Multi-resource investment strategies: Operational hedging under demand uncertainty

Theory and Methodology

Multi-resource investment strategies: Operational hedging underdemand uncertainty

J. Michael Harrison a, Jan A. Van Mieghem b,*

a Graduate School of Business, Stanford University, Stanford, CA 94305-5015, USAb J.L. Kellogg Graduate School of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208-2009, USA

Received 1 May 1997; accepted 1 October 1997

Abstract

Consider a ®rm that markets multiple products, each manufactured using several resources representing various

types of capital and labor, and a linear production technology. The ®rm faces uncertain product demand and has the

option to dynamically readjust its resource investment levels, thereby changing the capacities of its linear manufacturing

process. The cost to adjust a resource level either up or down is assumed to be linear. The model developed here ex-

plicitly incorporates both capacity investment decisions and production decisions, and is general enough to include

reversible and irreversible investment. The product demand vectors for successive periods are assumed to be inde-

pendent and identically distributed. The optimal investment strategy is determined with a multi-dimensional news-

vendor model using demand distributions, a technology matrix, prices (product contribution margins), and marginal

investment costs. Our analysis highlights an important conceptual distinction between deterministic and stochastic

environments: the optimal investment strategy in our stochastic model typically involves some degree of capacity

imbalance which can never be optimal when demand is known. Ó 1999 Elsevier Science B.V. All rights reserved.

Keywords: Strategic planning; Capacity investment; Prices; Operational hedging; Multi-dimensional newsvendor model

1. Introduction

Consider a ®rm that markets n products thatare jointly manufactured using m resources.Facing uncertain product demand, the ®rm hasthe option to change its investment in the m re-sources, thereby changing the capacity of its

manufacturing process. Resources may bethought of as several types of labor and capital,and their increase usually comes at a certain cost,while a decrease may generate a revenue (e.g., inthe case of capital resources with resale value) oran expense (if the factor is costly to retire,e.g., labor).

Although multi-resource investment problemsmay seem natural, they have received little atten-tion in the literature. Economists traditionallyidentify two factors of production, capital and

European Journal of Operational Research 113 (1999) 17±29

* Corresponding author. Tel.: +1 847 491 5481; fax: +1 847

1220; e-mail: [email protected]

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 4 2 5 - 6

labor, and focus on single resource investment,viewing capital investment as mostly ®xed ± andthus irreversible ± and labor investments as vari-able and costlessly reversible. Operations re-searchers, inspired by the seminal work of Manne[1], have focused on capacity expansion and typi-cally study irreversible investment in a single re-source [2±4]. This emphasis on investment in asingle capital good does not remain justi®ed whenemployment changes are costly (the `Èurosclero-sis'' phenomenon). In addition, ®rms usually in-vest in a variety of resources with di�erent®nancial and operational characteristics. There-fore, it is desirable to consider simultaneous in-vestment in multiple resources, preferably underthe uncertainty that is crucial to most investmentproblems.

The operations research literature has studiedmulti-resource models to solve production andcapacity planning problems, typically by invokingmathematical programming techniques [5±7]. Re-cently an initial e�ort has been made to analyzetrue multi-resource investment dynamics: Dixit [8]studies the optimal investment dynamics in tworesources, capital and labor. Eberly and VanMieghem [9] present a general framework to studymulti-resource investment under uncertainty in adynamic, non-stationary environment. They showthat under rather general conditions the optimalinvestment strategy follows a control limit policy,and they provide closed-form solutions for aninvestment model where uncertainty is modeledby a geometric Brownian motion in continuoustime.

This article studies multi-resource investmentunder demand uncertainty, adding detail andspecial structure to the general framework devel-oped in [9] in order to make connection with thecontrol levers available to operations managers.While economics papers such as [8,9] take anoperating pro®t function as a primitive, we in-crease the level of detail by modeling explicitlyboth manufacturing capacity and production de-cisions, which means that the operating pro®tfunction is endogenous in our model. The for-mulation remains general enough to includeproblems of reversible and irreversible invest-ment. Having specialized the general structure

proposed in [9], we are able to sharpen thecharacterization of optimal investment strategiesdeveloped in that earlier article. We present agraphical interpretation of the optimal strategy,de®ned directly in terms of the manufacturingprocess yet simple enough to be easily taught andremembered.

The decision problems we solve are known inthe operations research literature as stochasticprograms with recourse. Traditional solutions forsuch problems use discrete stochastic mathemati-cal programming methods that can capture manypractical details, but often at the expense of ana-lytical tractability, so that one must resort to nu-merical methods. Our approach on the other hand,yields a parsimonious descriptive model that is amulti-dimensional generalization of the familiar``newsvendor model''. Like the traditional news-vendor model, our model is amenable to analyticsolution and graphical interpretation (but may betoo stylized for practical decision support sys-tems). Multi-dimensionality enriches the news-vendor model by incorporating product, demandand resource di�erentiation through price and costvectors, a technology matrix and a multivariatedemand distribution. In follow-up work, we applythe multi-dimensional newsvendor model that weintroduce here to study ¯exible technology [10]and subcontracting and outsourcing [11]. Bothmodels show how the capacity investment decisionnot only depends on correlation and risk inproduct demands, but also on price and cost dif-ferentials.

