Constrained service reliability under stochastic demand

Ira Horowitz*

City University of Hong Kong, Hong Kong

Received 1 July 1998; accepted 1 September 1999

Abstract

Boronico [Boronico JS. An investigation into the costs and bene®ts of reliability of service. Omega 1998;26(1):99±114] considers the determination of ex ante service reliability in a price-taking, expected-pro®t-maximizing enterprisethat must respect an expected-service±quality constraint in the face of stochastic demand. The constrained service±

reliability issue, however, is much more complex than his analysis reveals. That complexity comes to light once onerecognizes the additional need to consider the curvature of the underlying functions and the demands imposed uponthat curvature by the second-order conditions for a constrained maximum. The introduction of uncertainty into thepicture only adds to the issue's complexity and further complicates the problem of drawing unambiguous

inferences. 7 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Decision making/process; Management; Reliability

1. Introduction

Boronico [1, p. 112] summarizes his recent paper on

the costs and bene®ts of the reliability of service by

noting that his results ``o�er insight as to the solutions

regarding . . . service quality attributes for a pro®t

maximizing service provider operating under con-

ditions of stochastic demand. The pursuit of additional

research in this area will assist in increasing the e�-

ciency of service provided by ®rms in competition, and

help to insure that consumer expectations regarding

quality service are adhered to. This, as well as related

problems, pose a signi®cant and clearly important areafor continued research.'' Heeding this clarion call, thepresent paper revisits Boronico's model [1, pp. 101±2]

in order to address particular omissions and o�er ad-ditional insights.

2. The model, MARK II

Boronico posits a pro®t-maximizing, price-takingenterprise that sells X(r ) units of its service at the mar-ket price of p, where 1rrr0 is the ex ante declared re-

liability of the service. Since demand for the servicedepends only upon its reliability and X(r ) is strictlyincreasing in r, it is incumbent upon the ®rm to pro-

vide its customers with some assurance of that re-liability. To provide this assurance, the ®rm also sets

Omega 28 (2000) 361±369

0305-0483/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.

PII: S0305-0483(99 )00046-8

www.elsevier.com/locate/orms

* Correspondence address: Decision and Information

Sciences, Warrington College of Business Administration,

University of Florida, Gainesville, FL 32611-7169, USA. Tel.:

+1-352-392-8572; fax: +1-352-392-5438.

E-mail address: rohari@dale.cba.u¯.edu (I. Horowitz).

the level of some desirable local operating variable,such as installed capacity, so as to respect a service±

quality constraint, H. The level of the operating vari-able is denoted yr0: The service±quality constraint iswritten H�X�r�, r, y�R0: The total cost of providing the

service, the reliability, and the assurance is denotedC�X�r�, r, y�, where C�X�r�, r, y� is a strictly increasingfunction of X(r ) for given r and y, and r for given

X(r ) and y. That is, the ®rm's costs rise when it pro-vides more service or when it provides any given levelof service with a higher ex ante reliability, but not

necessarily when it establishes a higher level of thelocal operating variable (as, say, in the case of aneconomic lot size).The ®rm earns a pro®t of P�r, y� � pX�r� ÿ C�X�r�,

r, y�: All functions are assumed to be continuous anddi�erentiable in their arguments. Strictly concave orconvex functions are assumed to be at least twice dif-

ferentiable.

2.1. The certainty case1

The ®rm's problem is to set r and y so as to maxi-

mize P�r, y�, subject to H�X�r�, r, y�R0: Let gR0denote a Lagrange multiplier, let r�, y� and g� denotethe optimal levels of r, y and g, respectively, and write

the Lagrangian as

L�r, y, g� � P�r, y� � gH�X�r�, r, y�:The second-order conditions for a constrained maxi-

mum require the further assumptions that P�r, y� beconcave and that H�X�r�, r, y� be convex.It will also be assumed at the outset that the optimal

solution is an interior solution such that 0 < r� < 1, y�

>0 and P�r�, y��r0:2 These are cost-free assumptions

that reward us with the elimination of some unnecess-ary tedium.

Subscripts denote partial derivatives; e.g. @X�r�=@r �Xr and @C�X�r�, r, y�=@X�r� � CX, and similarly forsecond derivatives. Then, r�, y� and g� satisfy the

Karush±Kuhn±Tucker conditions:

pXr ÿ CXXr ÿ Cr � g�HXXr �Hr � � 0: �1a�

ÿCy � gHy � 0: �1b�

gH � 0: �1c�Assuming that Hy�X�r��, r�, y�� 6� 0, it immediately

follows from Eq. (1b) that

g� � Cy=HyR0:

The latter inequality opens up two possibilities, g�=0

and g�<0. These cases are considered in turn.

