlier now must be augmented with a fixed-pointcondition to solve for Nash equilibrium capacity-investment strategies. Complexity quickly mountsand great care must be exerted to specify a tractablemultiperson model. Most game-theoretic capacityanalysis has, therefore, been restricted to a station-ary setting where a single capacity investment is opti-mal, emphasizing capacity type and size, rather thantiming decisions.

13 For example, pharmaceutical Eli Lilly employs a full time staffof sophisticated stochastic capacity-portfolio planners that aid inthe strategic planning of new facilities for new compounds (privateconversations).



Single-resource, multiagent capacity models considernetworks with essentially one productive resourcecapability controlled by one (set of) agent(s) who inter-act with another set of agents without processing capa-bilities. The “nonprocessing” agents typically haveanother economic asset that is of value to the pro-ducer. Examples of such assets include market infor-mation and access, as in the typical manufacturer-retailers relationship; design and marketing capabili-ties, as in the more recent setting where OEMs haveoutsourced production to contract manufacturers; orthe buying decision, as in the direct relationship ofmanufacturer-customers. Cachon and Lariviere (1999)consider the manufacturer’s capacity investment andallocation decisions to several downstream retailersthat have private information. Armony and Plambeck(2002) consider the capacity investment of a manufac-turer that sells through two distributors. When sup-ply is scarce, customers may place duplicate orders,which leads the manufacturer to overestimate boththe demand and cancellation rates. This typicallyleads the manufacturer to purchase too much capac-ity, but estimation errors may lead to underinvest-ment when the cost of capacity is high. Plambeckand Taylor (2001) investigate the impact of bargain-ing and industry structure on capacity investment andprofitability when OEMs outsource manufacturing tocontract manufacturers. Caldentey and Wein (2003)present contracts that are linear in backorder, inven-tory, and capacity levels to coordinate a manufacturer-retailer production-inventory system, including thecapacity decision. While not a true multiagent capac-ity problem, Carr and Lovejoy (2000) analyze themanufacturer-customer relationship in a setting wheredemand management is relatively less costly thancapacity adjustment, so that a capacitated firm will“choose a demand distribution” from a set of poten-tial customer segments that is most profitable. Lovejoyand Li (2002) study how to best expand hospital oper-ating room (OR) capacity, which can be had by build-ing new ORs or extending the working hours in thecurrent ORs, acknowledging the conflicting prioritiesof patients, surgeons and surgical staff, and hospitaladministrators.Porteus and Whang (1991) consider a principal-

agent formulation where capacity and demand are a

function of the private effort of the manufacturingmanager and marketing managers, respectively. Theypresent an optimal incentive plan that is interpretedas requiring the firm’s owner to make a futures mar-ket for capacity, paying the manufacturing managerthe expected marginal value for each unit of capac-ity provided, receiving the realized marginal valuefrom the marketing managers, and on average losingmoney in the process. That framework was extendedby Kouvelis and Lariviere (2000), who allow prices toadapt as information evolves. In a newsvendor set-ting, managers in the first stage receive the expectedshadow price, whereas later agents are charged therealized shadow price of the resources they utilize.This linear transfer-pricing system simplifies the non-linear incentive scheme that coordinates the “globalnewsvendor” of Kouvelis and Gutierrez (1997),where two agents sell an identical product in twomarkets.Multiresource, multiagent capacity models consider

networks where several agents own processing capac-ity, which implies a capacity portfolio problem. Therelationship can be “vertical,” meaning that agentscontrol different “stations” in a supply chain, such asin multiechelon systems, “horizontal,” with parallelfirms supplying a common market, or a mixture ofboth. Most articles consider horizontal competition,typically with univariate uncertainty. For example,Loch (1991) considers price and capacity decisions fora duopoly in a competitive queuing model, which isextended by Lederer and Li (1997) to perfect compe-tition. Bashyam (1996) considers capacity expansionin a two-stage setting akin to a static newsvendornetwork but with private, Bayesian demand-information updating. Lippman and McCardle (1997)show how a newsvendor critical-fractile solutionextends to a competitive setting with multipleagents supplying a single market with fixed-priceand univariate demand uncertainty. That solutioncritically depends on the “splitting rules” that specifyhow initial demand is allocated among compet-ing firms and how any excess demand is allocatedamong firms with remaining capacity (or inven-tory). Van Mieghem and Dada (1999) discuss howthe relative timing of the three major operational



decisions—capacity, processing (inventory) quantity,and price—impact the sensitivity and profitabilityof those decisions under demand uncertainty for amonopoly, oligopoly, and perfect competition. Therelative value of postponement seems to increaseas the industry becomes more competitive. (Cf. VanMieghem and Dada 1999, for a review of earliercapacity and pricing models.)Subcontracting refers to the setting where both

the contractor and supplier have productive capac-ity. With outsourcing, on the other hand, the contrac-tor has no productive capacity and relies completelyon the supplier(s). Subcontracting is an example ofcombined vertical-horizontal relationships in the sup-ply chain if the subcontractor also has market access.Strategic capacity choice, by both subcontractor andmanufacturer, is analyzed by Van Mieghem (1999)under multivariate demand uncertainty. The coordi-nation potential of three contract types is investi-gated, including an incomplete bargaining contract.The impact of demand volatility and correlation onthe option value is presented, as well as conditions forwhen outsourcing—the extreme solution point wherethe manufacturer invests in zero capacity—is optimal.Cachon and Harker (2002) consider a duopoly wherefirms face scale economies with univariate uncer-tainty. Firms compete with two instruments: explicitprices and delivered operational performance. Thelatter includes quality of service, which is directlydriven by the capacity choice of the queue’s pro-cessor. They also allow each firm to outsource itsprocessing to a supplier and identify economies ofscale as a strong motivator for outsourcing. Bern-stein and DeCroix (2002) analyze a newsvendor-likecapacity game in a single-product modular assemblysystem. The final assembler moves first by setting aprice-only contract specifying the unit-price she willpay to subassemblers, who then set the unit-pricesthey will pay to their suppliers. In the second stage,all parties independently choose their capacity level.Finally, demand is observed and all parties producethe same number of units. Equilibrium capacities andprices are characterized and used to guide networkdesign.

5. Optimal Capacity Adjustmentover Time

This section reviews capacity investment problemsthat are dynamic, meaning they do not reduce to asingle initial investment.