A ®nal contribution of this article is to givequalitative insight into real-world capacity plan-ning and capital budgeting practices, which typi-cally involve two levels of decision making.Initially, lower-level production planners proposeleast-cost capacity adjustments necessary to en-able execution of forecasted production and salesplans, which are treated as deterministic. Ulti-mately, however, senior-level managers approvean investment plan that is a ``perturbed'' versionof the one recommended on the basis of suchdeterministic reasoning, seeking to minimize riskexposure due to demand uncertainties. Our modelcon®rms and quanti®es the optimal `òperationalhedge'': capacity levels should be set so as to

18 J.M. Harrison, J.A. Van Mieghem / European Journal of Operational Research 113 (1999) 17±29

balance `ùnderage costs'' and `òverage costs'',properly de®ned. The optimal hedging strategyoften leads to unbalanced capacities, in the sensethat no single demand scenario may exist thatimplies full utilization of all resources. This ob-servation, that optimal multi-resource investmentdecisions typically involve some degree of capac-ity imbalance when demand is uncertain, is notnew. On the other hand, it does not seem to besharply etched in the minds of either academics orpractitioners, and our exposition of the unbal-ancing phenomenon in the context of a stylizedmodel is intended to help remedy that situation.Under certain conditions common in high-techindustries, near-optimal hedging can be achievedby using a suitably de®ned one-dimensionalnewsvendor model to plan the capacity of thecritical resource, and then investing in ample ca-pacity for the remaining resources. The latterapproximation may be helpful to guide out-sourcing decisions.

The article is organized as follows. Section 2brie¯y reviews the dynamics of multi-resourceinvestment under uncertainty and specializes thegeneral model developed in [9] to linear jointproduction of n products on m resources. Sec-tion 3 presents the multi-dimensional newsven-dor solution for the optimal investment levels,which is illustrated with a comprehensiveexample in Section 4. Section 5 shows how theoptimal investment can be interpreted as ahedging strategy against demand uncertainty andrelates this to capital budgeting practice andoutsourcing decisions. Section 6 summarizes theanalysis.

We conclude this introduction with somenotational conventions. We will not distinguishin notation between scalars and vectors. Allvectors are assumed to be column vectors, andprimes denote transposes, so u0v is the innerproduct of u and v. Vector inequalities shouldbe interpreted component-wise, as well asmax�0; u� and max�0;ÿu� which are denotedby u� and uÿ, respectively. As usual, P��denotes a probability measure and E isthe associated expectation operator. Finally,rg�� denotes the gradient of a di�erentiablefunction g��.

2. Linear manufacturing with demand uncertainty

Consider a ®rm that deploys m resources tomanufacture n products in periods t 2 f1; . . . ; T g.At the beginning of each period t, the ®rm mustdecide on a non-negative m-vector of resourcelevels Kt 2 Rm

� for period t production, before theproduct demand vector Dt 2 Rn

� for that period isobserved. After demand is observed, the ®rmchooses an n-vector xt 2 Rn

� of production quan-tities for the various products, constrained by itsearlier resource investment. This multi-stage deci-sion problem is characteristic of real option models[6,12]: ®rst invest in capabilities, then receive someadditional information, and ®nally exploit capa-bilities optimally, contingent on the revealed in-formation. This decision problem also is known asa stochastic program with recourse. As in Dixit'swork [8] on investment dynamics, successive peri-ods are linked by resource investment decisions,but not by inventories (which would make theproblem much more complicated). These as-sumptions are consistent with what is observed inpractice where `èach time period is of su�cientlength (e.g. one year) so that production levels canbe altered within the time period in order to satisfyas closely as possible the demand that is actuallyexperienced'' ([5], p. 519).