2.1.1. g� � 0

To obtain g�=0 requires that Cy � 0: In this case,then, and only in the situation in which total cost is astrictly convex function of the local operating variable,will the ®rm set the level of that variable so as to mini-

mize total cost, given the optimum level of reliability;Hy is irrelevant and, from Eq. (1c) and the constraint,we only require H�X�r��, r�, y��R0: If, however,

Hy�X�r��, r�, y�� equals zero in contradiction of theassumption that it does not, then from Eq. (1b) wewill directly obtain the cost-minimizing condition that

Cy is also equal to zero. Moreover, if H�X�r�, r, y� is astrictly convex function of y, then this function, too,will now be minimized with respect to the operating

variable, although not necessarily with respect to thelevel of reliability.From Eq. (1a) it directly follows that when g�=0

pXr � CXXr � Cr > 0:

Boronico [1, p. 102] refers to the left-hand-side term,pXr, as ``the marginal bene®t of increasing reliability''

and I defer to that expression. The right-hand-sideterm is the marginal cost of increasing reliability, giventhat the local operating variable has been set at its op-

timum, cost-minimizing level. Thus we arrive at a fam-iliar marginal-bene®t-equal-to-marginal-cost operatingprinciple.

2.1.2. g� < 0To obtain g� < 0 ordinarily requires that Cy$0, Hy

$0, and that the two terms be opposite in sign.3 From

Eq. (1c), the strict inequality on the multiplier alsorequires that H�X�r��, r�, y�� � 0, or the service±qual-ity constraint is binding. Thus, if at r� and y� a further

1 I take some minor liberties in this section, including the

de®nitions of marginal bene®t and marginal cost of increasing

reliability, as Boronico does not consider the situation in

which demand for the service is known with certainty. It is,

however, very useful to explore this case in detail prior to

introducing uncertainty into the picture.2 When both the cost function and the service±quality con-

straint are linear, the problem reduces to a trivial linear pro-

gramming problem. In that case, either (a) r�=0 or y�=0, or

(b) r�=1 and y�r0, or (c) there are an in®nite number of op-

timal interior solutions. In any event, the constraint holds as

an exact equality.3 In principle it is also possible to obtain g� < 0, when

Cy=0 and Hy=0. For this to occur, however, from Eq. (1c)

it must also be true that H = 0. Therefore we have three

equations in only two unknowns, r and y (because X is a

function of r ). These equations must have the fortuitous

property that all are satis®ed by the same (r, y ) couplet(s).

Still further, when that couplet is substituted into Eq. (1a),

the substitution must yield a negative g�.

I. Horowitz / Omega 28 (2000) 361±369362

increase in the operating variable will add to (subtractfrom) total cost, it must reduce (increase) the service±

quality function. The one thing we know for certain,however, is that if total cost is a strictly convex func-tion of the operating variable, then total cost will not

be minimized with respect to that variable.In the event that we can achieve the dual objective

of having nondegenerate Cy and Hy be opposite in

sign and the constraint be binding, it then followsfrom Eq. (1a) that

pXr � �CXXr � Cr� ÿ �Cy=Hy��HXXr �Hr�: �2�

It is the latter right-hand-side expression that Boro-

nico [1, p. 102] refers to as the marginal cost ofincreasing reliability, an appellation that costs us con-siderable insight. To fully appreciate the loss, consider

a bare-bones model of the single-product pro®t-maxi-mizing ®rm that takes in a total revenue of TR � f �q�when it sells an output of q, where f(q ) is a concave

and at least once di�erentiable function of q. The totalcost of producing q is given by the strictly convex andat least twice di�erentiable function TC � g�q�: The®rm's total pro®t, which is given by P � f �q� ÿ g�q�, ismaximized at the output q� where marginal revenue ofdf/dq is equal to marginal cost of dg/dq.Suppose the ®rm has a capacity constraint that

forces it to produce no more than qM units. Denote byfR0 a Lagrange multiplier. If q�RqM, the pro®t-maxi-mizing solution once again occurs where marginal rev-

enue is equal to marginal cost. If, however, q� > qM,then the optimum occurs at qM, a point at whichdf=dq � dg=dqÿ f�: Since f� < 0, the optimum occurs

where marginal revenue is greater than marginal cost,and f� is the marginal pro®t that is sacri®ced as aresult of having to impose the capacity constraint.Alternatively, suppose the ®rm operates under a com-

mitment to produce at least qm units and let jR0denote yet another Lagrange multiplier. If q� < qm,then the optimum occurs at qm, a point at which

df=dq � dg=dq� j�: Since j� < 0, the optimum nowoccurs where marginal revenue is less than marginalcost, and j� is the marginal loss that is incurred as a

result of being committed to produce in excess of thepro®t-maximizing output.Viewed in this light, the ®rst bracketed term on the

