make-to-order versus make-to-stock in a production–inventory system with general production times

Make-to-order versus make-to-stock in aproduction±inventory system with general production times

ANTONIO ARREOLA-RISA1 and GREGORY A. DeCROIX2

1Department of Information & Operations Management, Lowry Mays College & Graduate School of Business,Texas A&M University, College Station, Texas 77843-4217, USAE-mail: [email protected] School of Business, Duke University, Durham, NC 27708-0120, USAE-mail: [email protected]

Received October 1996 and accepted September 1997

We study the optimality of make-to-order (MTO) versus make-to-stock (MTS) policies for a company producing multipleheterogeneous products at a shared manufacturing facility. Manufacturing times are general i.i.d. random variables, and di�erentproducts may have di�erent manufacturing-time distributions. Demands for the products are independent Poisson processes withdi�erent arrival rates. The costs of managing the production-inventory system are stationary and include inventory holding andbackordering costs. Backordering costs may be $ per unit or $ per unit per unit time. We derive optimality conditions for MTO andMTS policies. We also study the impact of several managerial considerations on the MTO versus MTS decision.

1. Introduction

Competitive pressures and continuous improvemente�orts have led many ®rms to review their production±inventory practices. In particular, given the push towarda ``zero-inventory'' goal, world-class companies havebegun a search for conditions that determine when it isoptimal to hold a ®nished goods inventory and when it isnot. If a ®nished goods inventory is held for a product, wesay that the product is produced make-to-stock (MTS);otherwise, we say that the product is produced make-to-order (MTO). Note that our designation of a productas MTO merely indicates stockless production, and doesnot mean that the product is made to order due to customspeci®cations.

To address the optimality of MTS and MTO policies,we consider a company that produces multiple productsfacing random demands. A single manufacturing facilityproduces the products one at a time, and successivemanufacturing times for each product comprise a se-quence of general i.i.d. random variables. Di�erentproducts may have di�erent manufacturing-time distri-butions. Demands for the products are independentPoisson processes with di�erent arrival rates. The costs ofmanaging the production±inventory system are station-ary and include inventory holding and backorderingcosts.

Although the form of the optimal production±inven-tory policy for this system has not been established, a

base-stock (order-up-to) inventory policy has been shownto be optimal in a similar system under periodic review[1]. Also, the base-stock inventory policy has commonlybeen used in previous work analyzing other aspects ofsystems similar to the one considered here [2±9]. For thesereasons we adopt a base-stock inventory policy for oursystem. The other element that de®nes the production±inventory policy is the scheduling rule ± i.e., if multipleproducts are waiting to be manufactured, which productshould be produced ®rst? Some recent work [3±9] suggeststhat dynamic scheduling rules may outperform the ®rst-come ®rst-served (FCFS) rule. However, analysis of theMTO/MTS decision for dynamic scheduling rules be-comes intractable due to complex interactions amongproducts. Furthermore, the additional information andcoordination required at the production facility by dy-namic scheduling rules may counter the inventory savingsover the FCFS rule. As a result and given that FCFS iscommon in practice, we will adopt FCFS as the sched-uling rule for our system. We believe that the insightsregarding the MTO/MTS decision obtained under theFCFS scheduling rule should be suggestive of the be-havior of systems with more general scheduling rules, andin fact, we will show that this is the case for a number ofour results.The paper makes several contributions. Optimality

conditions for MTO and MTS policies are established.These conditions are derived for two backorder-costcases: $ per unit and $ per unit per unit time. The opti-

0740-817X Ó 1998 ``IIE''

IIE Transactions (1998) 30, 705±713

mality conditions turn out to be independent of manu-facturing times in the ®rst backorder-cost case but not inthe second case. We identify special circumstances for thesecond backorder-cost case where the MTO versus MTSdecision can be made entirely based on the ®rst momentsof the manufacturing-time distributions. We o�er evi-dence that manufacturing-time diversity may or may notfavor a MTO policy, depending on how diversity isconceptualized. Finally, these results are used to provideinsights into the impact of manufacturing-time random-ness reductions e�orts on the MTO versus MTS decision.