We model the ®rm's manufacturing process andproduction decisions as follows. In each period t,having chosen a resource vector Kt and observed ademand vector Dt, the ®rm chooses its productionvector xt so as to maximize pro®t in the followinglinear program (often called a product mix prob-lem):

maxxt2Rn

�p0xt �1�

s:t: Axt6Kt �capacity constraints�; �2�xt6Dt �demand constraints�; �3�

where p 2 Rn� is an n-vector whose jth component

represents the unit contribution margin for productj (that is, sales price minus variable cost of pro-duction), and A is an m� n technology matrixwhose �i; j� component represents the amount ofresource i required to produce one unit of productj. In accordance with standard terminologyin linear programming, Kt often will be called a

J.M. Harrison, J.A. Van Mieghem / European Journal of Operational Research 113 (1999) 17±29 19

capacity vector. Note that the contribution mar-gins and technology matrix are assumed for sim-plicity to be time invariant (not depending on t).Also, contribution margins do not depend on theproduction quantities chosen, so the ®rm implicitlyis assumed to be a price taker in both the outputand input market. The optimal objective value ofthe product mix problem (1)±(3) is the maximaloperating pro®t function, and is denoted byp�Kt;Dt� � p0x�Kt;Dt�, where x�Kt;Dt� is an asso-ciated optimal production vector. From linearprogramming theory we know that p��;Dt� isconcave, a fact that is important for the develop-ment to follow.

It is straightforward to incorporate demandshortage penalties into the model as follows. As-sume product j carries a shortage penalty costcP;j P 0 for each unit of demand that is not satis-®ed so that the total shortage cost is c0P�Dt ÿ xt��.Using Eq. (3), it follows that all results presentedin this article remain valid if we in¯ate unit con-tribution margins p to p � cP and decrease theoperating pro®t p�K;D� by c0PD. Similarly, re-source operating costs that depend on the installedcapacity Kt, rather than on actual production xt,can be included if they are convex as a function ofKt.

What connects successive periods in our modelis that resource or capacity levels carry overwithout change unless explicit action is taken.Thus, resources are neither created, depleted, nordestroyed by production activity. If the ®rm de-cides to adjust resource vector Ktÿ1 to Kt at thebeginning of period t, it incurs an adjustment costC�Kt ÿ Ktÿ1�, where C�z� � c0z� ÿ r0zÿ and c and rare m-vectors. Thus, by assumption, the cost ofadjusting any one resource level either up or downis linear in the size of the adjustment, and totaladjustment cost is additive over resources. It isperhaps simplest to think of both c and r as pos-itive, but disinvestment in some resources (for ex-ample, labor) may be costly, or even prohibitivelyexpensive (which means that investment is irre-versible), and investment in other resources may beso heavily subsidized as to generate a positive cash¯ow. In general, then, we allow components ofboth c and r to be either positive or negative, butwe focus on resources that are costly to adjust,

such that c and r are non-zero, subject to the ad-ditional requirement that c P dr, where dP 0 isthe one-period discount factor (see below). Ifc < dr, then used capacity would be worth morethan new capacity, and the ®rm could generate acash stream with arbitrarily large present value byjust making a large investment in one resource andreversing it in the following period. Readersshould note that resource adjustment costs areassumed to be time invariant.

Because the ®rm's investment problem is as-sumed to end after T time periods, we must furtherspecify the salvage value f �KT � associated with the®nal vector of resource levels KT . In the interest oftractability, we assume that f �KT � � r0KT , whichmeans that disinvestment in all resources is man-datory and the associated marginal revenues areexactly the same as in earlier periods. The ®rmseeks a multi-period strategy of investment andproduction that maximizes the expected presentvalue of operating pro®ts minus resource adjust-ment costs over T time periods, given an initial m-vector of resource levels K0 and a single perioddiscount factor dP 0. (One naturally thinks interms of the case where 0 < d < 1.)

The problem formulated above is a special caseof the multi-period investment problem analyzedin [9]. The general model in [9] takes as primitivean operating pro®t function pt�Kt;x� for eachperiod t, where x is the ``state of the world'' andpt�Kt;x� is measurable with respect to informationavailable in period t. In the current context, pt isassumed to depend on x only through the period tdemand Dt. Furthermore, we have added detail tothe model by explicitly recognizing productionquantities xt as decision variables, and then de®n-ing pt�Kt;Dt� as the maximal operating pro®tachievable in the linear programming problem (1)±(3). Because the vector p of contribution marginsand the technology matrix A appearing inEqs. (1)±(3) do not depend on t, the operatingpro®t function p��; �� is itself independent of t, andwe have further assumed that the marginal in-vestment cost vector c and the marginal disin-vestment revenue vector r are time invariant.

As a last and crucial element of the problemformulation in this article, let us further assumethe following special structure: the demand vectors


D1;D2; . . . ;DT are independent and identicallydistributed (IID), and there is no other relevantsource of uncertainty in the problem environment.Under these assumptions, the ``state'' of ourdynamic investment problem at the beginning ofperiod t is fully summarized by the incoming ca-pacity vector Ktÿ1. Because capacities are non-negative by de®nition, our ``state space'' is theorthant Rm