right-hand side of Eq. (2), CXXr � Cr, remains the

marginal cost associated with an increase in reliability.Nothing has occurred that would alter this interpret-ation of that expression. The product of the bracketed

terms is therefore either (a) the marginal pro®t that theenterprise is forced to sacri®ce because it must respectthe service±quality constraint, or (b) the marginal loss

that it incurs as result of having to respect that con-straint; otherwise it is zero. Which of the three it is,and whether the marginal bene®t obtained at r�

exceeds, is less than, or is equal to the marginal costincurred at r�, depends upon the sign of the product of

these two bracketed terms.The marginal bene®t and marginal cost terms can

only be equal when HXXr �Hr � 0: Suppose H�X�r�,r, y� � H 0�r, y� is a strictly convex function of r. Then,the further implication of a pro®t-maximizing solutionthat results in a marginal bene®t±marginal cost equal-

ity is that H�X�r�, r, y�� is simultaneously minimizedwith respect to r, and that it takes on the value of zeroat its minimum.

Since Cy and Hy are of opposite sign, the marginalbene®t of increasing reliability exceeds the marginalcost associated with such an increase whenHXXr �Hr > 0: In this case, �Cy=Hy��HXXr �Hr� is

the marginal pro®t that is sacri®ced because of the ser-vice±quality constraint. The implication is that the ser-vice±quality constraint leads the enterprise to o�er a

lower level of service reliability than would be o�eredin its absence. The marginal bene®t of increasing re-liability will be less than the marginal cost associated

with such an increase when HXXr �Hr < 0: In thiscase, �Cy=Hy��HXXr �Hr� is the marginal loss incurredbecause of the service±quality constraint. The impli-

cation is that the service±quality constraint leads theenterprise to o�er a higher level of service reliabilitythan would be o�ered in its absence.In sum, the interrelationships here are far more com-

plex and richly interpreted than is implied in Boronico[1]. This is so even in the simplest possible casewherein management has no uncertainty about

demand. That complexity is revealed once we payattention to the second-order conditions for a con-strained maximum and the curvature of the functions,

which Boronico neglects. The issues become even moreconfounded when we turn to the situation of stochasticdemand, which is the sole bene®ciary of Boronico'sministrations.

2.2. The stochastic case

Let m denote a random variable with mean Efmg andvariance s2 > 0, where E denotes the expectation oper-ator. To introduce demand uncertainty into the model,the service demand function may be rewritten asX�r, m�, where X�r, Efmg� � X�r�: All the other func-

tions remain the same and retain the same propertiesas in the certainty case.Management of the enterprise is assumed to be risk

neutral and therefore to have the objective of deter-mining r and y so as to maximize expected pro®ts ofEfP�r, y, m�g: This maximization now takes place sub-

ject to an expected-service±quality constraint ofEfH�X�r�, r, y�gR0: It is assumed that the ®rm willsatisfy the demand for the service at a level of X�r�, m�,

I. Horowitz / Omega 28 (2000) 361±369 363

even when P�r�, y�, m� < 0 and when the service±qual-ity constraint is violated, for any realization of m.4

Let l R 0 denote a Lagrange multiplier, and writethe Lagrangian as

L�r, y, l� � EfP�r, y�g � lEfH�X�r�, r, y�g:

It will once again be assumed that the optimal solution

is an interior solution such that 0 < r� < 1, y� > 0,and, in a modest revision, that EfP�r�, y��gr0: Therevised Karush±Kuhn±Tucker conditions are:

Ef pXrg ÿ EfCXXr � Crg � lEf�HXXr �Hr�g � 0: �3a�

ÿEfCyg � lEfHyg � 0; �3b�

lEfH g � 0: �3c�

As in the certainty case, with the notation modi®ed

only by the addition of the expectation operator,assuming now that EfHy�X�r��, r�, y��g 6� 0, it immedi-ately follows from Eq. (3b) that

l� � EfCyg=EfHygR0:

The latter inequality again opens up two possibilities,l�=0 and l�<0. These cases are considered in turn.

2.2.1. ll�=0To obtain l�=0 requires that EfCyg � 0: In this

case, then, and only in the case in which total cost is a

strictly convex function of the local operating variable,will the ®rm set the level of that variable so as to mini-mize expected total cost, given the optimum level of re-

liability; Hy and its expectation are irrelevant, andfrom Eq. (3c) and the constraint, we only requireEfH�X�r��, r�, y��gR0: When EfHy�X�r��, r�, y��g � 0,we have a result that is analogous to the certainty case

wherein Hy�X�r��, r�, y�� � 0:

From Eq. (3a) it directly follows that

pEfXrg � �EfCXg��EfXrg� � EfCrg � s1: �4�

s1 is the covariance between CX and Xr. That covari-ance may be written as [2, p. 232]:

s11�CXXXm��Xrm�s2:

Each of the terms in the latter expression is evaluated

at E{m }.The term pEfXrg is the expected marginal bene®t of

increasing reliability; the term �EfCXg��EfXrg� �EfCrg � s1 is the expected marginal cost of increasing

reliability, given that the local operating variable is setat its optimum level. Thus we again arrive at a form ofa marginal-bene®t-equal-to-marginal-cost operating

principle. In the uncertainty case, however, theexpected marginal cost depends critically upon the var-iance in the random variable, or in e�ect on the var-

iance in demand, because of the s1 term. Moreover,there is no a priori basis for a conjecture as to whethera ceteris paribus increase in the variance will increaseor reduce the expected marginal cost. If, for example,

total cost is a linear function of service provided (X ),then CXX=0 and the variance is irrelevant. If, how-ever, total cost is a strictly convex function of service

provided, then CXX > 0: In that case, the sign of s1depends upon the signs of Xrm and Xm, neither one ofwhich is transparent.

Assuredly, when the source of the uncertainty isexplicitly de®ned it may be possible to deduce the signsof Xrm and Xm, but it is not possible to make a univer-

sally applicable statement about them. When, forexample, price is the source of the uncertainty so thatp � m, it is straightforward that Xm < 0: That is, higherprices reduce demand. One might also conjecture that

price and reliability are substitute attributes for consu-mers in the sense that a higher price reduces the ad-ditional demand e�ected by a greater declared

reliability, and Xrm < 0: Therefore �CXX��XmXrm�s2 > 0so long as total cost is a strictly convex function of theservice provided. Suppose, then, that the variance

increases. This increase implies a greater expected mar-ginal cost at r� and y�, which in turn implies a greaterexpected marginal bene®t in order to maintain theequality in Eq. (4). Now, however, the expected mar-

ginal bene®t is given by �Ef pg��EfXrg� � s2, where s2,as the covariance between p and Xr, is approximatelyequal to Xrms2 < 0, evaluated at the expected price.

Thus, the increased variance causes an automatic re-duction in the expected marginal bene®t at r�, ratherthan the requisite increase. The equality of Eq. (4) can

be restored in the most straightforward manner bychanging r so as to raise the expected marginal bene®t,reduce the expected marginal cost, or some combi-

4 It is only in the expected sense that the service±quality

constraint cannot be violated, at least as the problem is for-

mulated here. For example, as its service±quality constraint a

®rm might elect to keep an inventory level, including a safety

stock, in excess of some amount I�. The ®rm starts a pro-

duction period with a beginning inventory of I0, produces y,

and sells X. The end-of-period inventory is I0 � yÿ X: In a

world of certain demand, the ®rm can always adjust its oper-

ating variable y to assure that I0 � yÿ XrI �: When demand

(X ) is uncertain, however, and the ®rm produces y in advance

of knowing X, no such guarantee is possible. There may in

fact be times in which demand is so great that not only does

the ®rm's inventory level fall below I�, but it might be necess-

ary to backlog orders if at all possible. Yet, even under these

circumstances the ®rm can determine y so as to satisfy the

expected service quality constraint of EfI0 � yÿ X grI �:

I. Horowitz / Omega 28 (2000) 361±369364

nation of the two. When total cost is a strictly convexfunction of reliability, expected marginal cost can be

reduced by setting r at a lower level than r�. Simul-taneously, when demand is a strictly concave functionof reliability the expected marginal bene®t will increase

when r is set at a lower level than r�. In tandem theseobservations imply that an increase in the variance ofthe random variable, when price is the random vari-

able and the total cost and demand functions arestrictly convex and strictly concave, respectively, willresult in a lower level of ex ante declared reliability.

More critically, these observations also demonstratethat even for a risk-neutral enterprise the variance inthe random variable can play an important role in thereliability decision-making process.

2.2.2. ll�<0

To obtain l� < 0, ordinarily requires that EfCyg 6� 0,EfHyg 6�0, and that the two terms be opposite in sign.From Eq. (3c), the strict inequality on the multiplier

also requires that EfH�X�r��, r�, y��g � 0, or theexpected-service±quality constraint is binding. Thus, ifat r� and y� a further increase in the operating variable

will add to (subtract from) expected total cost, it mustreduce (increase) the expected-service±quality function.The one thing we know for certain, however, is that if

total cost is a strictly convex function of the operatingvariable, then expected total cost will not be minimizedwith respect to that variable.In the event that we can achieve the dual objective

of having nondegenerate E{Cy } and E{Hy } be oppo-site in sign and the constraint be binding, it thenfollows from Eq. (3a) that