A search of the inventory literature reveals that ana-lytical treatments of the MTO versus MTS decision arelimited to Li [10] and Arreola-Risa [2]. Li has looked atthe optimality of MTO/MTS in a multi-®rm marketwhere companies compete for customers based on deliv-ery time of orders. He considered the one-product caseand assumed that unit manufacturing times are expo-nentially distributed. The inventory cost function is apresent value where the interest rate is compoundedcontinuously. Arreola-Risa has considered a production-inventory system similar to the one in this paper, butwhere manufacturing times for the di�erent products areidentical and gamma distributed, and backorder costs areincurred $ per unit per unit time. He determined the valueof the squared coe�cient of variation of manufacturingtime at which the ®rm is indi�erent to produce a productMTO or MTS.

The contents of the paper are organized as follows.Section 2 contains a derivation of the cost model. InSections 3±5 we establish MTO/MTS optimality condi-tions and also explore several factors that may a�ect theMTO versus MTS decision. Section 3 deals with $ perunit backorder costs and Sections 4 and 5 deal with $ perunit per unit time backorder costs. In turn, Section 4considers homogeneous manufacturing times while Sec-tion 5 looks at the more general case of heterogeneousmanufacturing times. Conclusions and directions forfurther research are summarized in Section 6.

2. The cost model

In this section we develop the cost model and note someof its basic properties to be used for optimization pur-poses. Let:

E�� = expected value operator;fA�� = probability function of random variable

A;FA�� = distribution function of random variable

A;n = number of di�erent products;

Di = demand per unit time for product i;ki � E�Di� = average demand per unit time for

product i;

k �Pni�1 ki = average arrival rate of orders at the

manufacturing facility;di � ki=k = fraction of orders at the manufacturing

facility coming from product i;qi � kiE�Mi� = load o�ered by product i;q �Pn

i�1 qi = capacity utilization of the manufacturingprocess;

Mi = unit manufacturing time for product i;hi = unit inventory holding cost rate;

OHi = on-hand inventory for product i;K = average inventory holding and backor-

dering cost per unit time;Ri = base-stock level for item i;R�i = the value (or values) of Ri; i � 1; 2; . . . ; n,

which minimize K.

Inventory backordering costs may be incurred in avariety of ways. For a discussion of backorder costingthe reader may consult, for example, Silver and Peterson[11]. In this paper we will consider the following twotypes of backorder costs: $ per unit, and $ per unit perunit time. If inventory backordering costs are incurred $per unit, for each product i, let pi denote the unitbackordering cost and BRi denote the backorders rate(number of backorders per unit time).Then from basicprinciples:

K�p��R1;R2; . . . ;Rn� �Xn

i�1hiE�OHi� � piE�BRi�f g;

where K�p�� is the average cost per unit time under $ perunit backorder costs.If inventory backordering costs are incurred $ per unit

per unit time, for each product i, let pi denote the unitbackordering cost rate and BLi denote the backorderlevel. Again from basic principles:

K�p��R1;R2; . . . ;Rn� �Xn

i�1hiE�OHi� � piE�BLi�f g;

where K�p�� is the average cost per unit time under $ perunit per unit time backorder costs.Under a FCFS scheduling rule, the quantities E�OHi�,

E�BRi�, and E�BLi� are independent of Rj, for j 6� i. Thisindependence also holds for any scheduling rule that isindependent of the base-stock levels. (The ``lowest-in-ventory-level-®rst'' scheduling rule for homogeneousproducts with equal base-stock levels [4,8,9] is an exam-ple of such a rule.) Thus we can minimize K�p�� orK�p�� by minimizing K�p�i �Ri� or K�p�i �Ri�, respectively,for each i, where:

K�p�i �Ri� � hiE�OHi� � piE�BRi�;K�p�i �Ri� � hiE�OHi� � piE�BLi�:

Let OOi denote the number of orders outstanding forproduct i. It is not di�cult to show that E�OHi� �

706 Arreola-Risa and DeCroix

PRix�0 xfOOi�Ri ÿ x�; E�BRi� � ki

P1x�0 fOOi �Ri � x�; and

E�BLi� �P1

x�0 xfOOi�Ri � x�: Hence:

K�p�i �Ri� � hi

XRi

x�0xfOOi�Ri ÿ x� � piki

X1x�0

fOOi�Ri � x�:

�1�

K�p�i �Ri� � hi

XRi

x�0xfOOi�Ri ÿ x� � pi

X1x�0

xfOOi�Ri � x�: �2�

3. MTO versus MTS for K�p�i �Ri�

The next proposition establishes the optimality of MTO/MTS policies when backorder costs are incurred on a $per unit basis. Note that this result holds not just forFCFS, but also for any scheduling rule that is indepen-dent of the base-stock levels. All proofs are included inthe Appendix.

Proposition 1. It is optimal to produce product i MTO ifand only if:

hi

pi� ki; �3�

otherwise, it is optimal to produce product i MTS.

Proposition 1 shows that the factors that play a role inthe optimality of MTO/MTS for product i are the ratio ofits economic parameters �hi=pi� and its average demandper unit time (ki). Surprisingly, the optimality of MTO/MTS is independent of unit manufacturing times (Mi's),and hence of o�ered loads (qi's) and the capacity utili-zation of the manufacturing process (q). Before closing,notice that inspection of (3) indicates that small values ofki or large values of hi=pi would favor a MTO policy forproduct i, and large values of ki or small values of hi=piwould favor a MTS policy.

4. MTO versus MTS for K�p�i �Ri�: homogeneous Mi's

When backorder costs are incurred on a $ per unit per unittime basis, we will demonstrate that the optimal decisionbetween MTO and MTS is fundamentally di�erent andmore complicated than when backorder costs are incurredon a $ per unit basis. In particular, the MTO/MTS opti-mality conditions for each product now depend on theunit manufacturing times of all products in the system.

In this section we will consider homogeneous manu-facturing times ± i.e., manufacturing times for the variousproducts are identically distributed, where Mi � M forall i. Homogeneous manufacturing times may arise in asystem where the manufacturing process is similar for allproducts. Note, however, that the assumption of homo-

genenous manufacturing times does not imply that allproduct parameters are the same across products. Dif-ferences in price, component quality, con®guration, etc.,may result in di�erent values of ki, pi, and hi even thoughmanufacturing-time distributions are the same.Consideration of homogeneous manufacturing times

will allow us not only to pave the way for heterogenousmanufacturing times, but also to isolate the impact on theMTO versus MTS decision of e�orts to reduce manu-facturing-time randomness. More general systems con-sisting of products with di�erent manufacturing-timedistributions will be considered in Section 5.

4.1. MTO/MTS optimality conditions

The next theorem establishes the optimality of MTO/MTS policies when backorder costs are incurred on a $per unit per unit time basis. Let r�p�i � pi=�hi � pi�. (Notethat r�p�i is the well-known newsperson fractile.)

Theorem 2. It is optimal to produce product i MTO if andonly if:

E�eÿkiM � � �qÿ qi�r�p�i

qr�p�i ÿ �1ÿ q�qi

; �4�


Theorem 2 reveals that the factors that play a role inthe optimality of MTO/MTS for product i are its averagedemand per unit time (ki), its o�ered load (qi), its eco-nomic ratio (r�p�i ), the unit manufacturing time (M), andthe capacity utilization of the manufacturing process (q).Contrasting Theorem 2 with Proposition 1 we note thatmore factors in¯uence the MTO versus MTS decisionwhen backorder costs are $ per unit per unit time ratherthan $ per unit. Notably, these factors not only include®rst-moment information of M (via qi and q) but also theprobability distribution of M (via E�eÿkiM �). Applyingcalculus to (4) veri®es that small values of q, qi, orr�p�i would favor a MTO policy for product i, while largevalues of these parameters would favor a MTS policy.Since the right-hand side of the inequality in (4) is a