�. In [9] it was shown that there exists anoptimal policy having an ISD (Invest/Stay put/Disinvest) structure. Roughly speaking, this meansthat in each period t, the state space is partitionedinto various domains, including a ``continuationregion'' St (see Fig. 1) where no adjustment needbe made in the vector of current resource levels.From any one of the domains outside this con-tinuation region, the optimal investment action isto adjust the vector of resource levels to a newspeci®ed point on the boundary of the continua-tion region. In the special case treated here(problem data are time invariant, the salvagefunction is f �KT � � r0KT , demands are IID, and noother sources of relevant uncertainty exist), the

optimal ISD policy is shown in [9] to have thefollowing additional special structure. Denote theexpected operating pro®t by P, where

P�Kt� � Ep�Kt;Dt�for all Kt 2 R�, which also is concave and is as-sumed to be smooth. 1 According to [7], Proposi-tion 1, where the function P was taken as aprimitive, we have:

Proposition 1. Starting from any initial capacityvector K0 2 Rm

�, the optimal investment strategymakes no capacity adjustments after the beginningof period 1 (that is, K1 � K2 � � � � � KT under theoptimal strategy). Moreover, the \stay-put region",

Fig. 1. Structure of the optimal ISD investment policy for m � 2 resources.

1 This is the case if Dt is a continuous random variable (see

Proposition 2). In general, there may be a set of Lebesgue

measure zero where the concave function P is not di�erentiable.

However, its left and right partial derivatives always exist and

all following di�erential statements should be interpreted in

subdi�erential form.


or continuation region, characterizing the optimalISD policy for period 1 is given by

S1 �

K 2 Rm�: �1ÿ d�r6rP�K�6 1ÿ d

1ÿ dT �cÿ dT r��

:

�4�E�ectively, the multi-period solution in our case

can be found by solving a model with a singleperiod of length T with equivalent discount ratede � dT and equivalent marginal adjustment costs:

re � 1ÿ d

1ÿ dT r and ce � 1ÿ d

1ÿ dT c; �5�

which represent the total marginal cost of (dis)in-vesting in real terms. Because we have shown howto map this multi-period problem into an equiva-lent single period formulation, we will focus forthe remainder of the article on the single periodmodel and its continuation region S.

The fact that our multi-period problem e�ec-tively collapses to an equivalent single periodproblem is of course a result of our highly stylizedmodel. In general, the form of an optimal invest-ment strategy is not this simple (e.g., see theBrownian dynamic investment model in [9]).However, the qualitative insights regarding oper-ational hedging that we will derive from our styl-ized model can be extended to more complexsettings with non-stationary problem data, de-pendent demands, and additional sources of un-certainty, but no attempt will be made to provethat assertion in this article.

For ease of notation, we will suppress all time-subscripts and we will denote the components of avector v by vx; vy ; . . . to prevent confusion.

3. The multi-dimensional newsvendor solution

Continuing with the equivalent single periodformulation, the continuation region S of the opti-mal policy for linear production according toEqs. (1)±(3), can be expressed in terms of the shad-ow values or dual prices of the capacity constraints:

Proposition 2. Let D be a continuous randomvariable that is ®nite with probability 1. Then, the

expected operating pro®t function P�� Ep��;D�is di�erentiable and for each capacity choiceK 2 Rm

� and realized demand D 2 Rn�, the capacity

constraints of the linear program (1)±(3) haveoptimal shadow values k�K;D� 2 Rm

� andrP�� Ek��;D�. The continuation region of theoptimal ISD investment policy can be described interms of these shadow values:

S � K 2 Rm�: �1ÿ de�re6Ek�K;D�6 ce ÿ dere

� :

�6�(The proof is relegated to the appendix.) The

optimal investment solution is thus reduced to afundamental quantity of a linear program, ashadow value vector of the capacity constraints(2), which can be calculated explicitly for su�-ciently simple examples (see below). We interpretthis result as the multi-dimensional generalizationof the solution to the familiar newsvendor problemthat considers a newsvendor who must purchase Knewspapers at a cost ce in anticipation of an un-certain demand D. Given that newspapers sell atp � ce (thus, p is the unit contribution margin) andhave no resale value (re � 0), the newsvendor'sproblem is to determine how many newspapersshould be purchased in advance. This is the singleproduct, single resource, single period case of ourlinear manufacturing model where A � 1. The LPcan be solved by inspection: k�K;D� � 0 for D < Kand k�K;D� � p for D P K, and the optimal pur-chase quantity KI is the well-known critical fractilesolution:

Ek�KI;D� � pP�D P KI� � ce:

The critical fractile is found by balancing theexpected `ùnderage cost'' (e.g., the weightedprobability of the area fD P KIg) with the `òver-age'' or adjustment cost (ce). Proposition 2 greatlyenhances the intuitive content of the model byproviding a similar solution technique with agraphical interpretation to the multidimensionalcase: calculate the shadow values k�K;D� by solvingthe linear program parametrically in terms of K andD. There are a ®nite number of di�erent values forthe shadow values. Thus, one can draw polygonalconvex domains in the demand space in whichk�K;D� is constant and the expected shadow values


can easily be calculated by summing weightedprobabilities of areas. The optimal investment levelis found by adjusting the capacity vector K suchthat the sum of the areas equals the marginal ad-justment costs. One can think of this result as sayingthat it is optimal to invest up to a critical ``fractile''of the multidimensional demand distribution.