5.1. Generic Dynamic Capacity Strategies andTrade-offs

In addition to the portfolio configuration decisions ontype and amounts reviewed above, dynamic capacityproblems must also decide on the timing of capacityadjustments. Timing involves some general trade offs.For example, if one expects demand to increase, whenshould one increase capacity and by how much? Ina single-product setting, the generic choice is a com-bination of the polar extremes of a capacity-leadingstrategy, where there are never demand shortages,and a capacity-lagging strategy, where there is neverunderutilized capacity, as shown Figure 3. Capacitylagging has the advantages of eliminating capacity-overage risk, being less dependent on accurate fore-casting, and delaying capital expenditures. On thedownside, however, it incurs lost sales and dissatis-fied customers, which can invite entry by competi-tors; it has no ability to exploit the “upside” of aforecast; and it is very sensitive to any start-up prob-lems with new capacity additions. Neither of the twopolar extremes employ inventory. A hybrid strategyuses initial excess capacity to build inventory to sup-

Figure 3 Managing Capacity Over Time Involves Deciding theCapacity Adjustments’ Timing and Magnitudes

Dt

t

Amounts

Kt

(capacity-lagging)

Kt

(capacity-leading)



ply later undercapacitated periods. While that is hardto do in a service setting (unless part of the servicecan be performed in advance of customer demand), itcombines the advantages of high utilization and cap-turing all demand, at the cost (and risk) of holdinginventory. Clearly, the trade-off is between the capac-ity investment costs and the inventory holding cost,and appropriate levels of both are driven by the trade-off between cost of underage versus overage (cf. thenewsvendor-network discussion above). The appro-priate strategy may also depend on where productsare in their life cycle: Product-introduction requirescapacity-leading, whereas maturity may move moretowards inventory-smoothing or even lagging.The magnitude of the capacity adjustment and its

timing are also interdependent. A key question iswhether one should have many small adjustments orfew large adjustments, their polar extremes being con-tinuous adjustment or a single, discrete adjustment.Typically, many small adjustments are the result ofcontinuous improvement activities. Few, large adjust-ments, and, thus, discrete capacity changes, typicallyare caused by “frictions.” As we shall review, the lit-erature has identified several sources of friction: indi-visibility (i.e., “lumpy capacity”), irreversibility (i.e.,a kinked adjustment cost function as (3) with cK t >rK t) and nonconvexity (e.g., due to fixed cost andeconomies-of-scale in the adjustment cost function).Finally, in a multiresource setting, the adjustment

decisions of various resources are coupled and animportant question is whether there exists a simpledescription of the coupled dynamics and, perhaps, anatural ordering of resources, such that capacity i isalways adjusted before capacity j .

5.2. Structural Properties of Optimal CapacityPortfolio Dynamics

At each possible capacity adjustment time, the firmwill base its investment decision on the informationthen available and on its assessment of the uncertainfuture. Mathematically, information availability anduncertainty, which are crucial to any investment strat-egy, are modeled by a standard probabilistic frame-work with a probability space # P and filtra-tion = ⋃

t t as primitives. The filtration showshow information arrives and uncertainty is resolved

as time passes, with t representing the informationavailable at time t. Under the risk-neutral assump-tion, the firm’s objective is to maximize its expectednet present value, which is the discounted sum ofoperating profits tKt minus adjustment costsCtKt −Kt−1.To solve the dynamic investment problem, the

usual backward induction argument leads to Bellmanoptimality equations. Adopting a discrete-time for-mulation for simplicity, let VtKt−1 denote the opti-mal value function when starting at the beginning ofperiod t, with capacity vector Kt−1 given state of theworld . With concave operating-profit functions andconvex adjustment costs, the optimal value functionsVt· inherit the concavity of the operating-profitfunctions t (Theorem 1 in Eberly and Van Mieghem1997).Concavity makes the control problem well behaved.

Eberly and Van Mieghem (1997) show that concavityimplies that it is optimal to invest according to a cer-tain kind of control-limit policy that they call an ISD(Invest/Stay put/Disinvest) policy. Roughly speak-ing, an ISD policy adjusts each capacity i accordingto a control-limit policy defined by two critical num-bers or “triggers,” KL

t i ≤ KHt i, which are functions of

the critical numbers of other resources. These definethree action zones: Capacity i is increased to KL

t i ifKt−1 i < KL

t i (invest), decreased to KHt i if Kt−1 i > KH

t i

(disinvest), and not adjusted otherwise (stay put). AnISD policy for the two-resource case (n = 2) has thestructure shown in Figure 4.How is such an optimal ISD strategy found? For

ease of notation, let gtKt be the firm’s expectednet present value, evaluated at the beginning ofperiod t and conditioned on the available informa-tion, given that capacities have been adjusted to Kt

and an optimal (partial) investment strategy is imple-mented:

gtKt= tKt+Ɛ-Vt+1Kt t. (9)

Eberly and Van Mieghem (1997) show that if the solu-tion /tKt−1 to the concave optimization problemsupKt∈n+

gtKt−CtKt −Kt−1 is unique, then /t

is an ISD policy and is optimal for the general capac-ity investment problem for t, for every t ∈ 1 T



Figure 4 Structure of a Two-Dimensional ISD Policy Generated by aSupermodular Concave Function

Kt-1,1

St

Kt-1,2

Invest 1

Disinvest 2

Stay put 1

Disinvest 2

Disinvest 1

Disinvest 2

Disinvest 1

Stay put 2

Disinvest 1

Invest 2

Stay put 1

Invest 2

Invest 1

Invest 2

Invest 1

Stay put 2

Stay put Kt-1K

t

and ∈#. (Optimality extends to the infinite-horizoncase under mild additional conditions.)The concavity of gt yields the following additional

properties of an optimal ISD policy. The optimal ISDpolicy for period t is characterized by a connected setSt⊂ n

+ for each ∈#, where

St= K ∈ n

+ rK t ≤ gtK≤ cK t (10)

If Kt−1 ∈ St, no adjustments are made: Kt =/tKt−1 = Kt−1; otherwise, Kt−1 is adjusted to apoint /tKt−1 on the boundary of St. Thus, if thecapacity vector Kt−1 is within St, it is optimal “tostay put” on all dimensions and to continue with thesame capacity portfolio for the next period. Hence, theset St is also called the “region of inaction” or “contin-uation region.” Its boundaries are increasing (decreas-ing) if all operating profit functions, t·, and thesalvage function, f ·, are supermodular for each .The capacities are then economic complements (sub-stitutes) so that a higher optimal investment thresholdin resource k justifies a higher (lower) optimal invest-ment threshold in resource j , and vice versa.