combination of known and basic parameters, applicationof Theorem 2 depends on being able to evaluate E�eÿkiM �.The process of ®nding E�eÿkiM � can be simpli®ed by ob-serving that if M is modeled as a nonnegative randomvariable (e.g., exponential) or by an unrestricted randomvariable (e.g., normal), E�eÿkiM � is equivalent to theLaplace±Stieltjes transform of M evaluated at ki or themoment generating function of M evaluated at ÿki, re-spectively. Closed-form expressions for these transfomsand generating functions are available for commondistributions.If the behavior of M cannot be captured by a common

distribution, assuming that M is a ``reasonable'' random

Production±inventory system 707

variable (meaning that it has a rational Laplace±Stieltjestransform) FM�� can always be expressed as a Coxiandistribution [12]. Cox's method of stages representation isthe most general way of constructing a nonnegativerandom variable from independent exponential stages.Let k denote the number of stages, fj the parameter of thejth stage, pjÿ1 the probability of entering the jth stage forservice, and qj � 1ÿ pj the probability of terminatingservice after the jth stage. It can be demonstrated thatwhen FM �� is modeled as a Coxian distribution the left-hand side of (4) becomes:

E�eÿkiM � � q0 �Xk

j�1p0 � � � pjÿ1qj

Yj

m�1

fm

ki � fm:

4.2. Manufacturing-time randomness and the MTOversus MTS decision

In addition to providing practical conditions for identi-fying whether a product should be produced MTO orMTS, the preceding results provide a foundation forfurther insights into questions currently of interest tomanufacturers. For example, based on the work of Hall[13] and Schonberger [14], many manufacturers are fo-cusing on reducing randomness in their manufacturingprocesses in order to move towards MTO. Proposition 1clearly implies that if backorders incur costs on a $ perunit basis, while randomness-reduction e�orts may helpreduce optimal inventory levels, the decision for a prod-uct to be fully MTO is not dependent on manufacturing-time randomness.

If backorder costs are $ per unit per unit time, it fol-lows immediately from Theorem 2 that manufacturing-time randomness may have an impact on the MTO/MTSdecision. In fact, the condition in (4) can be used to es-tablish the general impact of manufacturing randomnessreductions on this decision. For this purpose, considertwo manufacturing-time random variables M and M 0.Suppose that M 0 is a manufacturing time with reducedrandomness, which we will represent by saying that M isstochastically more variable than M 0 ± i.e., E�g�M�� E�g�M 0�� for all increasing, convex functions g��. (Thispoint is discussed by Ross [15, p. 270].) Since M and M 0are nonnegative and E�M � � E�M 0�, it follows thatE�h�M�� E�h�M 0�� for all convex functions h�� (see Ross[15], Corollary 8.5.2), so in particular:

E�eÿkiM � � E�eÿkiM 0 �:As a result, Theorem 2 implies that, when the backordercosts are $ per unit per unit time, a reduction in manu-facturing-time randomness increases the likelihood thateach product should be MTO.

In addition to yielding this general result, Theorem 2also provides a means of testing the bene®ts of any spe-

ci®c reduction in manufacturing-time randomness. Forexample, if manufacturing time follows a normal distri-bution with parameters �h; r2�, we know that:

E�eÿkiM � � eÿ�kihÿ�k2i r2=2��:

If we ®x h and decrease r2, the right-hand side of (4)remains unchanged, so we can determine how much of areduction in the randomness of M is necessary for MTOto become optimal.

4.3. Extensions and other managerial implicationsof the MTO/MTS optimality conditions

Although Theorem 2 establishes simple and powerfulconditions for deciding whether a product should beMTO or MTS, using these conditions requires knowledgeof the entire distribution FM��. There may be caseshowever, where complete information on FM�� is notavailable ± e.g., in the design or reengineering of a pro-duction±inventory system. Consequently, in such cases itwould be useful to have alternative conditions implyingthe optimality of MTO or MTS that require only minimalinformation about the manufacturing distribution,namely its expected value E�M �. Corollaries 3 and 4provide such conditions.