The following generalizations of the one-di-mensional newsvendor problem are obtained.First, one will never invest so as to cover themaximum demand (``never go to the 100th per-centile''). Indeed, if all resources are costly to in-vest in (ce > 0), the expected shadow values Ekmust be positive. Because a shadow value is posi-tive only when the corresponding capacity con-straint is binding, all constraints must be bindingwith positive probability at the optimal investmentlevel. In addition, if marginal investment costs ce

decrease, expected dual prices Ek should decrease,lowering the probability that constraints are bind-ing, and one will invest so as to cover more de-mand (``go to a higher fractile''). Second, theoptimal investment level is increasing in the ratioof the scale of unit contribution margins to thescale of marginal adjustment costs. Indeed, theshadow values k are linear in p and if all marginsincrease proportionally without changing theprice-mix, i.e., there is a scalar h > 1 so that mar-gins p become hp, the shadow values Ek increaselikewise to hEk. Finally, we have the followingeconomic interpretation of the ISD policy: adjustinvestment levels only if the expected marginalbene®t of doing so, as measured by the shadowvalues, outweighs the marginal adjustment cost.

4. An example with two products and three resources

To illustrate the multi-dimensional newsvendorsolution and to set the stage for the next section,consider a ®rm that produces two products on twodedicated assembly lines with a joint ®nal test. Thissituation arises in the computer and disk drive in-dustries and in the back-end of semiconductormanufacturing (packaging and ®nal test). Manyoriginal equipment manufacturers in these hightechnology industries engage mainly in ®nal as-sembly and test operations, purchasing most sub-

assemblies and components from a network ofsuppliers. Their ®nal assembly processes often arecharacterized by relatively inexpensive assemblysteps followed by an expensive test step. Moreover,assembly capacity often is speci®c to a productmodel whereas test equipment is generic (test re-sources are typically computers with a changeabletest bed) and therefore shared by multiple products.

Denoting the capacity of the two dedicatedlines by Kx and Ky and the test capacity by Kz, the®rm's linear manufacturing process can be mod-eled by a technology matrix:

A �1 0

0 1

c 1

0B@1CA;

where we assume that the ratio of test capacityrequirement rates of product 1 to product 2 isc > 0. The demand forecast for both products ismodeled by a probability measure P on the de-mand (or sample) space R2

�, so formally,P : R2

� ! �0; 1�: We also assume that the unitcontribution margins p are su�ciently high tojustify production of both products and thatproducts are labeled such that px P py . It is clearthat one should not have more capacity in eitherassembly line than total test capacity and thatthere should not be more test capacity thantotal assembly capacity. We therefore have thatcKx6Kz, Ky 6Kz and cKx� Ky P Kz.

We want to determine the optimal investmentstrategy which follows an ISD policy whose con-tinuation region S is calculated with our multi-di-mensional newsvendor solution. The ®rst step is todetermine the optimal production decisions xcontingent on the demand vector D. To actuallywrite out the ®rm's optimal strategy in terms of theproblem parameters, it is useful to partition thedemand space R2

� for a given a capacity vectorK 2 R3

� as follows:

X0�K� � fD 2 R2�: AD6Kg;

X1�K� � fD 2 R2�: cDx6Kz ÿ Ky and Dy > Kyg;

X2�K� � fD 2 R2�: cÿ1�Kz ÿ Ky� < Dx6Kx

and cDx � Dy > Kzg;


X3�K� � fD 2 R2�: Dx > Kx and Dy > Kz ÿ cKxg;

X4�K� � fD 2 R2�: Dx > Kx and Dy 6Kz ÿ cKxg:

This partitioning is the direct result of the para-metric analysis of the product mix problem (1)±(3).Indeed, by applying the Simplex method to thatlinear program where K and D are parameters, thelinear inequalities that de®ne the domains Xi areautomatically manifested. Within each domain,there is an optimal basic solution with a corre-sponding optimal dual variable k�K;D�. It directlyfollows that the dual variables k�K;D�; represent-ing shadow prices for the three kinds of capacity ifthe state of the world D obtains, are constant overeach of the ®ve domains identi®ed above. Assum-ing px > py to insure a unique solution, the optimalvalues of the primal and dual variables are:

The optimal production decisions are displayedin demand space in Fig. 2. The second step of themulti-dimensional newsvendor solution calls onthe demand distribution to calculate the expectedshadow value vector Ek. Because the shadow valuevector is constant in each of the ®ve domains, itsexpectation is easily calculated and, using Eq. (6),the continuation region S of the optimal invest-ment strategy becomes