pEfXrg � �EfCXg��EfXrg� � EfCrg � s1

ÿ �EfCyg=EfHyg���EfHXg��EfXrg�

� EfHrg � s3�: �5�

The s3 term, as the covariance between HX and Xr, isapproximately equal to �HXXXm��Xrm�s2, evaluated atE{m }. Since ÿEfCyg=EfHyg > 0 and HXXR0, the same

inferences that were drawn above with respect to s1,and the same sorts of arguments that were made in thecertainty case with respect to HXXr �Hr, can now bemade with respect to �EfHXg��EfXrg� � EfHrg � s3:That is, the sign of the latter term re¯ects whether thenow-binding expected-service±quality constraint e�ectsa sacri®ce in expected marginal pro®t that would

otherwise be earned, or an expected marginal loss thatwould otherwise be avoided. The sign also re¯ectswhether the binding constraint implies a level of ex

ante reliability that is lower or higher, respectively,than would otherwise be the case. In addition, theextent of the sacri®ce or loss may depend upon the

variance of the random variable, but not in an unam-biguous and a priori predictable, fashion.

3. Illustrative example

3.1. The certainty case

By way of illustration, consider a price-taking ®rm

that sells X(r ) units of its product at the market priceof p. Management's policy is to replace defective unitsat no additional charge, on a ®rst-come, ®rst-served

basis, so long as replacement units are available.Because management does not want to risk making thesame mistake twice, it produces the units that it sells

to the market on one machine, and it produces thereplacement units, denoted y, on a second, higher-qual-ity machine. Letting F denote the ®xed cost associated

with operating both machines, the ®rm produces inaccordance with the short-run cost functionC�X�r�, y� � F�m1X

2 �m2y2, where m2 > m1 > 0:

The ex ante probability that a unit will be defective

is 1ÿr. In any short-run production period in whichthe ®rm has sold X units, the expected number ofdefectives is �1ÿ r�X: Acting out of caution, however,

management always understates the reliability of itsproduct. Nonetheless, it imposes the conditionthat yr�1ÿ r�X: Therefore H�X�r�, r, y� �ÿy� �1ÿ r�XR0:The demand for the ®rm's product is given by

X�r� � aeÿb�1ÿr�, where a > 0 and b > 0. Hence the

®rm earns a short-run pro®t of P�r, y� � pX�r�ÿFÿm1X

2 ÿm2y2: As observed in Appendix A

(Eq. (A.1)), P�r, y� is maximized where y�=0 and

r� � r�o � 1� �1=b��ln� p=2m1a��:

3.1.1. gg�=0

The case of g�=0 returns us to the unconstrainedoptimal solution of ro

�. From Eq. (A.1) and r� < 1, wesee that p < 2m1a: Even in that event, however, g�=0

also implies y�=0, in violation of the service±qualityconstraint. Therefore, g�< 0 and the constraint holdsas a strict equality.

3.1.2. gg�<0

As shown in Appendix A (Eq. (A.2)), wheny� � �1ÿ r��X, r� must satisfy

eÿb�1ÿr� � p=�2m1a��1� �m2=m1�f�1ÿ r�2 ÿ �1ÿ r�=bg�:The latter is a rather daunting equation in r, but a

numerical example will help.Let p=21.41, m1=2, m2=2.5, F=10, a=100

and b=10. In the unconstrained case,

I. Horowitz / Omega 28 (2000) 361±369 365

r�o � 1� 0:1ln�21:41=400� � 0:7072: Thus, ro� is mod-

estly greater than the r�c � 0:70 value that solves daunt-

ing Eq. (A.2). In this instance, then, when the service±quality constraint is imposed, the ®rm actually makesa less reliable product than otherwise. In compen-

sation, however, management seeks to mollify some ifnot all of those customers who get stuck with a lemon,by providing them with a free replacement unit. With-

out the constraint, the ®rm sells X(0.7072)=5.35 units,produces y=0 replacement units, takes in 114.54 inrevenue, incurs a total cost of 67.25, and earns a pro®t

of 57.29. With the constraint, the ®rm sellsX(0.7)=4.98 units, produces y=1.49 replacementunits, and earns a pro®t of 41.47. Thus, the replace-ment policy costs the ®rm 15.82, which amounts to a

27.6% pro®t reduction. As we could have anticipatedwith a little aforethought, the ®rm produces the mini-mum number of replacement parts that it permits itself

to produce. Moreover, the 1% percent reduction in re-liability results in a 7% reduction in units sold.The value g�=ÿ7.45 is the shadow price assigned to

the service±quality constraint. In e�ect, relaxing theconstraint by setting yr�1ÿ r�Xÿ D, for `small' D >0, would result in a pro®t increase of 7.45D Further,

since HXXr �Hr � aeÿb�1ÿr��b�1ÿ r� ÿ 1� is necessarilypositive, we could have anticipated from the theor-etical analysis that the binding service±quality con-straint would lead management to o�er a lower

level of reliability. Continuing in this vein,Cy=Hy � ÿ2m2�1ÿ r�aeÿb�1ÿr�: Hence, when evaluatedat rc