5.3. Optimal Investment Dynamics and Sourcesof Friction

When the time period becomes arbitrarily small, onecan instantaneously adjust capacity. Assuming suffi-cient regularity, a control problem in continuous timeobtains where the central region St may move contin-uously over time. Only the initial capacity investmentmay represent an “impulse” control to the bound-ary of S0 if the initial state K0 is outside S0. All sub-sequent optimal capacity adjustments are “barrier”or “instantaneous” controls. No control is needed aslong as Kt remains inside St and a period of inac-tion ensues. When Kt “hits” the boundary of St , theminimal amount of control is exercised to preventKt from “leaving” St . Thus, the optimal investmentdynamics are similar to the dynamics of a point, Kt ,floating inside a (typically moving) domain, St , beingreflected on its boundaries. This succession of periodsof inaction followed by bursts of capacity adjustmentcoincides with empirically observed patterns.The width of the continuation region along coor-

dinate i is an increasing function of the amountof irreversibility cK t i − rK t i of resource i invest-ment. The continuation region shrinks to a singlepoint, representing the optimal capacity vector at t,if investment is reversible (i.e., rK t = cK t). With such“frictionless” investment, capacity is almost alwaysadjusted because the adjustment can be “undone” atno cost in the future. If investment is irreversible (i.e.,rK t < cK t), St represents a hysteresis zone where opti-mal capacity at time t depends on previous capac-ity at time t − 1, and capacity is adjusted only ifperiod t’s operating profit and information outlookare sufficiently different from the previous period.Thus, irreversibility introduces “friction” in the opti-mal capacity-investment policy which exhibits peri-ods of action (invest or disinvest) interspersed withperiods of inaction.Indivisibility, such as discreteness in possible capac-

ity adjustment sizes—“lumpy capacity”—are a sec-ond source of nonlinearities and friction that mayincrease the region of inaction and distort the struc-ture of the optimal policy. Narongwanich et al. (2002)consider a single-product setting where new productgenerations are introduced stochastically over time,



and study whether one should invest in product-dedicated capacity, which is only used for one prod-uct life cycle, or reconfigurable capacity. They showthat the ISD policy remains optimal if all resourceshave identical adjustment sizes. With different adjust-ment sizes, the optimal policy is ISD-like but withperturbations around its boundary, probably reflect-ing adjustments from integer restrictions.Fixed costs in the capacity-adjustment cost function

are a third source of friction and introduce additionalnonlinearities that increase the region of inaction, asshown in Abel and Eberly (1998). Fixed capacity-adjustment costs have a similar effect as fixed order-ing costs in inventory theory; indeed, ISD policiesare related to a multidimensional generalization ofthe famous two-critical number S s control policy.In general, nonconvex costs or nonconcave operat-ing profits introduce frictions that will lead a firm’soptimal policy to exhibit occasional large changes, ordiscontinuities, of the kind that arise with impulsecontrols. For example, Dixit (1995) considers a convexadjustment-cost function, but the production func-tion is convex-concave, exhibiting initially increasingreturns to scale, and then decreasing returns to scale.This produces similar effects to nonconvex capacitycosts: The optimal capacity “jumps over” the increas-ing returns portion but is constrained by the eventualdecreasing returns.

5.4. Dynamic Capacity Models:Univariate Uncertainty

Characterizing the continuation region, St , and itsdynamics is a formidable problem. Only when uncer-tainty is generated by “nice and amenable” stochas-tic processes can it be analytically determined. Whilecapacity-portfolio studies with multivariate uncer-tainty are typically restricted to an i.i.d. setting,dynamic studies are typically restricted to univariateuncertainty. In addition, most models assume a uni-variate stochastic process X = Xt ∈ # t ≥ 0with independent increments. (While scenario-basedor stochastic-programming models can also capturedynamics, here, attention goes to papers that analyti-cally describe the investment dynamics.) Often, X is aMarkov process, such as a Markov chain or Brownianmotion, to maintain tractability.

Dynamic single-resource capacity models with stochas-tic demand seem to have started with the semi-nal work of Manne (1961), where demand follows aBrownian motion with positive drift and only capac-ity expansions are considered. Hence, a regenera-tive process obtains and it is optimal to alwaysadd capacity in the same increment 4K, which isan increasing function of variance, whenever thedemand backlog hits a given “trigger” level (the crit-ical ISD number). The associated expansion timescan be expressed in terms of the “hitting time” ofthe Brownian motion. Manne (1961) showed that thestochastic problem “does little—if anything—to com-plicate matters” compared to the deterministic prob-lem with known demand. Specifically, he showedthat the stochastic problem where capacity shortagesare not allowed can be transformed into what hasbecome known as an “equivalent deterministic prob-lem” by replacing the discount rate by a decreas-ing function of the variance rate 52 of the demandprocess. Thus, the entire effect of uncertainty is cap-tured by a single number: The modified or “equiv-alent” discount rate r∗ =

√1+2r52 − 1/52, where

r is the riskless rate. (Such equivalent determinis-tic problem, however, does not exist for the multi-resource capacity-portfolio problem, as discussed in§4.2.) Equally important to practice, Manne (1961)showed that the cost function is relatively insensitiveto the capacity increment, 4K, so that errors in para-meters are rather inconsequential (similar to the EOQmodel).Various extensions and modifications followed. For

example, Giglio (1970) considered a different non-stationary, increasing demand structure of the formDt= at+7t , where Ɛ7t = 0. Thus, demand mean andvariance are independent of each other, and the vari-ance of 7t is either constant or linearly increasing in t.Freidenfelds (1981) considers a series of models wheredemand is Markovian and represented by birth-deathprocesses. Bean et al. (1992) consider capacity expan-sion when demand is a semi-Markov process and nodemand shortages are allowed. They define the equiv-alent deterministic problem and show that it existswhenever the demand is a transformed Brownianmotion or a regenerative birth-and-death process. Theequivalent deterministic problem, again, uses a lower