Corollary 3. If:

q � 1ÿ r�p�i �5�then it is optimal to produce product i MTO.

Corollary 4. If either of the following conditions holds, thenit is optimal to produce product i MTS:

qi � 1ÿ r�p�i ; �6�

q >qi�eqi r�p�i ÿ 1�

eqi r�p�i ÿ r�p�i ÿ qi

: �7�

In addition to being useful as simple screens for theoptimality of MTO or MTS, Corollaries 3 and 4 provideinsight into the capacity and economic trade o�s thatdetermine the optimal inventory strategy for each prod-uct. Corollary 3 states that if the capacity utilization ofthe manufacturing process q is su�ciently low, thenproduct i should be MTO. Exactly how low the utiliza-tion must be is determined by the economic ratio r�p�i foreach individual product ± the smaller the economic ratio,the more likely the product should be MTO. These in-sights agree with our intuition: If the capacity utilizationof the manufacturing process or the ratio of backordercosts to holding costs is high, we are more likely to wantsome inventory as a bu�er.Corollary 4 states that if the load qi that product i

alone places on the system is su�ciently high, then that


product should be MTS. Again, the critical threshold forqi is determined by the economic ratio for that product ±the larger the ratio, the more likely the product should beMTS. Corollary 4 also provides a second conditionleading to the optimality of MTS: Even when the load qidue to product i is not su�ciently high to imply MTS, ifthe capacity utilization of the manufacturing process q issu�ciently high, then we can still conclude that product ishould be MTS.

By considering the proofs of the above corollaries, onenotices that the conditions in (5) and (6) imply the opti-mality of MTO and MTS, respectively, for systems usingany scheduling rule that is independent of the base-stocklevels. As a result, these conditions provide assistance inmaking the MTO/MTS decision not only in the absenceof complete information on FM��, but also in systemswith more general scheduling rules.

To estimate the likelihood that the conditions in Cor-ollaries 3 and 4 are su�cient for making the MTO versusMTS decision, we generated 540 cases by combining thefollowing parameter values:

qi=q � 0:90; 0:75; 0:60; 0:50; 0:40; 0:30;

0:20; 0:10; 0:05; 0:01:

pi=hi � 100; 50; 25; 10; 5; 2:

q � 0:9; 0:8; 0:7; 0:6; 0:5; 0:4; 0:3; 0:2; 0:1:

For example, the combination (qi=q � 0:10, pi=hi � 5,q � 0:7) is equivalent to a product whose o�ered loadrepresents 10% of the capacity utilization of the manu-facturing process, whose unit backordering cost rate is®ve times its unit inventory holding cost rate, and wherethe product is being produced in a manufacturing processwith a capacity utilization of 70%. We think that these540 cases represent products under a wide variety ofeconomic and production characterisitics, and thusshould be representative of most situations found inpractice.

By applying Corollaries 3 and 4 to the 540 cases, wedetermined that in 76.3% of the cases, ®rst-moment in-formation of M would su�ce to make the MTO versusMTS decision. This suggests that it should be more likelythan not to ®nd cases in practice where the importance ofhigher moments of M is small relative to E�M � when itcomes to deciding between MTO and MTS. By solelyapplying (5) and (6) to the 540 cases, we determined thatknowledge of E�M �was su�cient to make theMTO versusMTS decision in 69.7% of the cases even if the schedulingrule was of the more general type indicated above.

One ®nal use of the results in Corollaries 3 and 4 is tomitigate the problem of measuring the stockout-cost ratepi. Hadley and Whitin [16, p. 420] have written: ``. . .stockout costs are often the most di�cult to determine.They cannot normally be measured directly, since theyusually include such intangibles as good will losses.