S � K 2 R3�: �1ÿ de�re6

0

py

0

0B@1CAP �X1�K��

8><>:�

0

0

py


px ÿ cpy

0

py


�px

0

0

0B@1CAP �X4�K��6 ce ÿ dere

9>=>;:The continuation region S, a three-dimensional

volume whose exact shape depends on the demandprobability measure P , partitions the state space

x�K;D� k�K;D�if D 2 X0�K� D �0; 0; 0�if D 2 X1�K� �Dx;Ky� �0; py ; 0�if D 2 X2�K� �Dx;Kz ÿ cDx� �0; 0; py�if D 2 X3�K� �Kx;Kz ÿ cKx� �px ÿ cpy ; 0; py�if D 2 X4�K� �Kx;Dy� �px; 0; 0�:

Fig. 2. Production decisions x and shadow values k depend on the capacity K and demand D.


into nine domains. The optimal investment strat-egy makes no adjustment to an initial capacityvector that is contained in S. If the initial capacityvector is in any of the domains outside S, it isoptimal to adjust capacity to a point on theboundary of S with a domain-speci®c action,similarly to the actions shown in the two-dimen-sional example in Fig. 1. If we focus on the in-vestment decision starting with zero initialcapacity the optimal investment level KI is thelower left corner of the continuation region:

0

py

0

0B@1CAP �X1�KI��

0

0

py


�px ÿ cpy

0

py


px

0

0

0B@1CAP�X4�KI��

� ce ÿ dere: �7�

Thus, the optimal investment level KI can beinterpreted graphically as in Fig. 3 where wehave superimposed the joint correlated demand

distribution (represented by the elliptical area thatobtains as a level curve of the multivariate normaldistribution) onto the state space partitioning ofFig. 2: one has to adjust the thick lines of thefeasible region X0 (by adjusting K) such that theprobabilities of domains X1; . . . ;X4 o�set the totalmarginal investment cost ce ÿ dere as in the opti-mality equation (7).

In general, the optimal solution requires solvingthe characteristic equations simultaneously usingthe multivariate demand distribution. However, ifthere exists a critical resource like the expensivetest resource, a near optimal solution can be foundwith a one-dimensional newsvendor model as fol-lows. If test capacity is much more expensive thanassembly capacity (co

z ?cox ; c

oy ), optimality equation

(7) yields that P�X2� � P �X3�?P �X1�; P �X4�. ThusP�X1� and P �X4� should be very small in whichcase the graphical interpretation shown in Fig. 3yields P �X2� � P �X3� ' P �cDx � Dy P Kz�. Thenthe third optimality equation in (7) becomes atraditional one-dimensional newsvendor equation

pyP �cDx � Dy P Kz� ' coz : �8�

Fig. 3. Optimal investment KI is found by balancing `ùnderage costs'' with investment costs.


Thus the optimal operational hedging strategycan be decomposed: the capacity Kz of the expen-sive resource (testers) is determined with a one-dimensional newsvendor model (8) using the(univariate) total demand for test capacity cDx � Dy

(e�ectively assuming in®nite capacity at the up-stream assemblies) and the smaller contributionmargin py . Second, the optimal capacity of theinexpensive dedicated resources (assemblies) is theminimal capacity that does not decrease the ex-pensive test capacity (i.e., make P �X1� and P �X4�positive, but small, solving Eq. (7)).

5. Operational hedging versus coordinated planning

In many ®rms, incremental capacity invest-ments are decided upon through a two-stage de-cision process that can be roughly described asfollows using the language and notation intro-duced in Sections 2±4. First, lower-level produc-tion planners propose least-cost required capacityadjustments based on a deterministic masterproduction schedule. To arrive at these require-ments, the ®rm's operations division calculates thetechnology matrix A and current capacity vector Kusing product routing, resource consumption andresource availability information, all of which (inthe best case) reside in various engineering data-bases. Marketing and sales representatives developforecasts of the demand vector D and contributionmargins p for the coming period. Then, as dis-cussed in [13,14], sales managers and productionmanagers collaborate in the choice of a singlevector x of production quantities for the comingperiod, a choice which they deem to be the bestresponse to estimated market demand. The vectorx of production quantities is usually known as theproduction plan or Master Production Schedule(MPS), and production planners then choose aleast-cost capacity adjustment to enable executionof that plan. That is, using ®nancial informationcaptured in the capacity adjustment cost functionC, they choose a new capacity vector Kc whichminimizes adjustment costs subject to the re-quirement that the production plan x be feasibleunder Kc (Ax6Kc). We call this least-cost programof required capacity adjustments a coordinated