equivalent interest rate, and the equivalent deter-ministic demand can be found from a regression onobserved demand. Davis et al. (1987) take a differ-ent and quite unique approach: They suppose thatthe decision maker exercises day-to-day control overa capacity installation project by controlling the rateof investment. The demand is a Poisson process andis met by consecutive construction of identical expan-sion projects of given size. Sophisticated stochasticcontrol theory ultimately shows that the control is“bang-bang”: Either carry out construction at maxi-mal speed or do nothing and wait, similar to an ISDpolicy.Angelus and Porteus (2002) consider expansion fol-

lowed by contraction through a single product’s lifecycle. The driving stochastic process, X, representsdemand, which is first stochastically increasing overa known interval of time, after which it is stochas-tically decreasing. They explicitly solve for the twocritical numbers that define the continuation region,St , at each period. It is shown that it is optimal tochange the service level (i.e., the probability of meet-ing demand) over time: Provide the lowest servicelevel during the peak period and the highest duringcontraction periods. In addition, they provide initialinsights into the much more complex setting whereinventory is carried between periods.Dynamic multiresource or capacity portfolio models are

often continuous-time models and assume some spe-cial structure that simplifies the treatment of thesource of (typically univariate) uncertainty, X. Eberlyand Van Mieghem (1997) provide an example whereX is univariate Markovian. In addition, the gen-eral operating-profit function is assumed stationary(t = and Ct = C), Markovian (in the sense thatit only depends on the current capacity vector andcurrent state of nature so that t = KtXt), super-modular and linearly homogeneous in X and K, andincreasing in X. Homogeneity of degree 1 is power-ful in reducing complexity because all dynamics canbe described in a scaled coordinate system Ki/Xt orki = logKi/Xt. To appreciate the significance of thisresult, they show that the central domain is now fixedin K/X-space: SXt= K ∈ n

+ r≤ V K/Xt1≤ c.In k-space, the investment dynamics reduce to thoseof a “particle” moving along the line logK− logXt

in the interior of S and expanding or contractingcapacities when hitting a boundary of S. The impor-tant result here is that, because S is fixed, in equi-librium only a few of the 2n faces of S are everhit, and always in the same sequence. Therefore, theorder and frequency of capacity adjustments—or their“flexibility” in economic terms—is endogenous in thecapacity model: Some resources will be systemati-cally adjusted more often than other resources and,thus, are systematically “more flexible” than otherresources. That ordering follows the ordering of thewidth of S measured along the coordinate direc-tions. As discussed above, the width of the regionof inaction is increasing in the difference cK − rK .If the Markov process, X, is a univariate geometricBrownian motion and the operating-profit function,, is derived from a constant returns-to-scale Cobb-Douglas production function and a constant elastic-ity demand function, the dynamics can be solvedin closed form. It then is shown that the fractionsrK i/cK i endogenously determine the resource adjust-ment ordering or “flexibility.”The fixed flexibility ordering inspired the “bottle-

neck policies” of Çakanyildirim and Roundy (2002).Like Angelus and Porteus (2002), they consider a sin-gle product’s life cycle, but assume a multiresourceproduction process with lumpy capacity in a cost-minimization context. They show that the sequencein which resources are expanded in the first up-partof the cycle is the reverse of that in which they arecontracted in the down part. Clearly, in a single-product setting with lumpy capacity, the resourceadjustment sequence is easier to determine becauseonly capacity adjustment of bottleneck resources canbe optimal. However, the timing of those adjustmentsis nontrivial because it may be optimal to adjustseveral resources simultaneously. Çakanyildirim andRoundy (2002) provide an algorithm to calculate theadjustment times and, hence, determine the “clus-ters” of resources that are adjusted simultaneously.Hu and Roundy (2002) extend Çakanyildirim andRoundy (2002) and present an algorithm to calcu-late the adjustment times in a multiproduct, multire-source setting with stochastically increasing demand.In both papers, adjustment times are determined at



the beginning of the horizon (instead of dynami-cally over time). Narongwanich et al. (2002), reviewedabove, analyze the choice between lumpy dedicatedor reconfigurable capacity when new generations of asingle product are introduced stochastically over time.Analyses of specific instances of the general

dynamic capacity-portfolio investment analysis withmultivariate uncertainty, which is necessary to modelcorrelations between product demands or input sup-plies, seem virtually nonexistent. A likely reasonis that with multivariate uncertainty, the charac-terization of the continuation region in continuoustime typically reduces to a partial-differential equa-tion with nonstandard boundary conditions. Typi-cally, analytic solutions are not available, so one mustresort to numerical analysis. Discrete time does notseem much easier, except perhaps, if one considersonly a few periods. Kouvelis and Milner (2002) ana-lyze the two continuation regions in a two-periodmodel where a single product is provided througha multiresource process comprised of either in-houseproduction or outside sourcing. In addition to prod-uct demand, the outside supply capacity is alsostochastic.

6. Risk Aversion and Hedging inCapacity Investment

This section reviews how risk aversion is incorporatedin capacity problems and how financial and opera-tional hedging can reduce the risk associated withcapacity investments.

6.1. Moving Beyond Maximizing ExpectedPresent Values

Capacity investment often involves substantial cer-tain cash outlays in order to receive uncertain futurerewards. It seems natural to consider the variabilityin payoffs in addition to the average payoff. From aneconomic theory perspective, however, it is not obvi-ous that firms should care about risk. Indeed, thisquestion amounts to asking: What is the objective ofthe firm? Without attempting to summarize the fieldof corporate finance, let us discuss some importantissues. First, the objective depends on the ownershipstructure. For a privately owned company, the objec-