Consequently, the usual procedure, if stockout costs arespeci®ed at all, is for someone to make a guess as to whatthey are.'' For example, for qi=q � 0:1 and q � 0:8, i.e., aproduct i produced in a manufacturing process with acapacity utilization of 80% and o�ering a load of 8%,using Corollaries 3 and 4 we conclude that regardless ofFM��, product i should be produced MTO when pi=hi �0:25 and should be produced MTS when pi=hi � 3:97. Itis only when the decision maker has reason to believe that0:25 < pi=hi < 3:97, that complete information on FM��would be needed to decide if it is optimal to produceproduct i MTO or MTS.

5. MTO versus MTS for K�p�i �Ri�: heterogeneous Mi's

The prior section examined production±inventory sys-tems with homogeneous manufacuring times. Some sys-tems, however, exhibit heterogeneous manufacturingtimes ± i.e., manufacturing times for di�erent productsfollow di�erent distributions. In this section we extendmost of the results in Section 4 to such systems. We alsoexplore the impact of manufacturing-time diversity acrossproducts on the MTO versus MTS decision.

5.1. MTO/MTS optimality conditions

When manufacturing times are heterogeneous, the timerequired for the manufacturing process to make one unitof product i is the random variable Mi. Recalling that direpresents the fraction of overall demand coming fromproduct i, the composite manufacturing time at the pro-duction facility, denoted by bM , is a mixture of the indi-vidual manufacturing times, i.e.:

FbM �Xn

i�1diFMi :

The following theorem uses bM to provide optimalityconditions for the MTO versus MTS decision whenmanufacturing times are heterogeneous.

Theorem 5. It is optimal to produce product i MTO if andonly if:

E�eÿkibM � �Xn

j�1djE�eÿkiMj � � ��qi=di� ÿ qi�r�p�i

�qi=di�r�p�i ÿ �1ÿ q�qi

; �8�


Theorem 5 makes it clear that the factors in¯uencingthe MTO versus MTS decision given homogeneousmanufacturing times also in¯uence the decision whenmanufacturing times are heterogeneous. But in contrastto Theorem 2, the optimality conditions in (8) includeinformation about each manufacturing time (Mj) as well


as the fraction of orders coming from each product j (dj).However, the optimality conditions for heterogeneousmanufacturing times are not signi®cantly more di�cult toevaluate than their homogeneous counterparts. The workinvolved in evaluating the left-hand side of (8) is justcomputing a weighted sum of Laplace±Stieltjes trans-forms and/or moment generating functions, evaluated atki or at ÿki, as appropriate.

When the homogeneous FM �� is unavailable, in Section4 we developed several results that would allow managersto make the MTO versus MTS decision based solely onE�M �. For those situations in practice where manufac-turing times are heterogeneous and FbM�� is unknown, it isagain desirable and perhaps now more important to ob-tain conditions under which the MTO versus MTS deci-sion can be made based only on ®rst-moment informationabout manufacturing times. This is so because in theheterogeneous manufacturing times case, determiningFbM�� requires complete information of FMi��,i � 1; . . . ; n. Fortunately, with minor modi®cations, mostof the results of this type from Section 4 can be extendedto apply here as well. For example, it is easy to see thatCorollary 3 also holds for heterogeneous manufacturingtimes, and letting:

ci � kiE� bM �;it can be demonstrated that Corollary 4 becomes Corol-lary 6 for heterogeneous manufacturing times.

Corollary 6. If either of the following conditions holds, thenit is optimal to produce product i MTS:

qi � 1ÿ r�p�i ; �9�

q >qi�eci r�p�i ÿ 1�

eci r�p�i ÿ r�p�i ÿ qi

: �10�

A moment's re¯ection reveals that Corollary 3 and (9)not only are applicable in the absence of complete in-formation on FMi��, i � 1; . . . ; n (just the average manu-facturing times are needed) but also in systems using anyscheduling rule that is independent of the base-stocklevels.