�=0.70, the marginal bene®t of increasing re-

liability �pabeÿb�1ÿr�� exceeds the marginal costof such an increase �2m1a2beÿ2b�1ÿr�� by2m2�1ÿ r�a2eÿ2b�1ÿr��b�1ÿ r� ÿ 1� � 74:40:By inspection, higher values of p result in higher

levels of rc�, and higher values of m1 result in lower

values of rc�. The intuition behind this result is that

management wants to take full advantage of any price

increases by selling a greater amount of its product,and the way to do this is by increasing the product'sreliability. As production becomes more costly, how-

ever, management wants to produce and sell less of the

product. As a price taker, the ®rm's only recourse is toreduce the reliability of the product and hence the

demand for it. This observation is useful in interpret-ing management's reaction to uncertainty.5

3.2. The stochastic case

In the economics literature, uncertainty has typically

been introduced into demand functions in one of twobasic ways: by adding a stochastic term (see e.g. [3]) orby multiplying an independent variable by a stochastic

term (see e.g. [4]). More generally, when the demandfunction is estimated using some regression technique,uncertainty enters both through the random-error term

and through the parameter estimates, each of whichcomes with a standard error [5]. Thus, depending uponthe functional form, uncertainty may be both additive

and multiplicative. There are therefore ample pre-cedents for introducing uncertainty into the presentdemand function in any number of forms, three ofwhich are considered here. In the ®rst form, demand is

equal to X(r ) plus a normally distributed random vari-able, m, that has an expected value of Efmg � 0 and avariance of s2 > 0:6 In the second form, the exponent

in the demand function is equal to ÿb�1ÿ r� plus arandom variable, m, that has the aforementioned prop-erties. And in the third form, a is assumed to be the

random variable, where Efm � ag � 100:Table 1 summarizes the numerical results for both

certainty cases, as well as for the three stochastic cases.In the ®rst stochastic case, X�r, m� � aeÿb�1ÿr� � m and

so Xr is independent of m, which results in s1 � s3 � 0:We therefore have the equivalent of the certainty situ-ation and consequently the same numerical results. In

this situation, each covariance is equal to zero, becausethe marginal demand that is generated by increased re-liability (Xr) does not vary at all in response to changes

in the random variable. Consequently there is no pointin tinkering with the level of reliability in the face of

Table 1

The numerical results for the illustrative example

Certainty case r � X(r �) y � P g �

Unconstrained 0.707 5.35 0 57.29 0

Constrained 0.700 4.98 1.49 41.47 ÿ7.45

Stochastic case r � EfX�r�, m�g Ef y�g EfPg l�

X�r, m� � aeÿb�1ÿr� � m 0.700 4.98 1.49 41.47 ÿ7.45X�r, m� � aeÿb�1ÿr��m 0.615 4.29 1.65 38.13 ÿ8.25X�r, m� � meÿb�1ÿr� 0.691 4.55 1.41 41.06 ÿ7.05

5 To help keep the algebra manageable, for purposes of this

illustration r was not made an argument of C so that Cr � 0:One plausible way in which r might enter the cost function

directly is through m1, say by writing m1 � mr: Then, in the

unconstrained case the pro®t maximum is achieved where

p � 2maeÿb�1ÿr��r� 1=b�: Let m=2.828. Then, at the uncon-

strained optimum of r�o � 0:7072, m1 � mr once again is equal

to 2. Now, however, the optimum ex ante reliability is

r�o 0 � 0:6954: As one would expect, when an increase in re-

liability has a direct cost �Cr � mX 2), the ®rm o�ers a lower

level of reliability than otherwise.6 It is assumed throughout that s 2 is su�ciently small as to

make a negative demand virtually impossible.

I. Horowitz / Omega 28 (2000) 361±369366

the uncertainty that is introduced by the random vari-

able.In the second two cases, where X�r, m� � aeÿb�1ÿr��m

and X�r, m� � aeÿb�1ÿr�, respectively, the optimal re-

liability must satisfy Eq. (A.3) and Eq. (A.4), respect-ively.

Eq. (A.3) is a rather more daunting version of Eq.(A.2). Nonetheless, let s 2=1. Then by numerical ap-proximation we determine r�u10:615: When uncer-

tainty enters in additive form through the exponent,the optimal level of reliability declines. Moreover, thegreater is the variance, the greater will be the decline.

In this case, Xr depends upon m and the decrease inthe stated reliability is e�ected by what is a positive co-

variance between the marginal cost of production(CX ), and the marginal demand generated by increasedreliability.