plan because it typically implies full utilization ofresources under the production plan x. (In theexample of the previous section, a coordinatedplan corresponds to a rectangular feasible regionX0�Kc�.) By basing incremental capacity invest-ments on a speci®c production plan x, coordinatedplanning e�ectively makes a bet on a particularfuture demand. Ultimately, however, senior-levelmanagement ®nalizes a capital budget and ap-proves an investment plan which may be quitedi�erent from the coordinated plan. Acknowledg-ing intrinsic demand uncertainty (or, similarly,price uncertainty), senior executives ``perturb'' thecapacity requirements given by production plan-ning in order to minimize risk exposure (thus im-proving expected pro®t). The perturbation is basedon experience and individual appraisal of futureuncertainties and risks and yields a hedging 2

position by a counterbalancing investment in var-ious production factors. Following Huchzermeierand Cohen [16], we call this operational hedging todistinguish from ®nancial hedging, which uses ®-nancial instruments and capital markets to mini-mize a ®rm's risk exposure. A major US disk drivemanufacturer, facing an investment problem sim-ilar to the one described in the previous sectionalmost every year, reduces the investment adjust-ment for expensive test capacity compared to thelevel speci®ed by the coordinated plan while in-creasing less expensive ®nal assembly capacities. Inour stylized example, this ``cuts o� the corner'' ofthe rectangular feasible region that results fromcoordinated planning and yields a feasible regionas shown in Fig. 3. Similar practice is found in theautomobile industry where car manufacturers of-ten set the capacity of a body assembly line to lessthan the sum of the assembly line capacities of theV6 and V8 engines that will power the body.

Our model con®rms the soundness of opera-tional hedging: the optimal strategy minimizes therisk exposure of the investment by not committingto a single production scenario/plan. The model

2 The Oxford Dictionary [15] describes hedging as ``protec-

ting oneself against loss or error by not committing oneself to a

single course of action.''


also quanti®es the optimal hedge as the one thatobtains from setting investment levels so as tobalance the dual prices or `ùnderage costs'' ofresources against the investment or `òveragecosts'' of resource adjustment. (In our example ofSection 4, the optimal balance is given by Eq. (7).)Two conclusions 3 follows from this interpretation:First, one should expect the required capacity ad-justments as proposed in the coordinated plan tobe di�erent from the optimally hedged investmentsthat senior management tries to attain, no matterhow sophisticated the ®rm is in picking the pro-duction plan x. Second, it is optimal not to balancecapacities: the optimal operational hedging posi-tion may be unbalanced in the sense that theredoes not exist a single demand scenario that willkeep all resources fully 4 utilized. This implies thatthe performance of capacity planning softwarethat starts with a single deterministic demandscenario may be improved.

As shown in Section 4, near-optimal hedgingcan be achieved when there exists a critical re-source by using a suitably de®ned one-dimensionalnewsvendor model to plan the capacity of thecritical resource, and then investing in ample ca-pacity for the remaining resources. Then the opti-mal hedging solution can be applied successfully ina market mechanism where a ®rm can subcontractor outsource parts of its manufacturing process.Consider for example the high-tech example in theprevious section where the ®rm has the option tooutsource the assembly of product 2. The ®rm hasto commit to (and pay for) subcontracting capacityKy at a market 5 price of ce

y per unit before product

demand is realized. This is equivalent to capacityadjustment and this model gives the optimalquantity (or capacity) that the ®rm should out-source. Although optimally made simultaneously,this decision can be made sequentially after the®rm has decided on its crucial (test) capacity ac-cording to the above decomposition scheme.

Finally, it may be interesting to estimate byhow much senior management's operationalhedging improves the net value of the ®rm ascompared to the coordinated plan that derivesfrom simple least-cost adjustment calculations. Ageneral answer obviously depends on all parame-ters of the problem. To get a feeling for the mag-nitude of the improvement that may be obtained,we can compare operational hedging to coordi-nated planning in the simple example of the pre-vious section when starting with no initialcapacity. For simplicity, assume that both prod-ucts are substitutes (e.g., laptop and desktop disk-drives or V6 and V8 engines) and their total de-mand of volume is reasonably well know while theproduct mix is uncertain. Thus we can approxi-mate the demand distribution to be uniform on theline Dx � Dy � 1, and we will assume that bothproducts require equal amounts of test capacity(c � 1). Let �ce

z � py�1ÿ �cex=px� ÿ �ce

y=py��. If06 ce

z 6 �cez , the optimal hedging strategy 6 sets

Kx � 1ÿ �cex=px�, Ky � 1ÿ �ce

y=py�, and Kz � 1 (seeFig. 4). The corresponding expected value of the®rm under optimal hedging is

V � px

21ÿ ce

x

px

� �2

� py

21ÿ ce

y

py

� �2

ÿ cez :