5.2. Manufacturing-time diversity and the MTO versusMTS decision

The results of the preceding sections can be used, as inSection 4, to address the impact on the MTO/MTSdecision of variability in the unit manufacturing timefor an individual product. When manufacturing timesare heterogeneous, however, there is an additionalsource of variation in the system ± the diversity ofmanufacturing times across products. Even if manu-facturing times for individual products are determinis-tic, the composite manufacturing time of the system

will exhibit variation if di�erent products take di�erentamounts of time to produce. The impact of this sourceof variation on the MTO versus MTS decision is thefocus of this section.There is a vast number of ways that manufacturing-

time diversity could be interpreted. For example, it couldmean that manufacturing times for di�erent productshave di�erent means, di�erent variances, di�erent coef-®cients of variation, or di�erent distributional forms. Asa result, achieving general statements about the e�ect ofmanufacturing-time diversity on the MTO versus MTSdecision seems unlikely. Instead, we explore di�erentpossible e�ects by examining three manufacturing-timediversity scenarios. Since manufacturing times only a�ectthe left-hand side of (8), for each scenario below we an-alyze the e�ects of manufacturing-time diversity bycomparing the left-hand side of (8) for the system withmanufacturing-time diversity to the correspondingquantity under an ``equivalent'' system that has had themanufacturing-time diversity removed.

Scenario 1. Deterministic manufacturing times, di�erentmeansSuppose the time to manufacture product i is a deter-ministic value 1=li. Then the left-hand side of (8) forproduct i in this system with manufacturing-time diver-sity is:


j�1djE�eÿkiMj � �

Xn

j�1djeÿki=lj : �11�

If we strip out manufacturing-time diversity but preservethe other key attributes of the system (most notably thecapacity utilization of the manufacturing process q), weobtain a system where all products have a deterministicmanufacturing time 1=l �Pn

i�1 di=li. For this systemwithout manufacturing-time diversity, the left-hand sideof (8) is

E�eÿkiM̂ � � eÿki=l �Xn

j�1djeÿki=lj ; �12�

where the inequality follows from Jensen's inequality.The implication of (12) is that, for this scenario, diversityamong manufacturing times makes MTO less likely to beoptimal for every product in the system.

Scenario 2. Gamma manufacturing times, di�erentmeans, same scv'sSuppose the time to manufacture product i is gammadistributed with parameters ai and b, where the parame-ter b is shared across products so that all products havethe same squared coe�cient of variation. The left-handside of (8) for product i in this system with manufactur-ing-time diversity is:




Xn

j�1dj 1� ki=ajÿ �ÿb

: �13�

If we strip out manufacturing-time diversity, and replaceeach manufacturing time with a common gamma distri-bution with the same squared coe�cient of variation 1=band the same system utilization q, we obtain a systemwhere all products have a gamma-distributed manufac-turing time with parameters a � k=

Pnj�1�kj=aj� and b.

For this system without manufacturing-time diversity, theleft-hand side of (8) is:

1� di

Xn

j�1�kj=aj�

!ÿb

: �14�

De®ne a discrete random variable J that takes on thevalue 1� ki=aj with probability dj and a functiong�z� � zÿb. Since g�� is convex, Jensen's inequalityimplies that

E�g�J�� g�E�J ��: �15�By noting that the left-hand side of (15) is equal to thequantity in (13) and the right-hand side of (15) is equal tothe value in (14), we again observe that, in this scenario,diversity among manufacturing times makes MTO lesslikely to be optimal for every product in the system.