tive may be the maximization of the owners’ expectedutility. The appropriateness of that objective, however,depends on whether the owner is diversified or not.If a majority of an owner’s assets are tied up in thefirm, the capacity-investment decision may materi-ally impact the owner’s utility. If, on the other hand,the owner is well diversified,14 the total variability ofher asset portfolio will impact the owner’s utility. Inthat case, the impact of the capacity-investment deci-sion on expected utility will be driven by the covari-ance of the returns with those of the market. For apublicly owned company, the objective is typicallystated as maximizing the market value of cash flows,where that value is priced in competitive markets.If the market is “perfect,” the capital-asset pricingmodel dictates that the value is linear in the expectedreturn if the firm’s returns are uncorrelated with themarket returns,15 in which case, risk-hedging doesnot enhance value. If the market has imperfections,however, nonlinearities enter into the objective andfirms do care about risk. Typical market imperfectionsarise from the cost of bankruptcy and related finan-cial distress, cost of raising external capital, taxes, etc.In addition, agency concerns lead to risk aversion:To motivate risk-averse managers, the firm’s ownersmay give them a stake in the firm. Finally, it is welldocumented that “executives (including those whoare risk-seeking) make substantial effort to reduce oreliminate risk, usually by delaying decisions and bycollecting more information,” according to Pindyckand Rubinfeld (1989).Given the significance of the cash outlays, the num-

ber of capacity-investment articles incorporating riskis surprisingly small. While Caldentey and Haugh(2003) present a more general model setting, themajority of the few available risk studies in opera-tions management are done in an inventory context,as reviewed by Chen and Federgruen (2000), Dingand Kouvelis (2001), and Gaur and Seshadri (2002).

14 Actually, diversification eliminates only “diversifiable risk,” andnondiversifiable or “systematic” risk remains because the capacityreturn may depend on the overall economy.15 If the firm’s returns are correlated with market returns, firm valuecan still be constructed to be linear in expected returns, if the expec-tation is taken using risk-adjusted state probabilities; i.e., under theequivalent Martingale measure (see later).



This is, however, an active research area and one canexpect substantial future activity.When firms care about risk, a fundamental prob-

lem is how to model risk behavior. According toKreps (1990), the predominant models for choiceunder uncertainty are the von Neumann-Morgensternpreferences and expected utility theory, where prob-abilities are objective, and the Savage (or Anscombe-Aumann) model, where probabilities are subjective.Here, we adopt the von Neumann-Morgensternparadigm that simplifies the choice over outcomes ofterminal wealth to the maximization of the expectedutility of those outcomes. Several new important con-siderations arise when adopting this paradigm:First, endowment or wealth enters the model. While

one maximizes the expected utility of terminal wealth,typically, the model primitive is initial endowment orwealth W (or a budget constraint). The investmentoptions and returns link initial wealth to terminalwealth. In line with real options theory, it is impor-tant to incorporate all available investment options.At a minimum, the investor can choose to keep someof her money in a riskless asset with certain (risk-free) return r . For simplicity, consider the one-periodinvestment problem,16 in which case, terminal wealthequals K+ 1+ rW −CK.Second, can one invest beyond the initial endow-

ment? Theoretically, this possibility is automaticallyincorporated if one assumes a perfect capital mar-ket in which one can borrow without limitations atthe risk-free interest rate. While we shall assume sofor simplicity, in practice this does not hold, and thedebate then moves to how one finances the invest-ment. Choosing the “right” mix between new equityor debt defines the capital structure of the firm, whichis beyond the scope of this paper. Stochastic capac-ity models assume (often implicitly) either a perfectcapital market or investment costs that are relativelysmall compared to the value of the firm, such thatcapital structure is unaffected. (Refer to Stenbacka andTombak 2002, for an overview of financing compli-

16 Incorporating risk considerations over sample time paths is vastlymore complicated and beyond the scope of this paper. Classicreferences include Merton (1971) and Kreps and Porteus (1978).Schroder and Skiadas (1999) present state-of-the-art formulationsand results.

cations.) Then, a risk-sensitive investor will choose acapacity vector that maximizes the expected utilityUK of terminal wealth, where

UK= ƐuK+ 1+ rW −CK (11)

and the utility function u· is strictly increasing (suchthat the investor prefers “more over less”) and con-cave (such that the investor is “risk averse,” mean-ing that the expected value is preferred over the riskyoutcome).Third, what is the form of the utility function u?

Among the myriad forms, ux = −e−zox (zo > 0) istheoretically and mathematically appealing. It modelsa decision maker with constant coefficient of abso-lute risk aversion zx = −u′′x/u′x = zo. One ofits appealing features is that the capacity decisionbecomes independent of initial wealth (if one can bor-row without limitations). Indeed, the expected utilityin (11) simplifies to

UK> z0 = −e−zo1+rW

·Ɛexp(−zoK− 1+ rCK)

A second appealing simplification happens if theoperating profits K are normally distributed.Denote its mean and variance by K and 52K.Invoking the characteristic function of the normal dis-tribution ?t= Ɛexpi= expit−1/252t2 thenyields:

UK>z0

=−e−zo1+rW−CK?izo

=−exp(−zo1+rW−CK−zoK+ 125

2Kz2o)

In summary, under the assumptions of per-fect capital markets, constant absolute risk aver-sion, and normally distributed operating profits, thevon Neumann-Morgenstern framework of maximiz-ing UK is equivalent to maximizing

UMV K> z= K−z52K (12)

where z = zo/2 is called the “risk parameter,” and and 52 are the mean and variance of firm valueexpressed in end-of-period monetary units:

K = ƐK− 1+ rCK=K− 1+ rCK

52K = Ɛ2K− ƐK2



(Recall that = 1 + r−1, so that these defini-tions are equivalent to (6), which was expressedin present-value monetary units.) The objective (12)mirrors the celebrated mean-variance formulation offinancial-portfolio theory developed and reviewed byMarkowitz (1991).Mean-variance formulations have two significant

benefits: They are implementable (i.e., only twomoments are required, which can be estimated) anduseful in the sense that they provide “good recom-mendations,” even when the decision maker does notknow her utility function. The benefit of usefulnessrelates to the important concept of the Pareto-optimal,or efficient, frontier, which can be defined as follows.For generality’s sake, let K denote a measure ofrisk of the capacity investment K and assume the fam-ily of “quasi-utility” functions

UK> z= K−zK (13)

with risk parameter z≥ 0. Review the traditional def-initions:Definition 2. Capacity portfolio K is -efficient if

and only if there does not exist another portfolio K′

such that either K′ > K while K′=K, orK′ <K while K′= K.Definition 3. The efficient frontier is the set of

risk-return pairs of -efficient portfolios:

= KK K is -efficient

Loosely speaking, under mild regularity condi-tions, the frontier is the northwest boundary of theset of all risk-return pairs KK K ∈ .Depending on the risk measure, process structure,and uncertainty, the frontier may be a concave func-tion. Then there is a one-to-one correspondence to effi-cient investments and investments that maximize the(quasi-)utility UK> z, as illustrated in Figure 5. Inthat case, we can define Kz as the efficient capacityportfolios that maximize UK> z:

Kz = argmaxK∈R

UK> z and

Uz = UKz> z (14)

If the quasi-utility UK> z is strictly concave in K,the maximizer, Kz, is unique. Then, K· is a path

Figure 5 The Risk-Return Frontier and Its Correspondence to theEfficient Capacity Portfolio Kz and Maximal Value Uz ofthe Quasi-Utility Function U· z for a Fixed RiskParameter z

Risk of capacity portfolio value R(K)

Mean capacity portfolio

value µ(K)

Best minimal

risk K0

Risk-neutral

(newsvendor) K*

z

µ(K(z))

UR z

R(K(z))

Efficient

Frontier FR

Risk

parameter

or curve in K-space that summarizes everything oneneeds to know to implement the optimal invest-ment recommendation. Hence, we shall refer to Kz

as the efficient capacity path or the relevant set of“good recommendations” referred to above, whichare those that are on the efficient frontier. To con-clude the risk-averse capacity portfolio problem, allthat is needed is a reasonable estimate of the deci-sion maker’s risk attitude z. Thus, the efficient pathand frontier embody the trade-off between expectedreturn and risk. In practice, the decision maker canbe presented with (a subset of) the efficient invest-ment path, starting from the risk-neutral solution K∗

and moving to the lowest risk efficient plan bytightening the risk parameter z. Moving from von-Neumann-Morgenstern preferences to mean-variancepreferences, however, cannot be done, in general.Indeed, the expectation of u, typically, cannot beexpressed in terms of the first and second momentonly, except under the special assumptions above orwhen the utility function u is concave quadratic. Adebate has been held on the appropriateness of mean-variance formulations. Markowitz (1991) states:

So, equipped with database, computer algorithmsand methods of estimation, the modern portfolio the-orist is able to trace the mean-variance frontiers forlarge universes of securities. But is this the right thingto do for the investor? In particular, are mean andvariance proper and sufficient criteria for portfoliochoice?



We seek a set of rules which investors can fol-low in fact—at least investors with sufficient com-putational resources. Thus we prefer an approximatemethod which is computationally feasible to a pre-cise one which cannot be computed. I believe thatthis is the point at which Kenneth Arrow’s work onthe economics of uncertainty diverges from mine. Hesought a precise and general solution. I sought asgood an approximation as could be implemented. Ibelieve both lines of inquiry are valuable.

Thus, while mean-variance preferences are consistentwith utility theory only under very limiting assump-tions, the crucial question is: How much utility dowe “give up” when maximizing the mean-variancequasi-utility? Levy and Markowitz (1979), and Krollet al. (1984) studied this question in the followingsense: If you know the expected value and varianceof a probability distribution of return on a portfo-lio, can you guess fairly closely its expected utility?They found that the correlation between the predictedexpected utilities and the actual expected utilities wasextremely high, usually exceeding 0.99. (This was thecase for a variety of utility functions and nonnor-mal probabilities.) In other words, there is evidencethat mean-variance formulations suggest “good rec-ommendations” in the sense that they yield close-to-maximal utility.Nevertheless, while widely used and appealing to

practice, mean-variance preferences have serious lim-itations. Most importantly, they treat positive devia-tions from the mean (“upside”) symmetrically withnegative deviations (“downside”). Indeed, variancesgive equal weight to deviations above or below theaverage. This implies that dominating strategies maynot lie on the frontier: There may be investment plansthat yield returns higher, or at least as high, in eachstate of nature as an investment plan on the frontier.That has led to the suggestion of other approachesto incorporate risk, including asymmetric preferencessuch as reviewed in Nawrocki (1999):

1K= ƐK−K+2

“below-mean semivariance,”

2K= Ɛt−K+2

“below-target t semivariance,”

3K= Ɛt−K+

“expected below-target t risk.”

Eppen et al.’s (1989) argument for their use ofexpected downside loss, 3, is based on the fact thatthe histogram of profits with actual data show that thedistribution of profits was not nearly normal or evensymmetric and that 3 only adds a simple linear con-straint (while variance makes the problem nonlinearand is computationally less tractable). Another relatedapproach, called robust planning (cf. Paraskevopouloset al. 1991, Laguna 1998, Malcolm and Zenios 1994) isto have the risk measure, say4, denote an increasingfunction in the “uncertainty-sensitivity of the returnK.” For example, with multivariate uncertainty(e.g., represents a “noise” vector), one suggestionfor 4K is J ′CJ, where J = K =0 and C

is the covariance matrix of . A closer view revealsthat robust planning with this sensitivity term isintimately connected to mean-variance analysis. Itsaim was to provide a practical approach for han-dling noisy data and uncertainty and it originatedfrom a stochastic control, as opposed to utility max-imization, perspective. The ideas of robust planninghave recently been adopted in economics by Hansenand Sargent (2001) and others, as summarized in themodule on “robustness to uncertainty” (May 2001issue of American Economic Review), which also givesarguments against this more heuristic approach touncertainty and ambiguity.Instead of direct maximization of expected utility

or other more heuristic risk objectives, risk sensitivitycan also be incorporated through an option valuationframework in a financial market equilibrium model ifan arbitrage-free market for trading and short-sellingcapacity assets exists. This approach is discussed next.