Eckel and Smith [6, p. 61] observe that the covari-ance may be di�cult to interpret. In the present

instance, however, the interpretation is quite straight-forward: both marginal cost and marginal demandmove in the same direction in response to changes in

the random variable. Management's reaction to theintroduction of uncertainty will therefore be con-ditioned by whether a greater marginal demand, say,

will produce su�ciently higher revenues as to o�set theaccompanying increase in the production costs needed

to satisfy that greater demand.Speci®cally, higher values of the random variable m

result in a stronger positive response of demand to

increases in the product's reliability. As both m and rappear with positive signs in the exponent, increases inm move demand in the same direction as do increases

in r. At the ®xed price any such increases result inhigher revenues. To satisfy those higher demands the

®rm must produce greater outputs. Those greater out-puts imply higher marginal costs. The higher revenuese�ected by that higher marginal demand, however, do

not compensate for the increases in marginal cost. Thisis why the ®rm stopped increasing rc

� beyond 0.70 inthe m=0, certainty-equivalent situation. To guard

against the unwelcome prospect of higher unantici-pated demands, management reduces the level of ser-

vice reliability. The lower reliability in turn reduces theexpected demand. In tandem, the latter reductionsresult in an expected pro®t that is 8.1% less than the

certainty-equivalent pro®t. Although the expecteddemand is reduced, the expected number of replace-ment parts increases, because of the reduced reliability

of the original parts. Accompanying the increase in theexpected number of replacement parts is a higher cost

to respecting the constraint (l�=ÿ8.25 as opposed tog�=ÿ7.45).The only di�erence between Eq. (A.4) and Eq. (A.2)

is in the addition of the s 2/a 2 term on the right-handside. To assure a negligible probability of X < 0, s

must be small relative to Efmg � a: With a=100,s=30 would be about as large a value as one could

justify, and even in that case, s2=a2 is only 0.09. Hencewe can infer that when uncertainty enters through athere will be minimal impact on reliability. The num-

bers of Table 1 con®rm that inference. In the presentinstance, r�u10:691: The latter reduces the expectedpro®t to a level that is less than 1% below the cer-

tainty-equivalent pro®t level. The expected number ofreplacement parts drops slightly below the certainty-equivalent level, as a consequence of which the cost to

respecting the constraint is also reduced.The covariance s3 is equal to zero in all three cases,

because HXX=0. Thus, the HX term that plays a rolein the determination of the service-level optimum is

una�ected by any unanticipated changes in demandthat result from the uncertainty.Nevertheless, once again when uncertainty is intro-

duced into this particular setting the optimal level ofstated reliability is reduced. And, as was the case withthe previous form of uncertainty, the reduction occurs

because of the positive covariance. Speci®cally, highervalues of m mean greater marginal demand achievedthrough increased reliability, as well as increased mar-

ginal costs resulting from increased production; or,marginal demand and marginal cost move in the samedirection in response to changes in the random vari-able. But each additional unit sold brings in a ®xed

revenue of p = 21.41. Marginal production costs forthose additional units are equal to 4X; or, marginalcosts are increasing. Absent the uncertainty about a,management stopped increasing the stated reliability ofits product at rc

�=0.70 because it was not pro®table toproduce the additional units that would be demanded

at a higher level of ex ante reliability. In response tothe present form of uncertainty, management compen-sates for the positive covariance by reducing the pro-duct's reliability, albeit not by very much, in order to

discourage the additional demand that would resultfrom values of a that exceed its certainty-equivalentlevel of 100.

4. Conclusion

Service reliability is indeed an important and inter-

esting issue. Boronico [1] has demonstrated the neces-sity of considering that issue in conjunction with aservice±quality constraint. The constrained service±re-

liability issue, however, is much more complex than hisanalysis reveals, even when there is no uncertainty andwhen the price-taking assumption is retained. That

complexity comes to light once one recognizes the ad-ditional need to consider the curvature of the under-lying functions and the demands imposed upon that

I. Horowitz / Omega 28 (2000) 361±369 367

curvature by the second-order conditions for a con-strained maximum. The introduction of uncertainty

into the picture only adds to the issue's complexityand further complicates the problem of drawing unam-biguous inferences. But life is rarely easy, particularly

when either, to say nothing of both, stochastic el-ements and Mr. Karush, Mr. Kuhn, and Mr. Tuckerare involved.