The best coordinated plan Kc has Kcz � Kc

x � Kcy

with constrained optimality equations Eki�Kc;D��Ekz�Kc;D� � ce

i � cez for i � x; y or, because

P�X2�Kc�� 0, piP �Di P Kci � � ce

i � cez and thus

Kci � 1ÿ �ce

i � cez�=pi. The ®rm's expected value

under the best coordinated plan is

V c � px

21ÿ ce

x � cez

px

� �2

� py

21ÿ ce

y � cez

py

� �2

:

3 These conclusions hold almost always (i.e., whenever the

solution K to the optimal investment equations yields capacity

constraints that do not cross in a single point, that is there exist

no x such that Ax � K).4 The optimal contingent production vector is either inside

the feasible region, in which case no resource is fully utilized, or

on the boundary of the feasible region, in which case not all

resources are fully utilized because the capacity constraints do

not intersect.5 If no e�cient market for capacity exists, the capacity

investment decision with subcontracting becomes ``relationship

speci®c'' and will depend on the capacity decisions of the

subcontractor as shown in [11].

6 Under this assumption on cez , the optimality equations (7)

do not have a solution if Kz > 1 (and thus P�X2� � P �X3� � 0)

or if Kz < 1 (and thus P �X1� � P �X2� � P �X3� � P�X4� � 1).


Therefore, the improvement of operationalhedging compared to all other coordinated plans isat least

V ÿ V c � cez 1ÿ ce

x � 0:5cez

pxÿ ce

y � 0:5cez

py

� �> 0;

and even compared to the best coordinated plan,the improvement may be signi®cant:

06 V ÿ V c

V c 6pxpy�px � py� 1ÿ ce

xpxÿ ce

y

py

� �2

p3x 1ÿ ce

xÿcey

px

� �2

� p3y 1ÿ ce

yÿcex

py

� �2

6 1

2 1� �pxÿpy�2pxpy

� � 6 50%;

where all bounds are tight (lower and upperbounds are attained when ce

z � 0; cez � �ce

z ;ce

x � cey � 0; ce

z � �cez ; and ce

x � cey � 0; ce

z � �cez ;

px � py , respectively).

6. Conclusion

This paper has presented a model to determinethe optimal investment strategies for a manufac-

turing ®rm that employs multiple resources tomarket several products to an uncertain demand.The model incorporates explicitly the productionprocess and the production decisions and is ableto consider reversible and irreversible investment.The optimal investment is determined with ourmulti-dimensional newsvendor solution usinga demand distribution, a technology matrix,prices (product contribution margins), and mar-ginal investment costs, and can be representedgraphically.

The optimal investment position can be inter-preted as a hedge against demand uncertainty.This has led us to four managerially relevantconclusions: First, the model explains currentpractice and quanti®es the optimal operationalhedge. Second, one should expect the requiredcapacity adjustments as proposed in the coordi-nated plan by lower-level capacity planners to bedi�erent from the optimal hedged investmentsthat senior management tries to attain. Third, it isoptimal not to balance capacities. Finally, near-optimal hedging can be obtained by planningcapacity of the critical resource with a suitablyde®ned one-dimensional newsvendor model and

Fig. 4. Comparing the optimal hedge K with the best coordinated plan Kc.


by investing in ample capacity for the remainingresources.

Appendix A. Proof of Proposition 2

Consider the (single period) dual of the linearprogram (1)±(3):

mink2Rm

�;l2Rn�

K 0k� D0l;

s:t: A0k� lP p:

If D is ®nite, the primal program has a ®nite op-timal solution and its corresponding objectivefunction is equal to the optimal objective functionof the dual according to the Duality Theorem oflinear programming. Let k�K;D� and l�K;D�denote an optimal solution of the dual. Fixa Ko 2 Rm

�. Then, for any K 2 Rm�, it follows

directly that p�K;D�6K 0k�Ko;D� � D0l�Ko;D�:Combining this with p�Ko;D� � Ko0k�Ko;D��D0l�Ko;D�, directly yields the familiar result thatk�Ko;D� is a subgradient of p��;D� at K � Ko:

p�K;D�6p�Ko;D� � k�Ko;D�0�K ÿ Ko�; �9�

or, interpreting r as a subgradient, rp��;D� �k��;D�. In essence, we now must show that wecan interchange di�erentiation and integration(a.k.a. Leibniz's rule) to yield rEp��;D� �Ek��;D�. Use the arguments in [17], pp. 97±98 asfollows: Because D is ®nite w.p. 1, taking expec-tations in Eq. (9) yields that Ek�Ko;D� is a sub-gradient of P�� Ep��;D� at K � Ko. Also,because p��;D� is concave, it is di�erentiable ev-erywhere except on a possible set L of Lebesguemeasure zero. Thus, k is single valued except onL. If D is a continuous random variable, L has P -measure zero. Thus the subgradient Ek�K;D� isunique for all K 2 Rm

� so that P�� is di�erentia-ble everywhere.

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Multi-resource investment strategies: Operational hedging under demand uncertainty