6.2. Mitigating Capacity-Investment Risk withFinancial Hedging

Risk-averse decision makers may be interested inmitigating risk in the capacity-investment decision.Mitigating risk, or hedging, involves taking coun-terbalancing actions so that, loosely speaking, thefuture value varies less over the possible statesof nature. If these counterbalancing actions involvetrading financial instruments, including short-selling,futures, options, and other financial derivatives, wecall this financial hedging. If, on the other hand, no



financial instruments are involved in the counterbal-ancing actions, we speak of operational hedging, whichis the focus of the next section.The relationship between investment in financial

instruments and capacity-investment is immediate byrecognizing that the effect of capacity is identicalto selling a call option to the firm’s demand aboveits capacity. Birge (2000) shows how this analogy isused to value capacity in the presence of demanduncertainty, and how it can be integrated in a linearcapacity-investment model. Financial hedging yieldsan elegant approach to price present values usingrisk-neutral discounting and to incorporate risk with-out having to resort to utility functions. The basic ideais to construct a “perfect hedge,” which is a portfo-lio that provides a constant future value in any stateof nature and, therefore, can be priced using risk-freediscounting.17

The classic example of a perfect hedge is to buya risky asset (e.g., shares in a company’s stock atshare price S0) today, and to sell a ticket that enti-tles its bearer to buy one share of stock at a terminaldate T , if she wishes, for a specified “strike” price s.(Thus, the ticket is a European call option with futurevalue Xc = ST − s+, where ST is the stock price attime T .) For simplicity, consider a two-date economywith dates indexed t = 0T and with uncertainty attime T modeled by two states of nature. Call the twostates “up” and “down,” so that ST is either S0u orS0d, where u > d are the up and down percentages.The only interesting case is to specify a strike price s

such that S0d≤ s ≤ S0u. The future value of the portfo-lio then is either S0u− S0u− s in the up state whenwe pay S0u− s to the owner of the call, or S0d in thedown state when the call is worthless. Clearly, bothpayoffs are equal if equals ∗ = S0u−s/S0u−S0d,which yields a perfect hedge. With the perfect hedge,the portfolio value is riskless and, in an arbitrage-free market, its expected value must increase at thesame rate as a risk-free asset with return r . There-fore, the present value of the portfolio is found using

17 The theoretical justifications come from the fundamental theo-rems of asset pricing stating that (1) such perfect hedge is possibleif the market is “complete,” and (2) its price is unique if the marketis “arbitrage-free.”

risk-free discounting to be ∗S0d, where, as before, = 1+ r−1 is the period’s risk-neutral discount fac-tor. In market equilibrium, there is no “free lunch,”so that the present value of the portfolio must equal∗S0 − pc, the present value of ∗ shares and a soldcall. Thus, this arbitrage condition prices the call atpc = ∗S01+ r−d.The interesting facts here are that: (1) This price

is independent of the particular risk-attitude of thebuyer of the call, and (2) the perfect hedging share∗ is independent of uncertainty. Pursuing the lat-ter fact, one can choose any probability measure overthe future states without affecting the option price.A most useful choice is to adopt the famous equiva-lent Martingale measure18 of Harrison and Kreps (1979),whose expectation operator Ɛ∗ is such that the pricepX of any claim with future value X is pX = Ɛ∗X. Inother words, the equivalent Martingale measure sepa-rates risk from time valuation by adjusting state prob-abilities such that its expectation of future value isrisk adjusted while discounting can be risk neutral. Inour example, the equivalent Martingale measure sim-plifies to the “risk-neutral state probabilities” p∗ and1− p∗ for the up and down state, respectively. Oneeasily verifies that there exist a unique p∗, such thatpc = Ɛ∗Xc= p∗Su−s and S0 = Ɛ∗ST = p∗S0u+1−p∗S0d. The results from this basic idea extend tocontinuous time and continuous state-space, yieldingthe celebrated Black-Scholes formula.To summarize, option pricing of financial hedging

provides a powerful tool to value risky assets via risk-neutral discounting using the equivalent Martingalemeasure. It circumvents the need to specify utilityfunctions or risk-adjusted discount rates. Such pow-erful results come at a cost, however. In our setting,it requires the existence of a market that: (1) tradesoptions on the firm’s and all its competitors’ capaci-ties, (2) is perfect (it allows continuous-time tradingwithout transaction fees and without restrictions onshort selling), (3) is arbitrage-free, and (4) is com-plete (there exists a self-financing trading strategy

18 The fundamental theorems of asset pricing can be restated interms of the equivalent Martingale measure (EMM) as (1) the mar-ket is arbitrage-free iff there exists an EMM, and (2) in an arbitrage-free market the EMM is unique iff the market is complete.



that replicates the firm’s cash flows so that a perfecthedge can be constructed). (While the option pricingframework is no longer consistent with utility maxi-mization when some of the conditions are relaxed, itstill provides practical use. For example, even imper-fect hedges are valuable and, thus, so is hedgingusing financial instruments that are correlated withthe firm’s profits, as shown by Gaur and Seshadri2002.) With the advent of the Internet, such capac-ity options markets may become reality, but probablyonly for certain specific industries. Financial hedg-ing requires writing an unambiguous contract thatspecifies capacity usages in a form that is divisible,tradeable, and enforceable. While possible in somesingle-commodity settings, such contract specificationis not obvious in an idiosyncratic multiproduct, mul-tiresource setting where actual capacity depends notonly on investment levels, but also on practically anyaspect of operating that process.

6.3. Mitigating Capacity-Investment Risk withOperational Hedging

Besides financial hedging, mitigating risk can also beaccomplished by operational hedging, which involvescounterbalancing actions that do not require trad-ing financial instruments. Operational hedging, some-times called natural hedging, includes various types ofprocessing flexibility such as dual-sourcing, compo-nent commonality, having the option to run overtime,dynamic substitution, routing, transshipping, or shift-ing processing among different types of capital, loca-tions, or subcontractors, holding safety stocks, havingwarranty guarantees, etc. An interesting characteris-tic of operational hedging is that it may allow oneto actually exploit uncertainty. For example, while afirm may hedge currency risk with forward contracts,a firm that has production locations in two countriesmay be able to ex-post shift production to the pref-erential country, as analyzed by Huchzermeier andCohen (1996). Similar benefits accrue to the “globalnewsvendor” of Kouvelis and Gutierrez (1997) withone production location but an ex-post transship-ment option between countries. Clearly, such oper-ational hedging may involve additional costs. Forexample, multilocation processing incurs a loss ofscale, requires procurement from a wider supply base,

slows down the learning curve process, and may pro-duce less-consistent quality.Obviously, a firm could simultaneously use both

financial and operational hedging. Allayannis et al.(2001) empirically support the hypothesis that sim-ply having multicountry production provides addi-tional value to financial hedging. (Their econometricmodel does not incorporate any ex-post operationaldecisions.) Ding and Kouvelis (2001) explore the useof both hedging instruments in a univariate, singleproduct, two-stage model. A domestic manufacturermust ex-ante invest in domestic capacity to producegoods that will be sold in a foreign market. In addi-tion to a financial currency forward contract, the firmcan ex-post decide its processing level. Chod et al.(2003) study the interdependence of operational hedg-ing, using postponement of capacity and pricing, andfinancial hedging, using a financial derivative contractwhose payoff is imperfectly correlated with the oper-ating profit. While intuition may suggest that opera-tional and financial hedging may be substitutes, theyshow that they actually can be complements.Sometimes, however, complementing operational