Appendix A

A.1. The certainty case

The ®rm earns a short-run pro®t of P�r, y� �pX�r� ÿ Fÿm1X

2 ÿm2y2: Assuming continuity, it is

readily veri®ed that P�r, y� is maximized where y�=0and r� � 1� �1=b��ln� p=2m1a��, at which point P�r, y�is a strictly convex function of r and y. It is alsoreadily veri®ed that H 0�r, y� is a linear (hence convex)function of y and a strictly convex function of r so

long as b�1ÿ r� > 2:To maximize pro®ts we need to determine values of

r� and y� that satisfy the Karush±Kuhn±Tucker con-

ditions derived from the following Lagrangian:

L�r, y, g� � P�r, y� � gH 0�r, y�:

A.1.1. g� � 0

The case of g�=0 returns us to the unconstrained

optimal solution of:

r�o � 1� �1=b��ln� p=2m1a��: �A:1�

The expression for the unconstrained optimum im-mediately tells us that p < 2m1a is required for r� < 1:Even in that event, however, g�=0 also implies y�=0,

in violation of the service±quality constraint.Therefore, g� < 0 and the constraint holds as a strictequality.

A.1.2. g� < 0

When y� � �1ÿ r��X, to determine r� we need `only'substitute the appropriate expressions into Eq. (2). The

required expressions are the following:

Xr � abeÿb�1ÿr�; CX � 2m1aeÿb�1ÿr�; Cy � 2m2y;

Hy � ÿ1; HX � 1ÿ r; Hr � ÿaeÿb�1ÿr�:

Since r enters C only through X�r), Cr � 0: Thus, Eq.(2) becomes:

pabeÿb�1ÿr� � 2m1a2beÿ2b�1ÿr�

ÿ �2m2y=�ÿ1����1ÿ r�abeÿb�1ÿr� ÿ aeÿb�1ÿr��:

Substituting y� � �1ÿ r�X and rearranging terms:

eÿb�1ÿr� � p=�2m1a��1� �m2=m1�f�1ÿ r�2

ÿ �1ÿ r�=bg�:�A:2�

A.2. The stochastic case

Taking advantage of what was learned in the cer-tainty case, let us assume at the outset that under

uncertainty the ®rm will want to set its operating vari-able, y, at the minimum acceptable level ofy� � �1ÿ r�X, so that l� < 0: In this case, r� mustsatisfy Eq. (5).

A.2.1. X�r, m� � aeÿb�1ÿr� � m

In this stochastic case, EfXrg � Xr and EfCXg � CX:Hence, the optimal solution can only di�er from thatin the corresponding certainty case if either s1 6� 0 ors3 6� 0: With this form of additive uncertainty, how-

ever, Xm � 1, so that Xmr � 0, which means that s1 �s3 � 0: Hence, when X�r, m� � aeÿb�1ÿr� � m, we havethe equivalent of the certainty situation, and r� � r�u �r�c � 0:70:

A.2.2. X�r, m� � aeÿb�1ÿr��m

For notational convenience, let z � s2=2: In thisstochastic situation, X�r, m� � aeÿb�1ÿr��m � aeÿb�1ÿr�em:It is well known that Efemg � ez [7, p. 541]. Thus,EfX g � aeÿb�1ÿr��z, EfXrg � abeÿb�1ÿr��z, EfCXg �2m1aeÿb�1ÿr��z, EfCyg � 2m2�1ÿ r�aeÿb�1ÿr��z, EfHXg �1ÿ r, EfHrg � ÿaeÿb�1ÿr��z and CXX � 2m1: Further-more, Xm � aeÿb�1ÿr� and Xrm � abeÿb�1ÿr�, where the

latter expressions are evaluated at Efmg � 0: Hence,s11�CXXXm��Xrm�s2 � 2m1a2beÿ2b�1ÿr�s2: Happily, how-ever, HXX � 0, so that s3 � 0: In this situation, then,

Eq. (5) becomes:

pabeÿb�1ÿr��z � 2m1a2beÿ2b�1ÿr��2z

��2m2�1ÿ r�aeÿb�1ÿr��z���1ÿ r�abeÿb�1ÿr��z

ÿ aeÿb�1ÿr��z� � 2m1a2beÿ2b�1ÿr�s2:

After dividing both sides of the equation byabeÿb�1ÿr��z and rearranging terms we are rewardedwith the following result:

I. Horowitz / Omega 28 (2000) 361±369368

eÿb�1ÿr� � � p�=�2m1a��1� �m2=m1�f�1ÿ r�2

ÿ �1ÿ r�=bg � eÿ2zs2�ez:�A:3�

A.2.3. X�r, m� � meÿb�1ÿr�

Since Efmg � a in this third form of uncertainty, theonly change beyond the certainty case again occursthrough the covariance term where CXX � 2m1,

Xm � eÿb�1ÿr� and Xrm � beÿb�1ÿr�: Hence,s1 � 2m1beÿ2b�1ÿr�s2: Repeating the earlier algebraicmachinations, we determine that:

eÿb�1ÿr� � p=�2m1a��1� �m2=m1�f�1ÿ r�2

ÿ �1ÿ r�=bg � s2=a2:�A:4�

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