hedging with financial hedging may not be possible.For example, the planning horizon for a productionfacility may exceed 10 years. While operational hedg-ing can be used, it is unlikely that financial hedgingover that time-horizon is available. Financial hedg-ing of capacity is also problematic if there does notexists a capacity futures market with the characteris-tics described above.In a capacity portfolio setting, capacity imbalance is

a natural form of operational hedging. As discussedin the Introduction and in §4.2, such imbalance isworthwhile even in a risk-neutral setting. The impactof operational hedging through capacity imbalancein a risk-averse, mean-variance setting is analyzed inVan Mieghem (2003b) and some new questions areaddressed. For example, of a practical concern, onewants to know the direction along which one shouldadjust the risk-neutral capacity vector, so as to tracethe frontier and achieve a risk-optimal operationalcapacity hedge. Related, from a design and perfor-mance analysis perspective, one wants to know howeffective operational hedging is for a given network,i.e., how much return must be given up for a decrease



in risk? An answer to these two questions is formu-lated mathematically and interpreted in Van Mieghem(2003b). That paper also gives the first evidence thatoptimal relative capacity imbalance increases in riskaversion, but less significantly so when correlation ishigh. Moreover, it shows that increasing risk aversionsometimes leads to an increase in some optimal capac-ity levels, which is not optimal in single-resourcemodels. Finally, its analysis under risk aversion high-lights the third concern, in addition to the two dis-cussed above in §4.2, that should be considered whenadopting a risk-neutral newsvendor capacity solution:The newsvendor solution K∗ defines the maximal-riskextreme point of the efficient frontier of risk-returncapacity configurations. Consider all possible capacityplans—that is any nonnegative capacity three-vectorK ∈ 3

+ in the example of the introduction—and plotits associated expected profit and variance. The resultfor the example is shown in Figure 6 using the capac-ity plans defined above in §4.2. By definition, the risk-neutral optimizing newsvendor-network capacity K∗

has highest expected profit and, thus, is the right-end-point of the frontier. Hence, a risk-averse deci-sion maker may find it attractive to deviate from thenewsvendor solution to trade-off some risk for return.

Figure 6 Risk-Return Scatter Plot and Efficient Frontier for theCapacity Portfolio Investment Problem of the Example

Newsvendor

K*

Total coverage

Kc

Best zero risk

K0

Balanced to mean and

most likely demand

Kb

Variance of capacity portfolio value σ2(K)

Mea

n c

apac

ity p

ort

foli

o v

alu

e (K

)

Frontier F

Note. Each point corresponds to a specific capacity investment vector K.

As shown, giving up a small percentage in returnscan cut risk by an order of magnitude.

7. Concluding RemarksWhat inferences can be made for future stochasticcapacity-portfolio investment? From an applicationand organizational perspective, the requirements ofstochastic capacity portfolio optimization on methodsand systems for capacity planning are substantial. Itrequires that forecasting expands its point forecastwith either supplemental scenarios, or with a covari-ance matrix. The coupled nature of the optimalityequations suggest a coordinated, firm-level approach.It would be interesting to see whether decompositionapproaches are feasible that would allow some decen-tralized decision making at the plant level.Related, research is needed to model the fact that

demand is partly endogenous and partly exogenous,as discussed above. Such models could assess the“efficiency loss” that results from adopting sales-plan driven capacity planning. The problem herewould be to determine the “best” deterministic salesplan assuming an optimal mechanism design for thesales force and manufacturing incentives. An agency19

problem is embedded: Sales and marketing may agreeon a plan, but it typically does not reflect an unbi-ased assessment of what will be sold. Kouvelis andLariviere (2000) may provide a starting point for suchresearch. In addition, sales faces a very nonsymmetricpenalty function with harsher punishment for fallingshort of the sales plan. Consequently, manufacturingdoes need to hedge its capacity portfolio somehow torespond to deviations from the sales plan. In short,it appears that the need for operational hedging bypurposely unbalancing the capacity portfolio, as dis-cussed in this paper, may be robust to this more real-istic model formulation. Future research is needed toconfirm this conjecture.From a theoretical perspective, capacity portfolio

research is still in its infancy and many models remainun- or under-explored, especially queuing formula-tions. Game-theoretic and risk-averse capacity portfo-lio analysis seem to rapidly gain in popularity. Much

19 The unbalanced optimal newsvendor capacity portfolio seems toimply that a first-best deterministic sales plan does not exist.



remains to be done and many obvious questions arestill unanswered. For example, it is unclear what intu-ition we can build to answer the obvious questions:Where should one put safety capacity in the process-ing network and how does that answer change whenrisk aversion increases? An obvious initial step is toconsider these questions in the most simple capacity-portfolio problems. Eventually, analysis will developintuition.Fortunately and unfortunately, capacity-portfolio

models rapidly become complex. Complexity isunfortunate because it often makes superior analyticalsolutions elusive. Thus, simulation-based optimiza-tion becomes the natural second-best option and isexpected to increase in popularity. At the same time,complexity is fortunate as study is worthwhile with apotential impact on practice. Compared to the impactof financial portfolio analysis, even a fraction wouldbe substantial.

AcknowledgmentsThe author is grateful to the former Editor-in-Chief LeroyB. Schwarz for encouraging him to start, improve, and finish (!)this review paper. The author also thanks the senior reviewers,Evan Porteus and Gerard Cachon, for many excellent suggestions.This paper also benefited from feedback by Philipp Afeche, SunilChopra, Jim Dai, Yi Ding, Avinash Dixit, Janice Eberly, StephenGraves, Michael Harrison, Martin Lariviere, Armony Mor, RogerMyerson, Erica Plambeck, Suresh Sethi, and Costis Skiadas.

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Received: July 11, 2002; days with author: 205; revisions: 2; average review cycle time: 30 days; Senior Editors: Evan Porteus and Gérard Cachon.


capacity management, investment, and hedging

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