Scenario 3. Gamma manufacturing times, identicalmeans, di�erent variancesSuppose the time to manufacture product i is gammadistributed with parameters ai and bi, where the productsshare a common mean manufacturing time bi=ai � 1=l.Then the left-handside of (8) for product i in this systemwith manufacturing-time diversity is:



Xn

j�1dj 1� qi=bj

ÿ �ÿbj : �16�

We now strip out manufacturing-time diversity, and re-place each manufacturing time with a common gammadistribution with parameters a and b such that the newdistribution preserves the common mean and has avariance equal to the weighted average of the variances ofthe individual manufacturing times, i.e.:

b=a2 �Xn

j�1dj�bj=a

2j �:

For this system without manufacturing-time diversity, theleft-hand side of (8) is:

1� qi

Xn

j�1�dj=bj�

!ÿ1=Pn

j�1�dj=bj�: �17�

Now de®ne a discrete random variable J that takes on thevalue 1=bj with probability dj and a function g�z� ��1� qiz�ÿ1=z. Since g�� is concave, Jensen's inequalityimplies that:

E�g�J�� g�E�J ��: �18�By noting that the left-hand side of (18) is equal to thequantity in (16) and the right-hand side of (18) is equalto the value in (17), we observe that, for this scenario,diversity among manufacturing times makes MTOmore likely to be optimal for every product in the system.The preceding three scenarios illustrate that diversity of

manufacturing times among products can certainly havean impact on the MTO versus MTS decision. However,although Scenarios 1 and 2 seem to reinforce the naturalintuition that more manufacturing-time diversity wouldtend to make MTO less attractive, Scenario 3 provides acounter-example to this notion. Clearly the relationshipbetween manufacturing-time diversity and the MTOversus MTS decision is a complex one. Although thework presented here provides tools and a framework forstudying manufacturing-time diversity, further analysis ofthis issue is beyond the scope of the paper.

6. Conclusions

We studied the optimality of MTO/MTS policies for amulti-product company with generally distributed man-ufacturing times. Optimality conditions were derived forthe backorder-cost cases of $ per unit and $ per unit perunit time. While in the ®rst backorder-cost case theoptimality conditions are independent of manufacturingtimes, in the second case they are not. When the dis-tributions of manufacturing times play a role in theoptimality of MTO/MTS, we identi®ed circumstanceswhere the MTO versus MTS decision can be made en-tirely on ®rst-moment information. Our results alsoprovide a framework to assess the impact of severalmanagerial considerations on the MTO versus MTSdecision. We discussed to what extent reducing manu-facturing-time randomness helps a product to be pro-duced MTO. We also discovered that depending on howdiversity is de®ned, manufacturing-time diversity acrossproducts may or may not be conducive to MTO pro-duction.Given the few analytical treatments of the MTO versus

MTS decision, further research on this issue may bepursued along several fronts. First, a generalization of thePoisson process to a renewal process as a model of de-mands would be worthwhile to pursue because of itspractical relevance. A second research direction would beto analyze a manufacturing facility with multiple pro-duction stages. Besides increasing the scope of the results,and perhaps even adding more realism, it would be in-teresting to see what role the multiple stages play in theMTO/MTS optimality conditions. Finally, simulationstudies may be conducted to formally study the e�ects onthe MTO versus MTS decision of various dynamicscheduling rules.


Acknowledgements

We want to thank Professor Yigal Gerchak, InventoryDepartment Editor, and the anonymous referees for theircomments which led to a much improved paper.

References

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[2] Arreola-Risa, A. (1996) On the existence of a manufacturingrandomness threshold. Working Paper, Department of Informa-tion & Operations Management, Lowry Mays College & Gradu-ate School of Business, Texas A&M University.

[3] Federgruen, A. and Katalan, Z. (1995) Make-to-stock or make-to-order: that is the question ± novel answers to an ancient debate.Working Paper, Department of Operations and InformationManagement, The Wharton School, University of Pennsylvania.

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[7] Wein, L.M. (1992) Dynamic scheduling of a multiclass make-to-stock queue. Operations Research, 40(4), 724±735.

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[9] Zipkin, P. (1995) Performance analysis of a multi-item produc-tion-inventory system under alternative policies. ManagementScience, 41(4), 690±703.

[10] Li, L. (1992) The role of inventory in delivery-time competition.Management Science, 38(2), 182±197.

[11] Silver, E.A. and Peterson, R. (1985) Decision Systems for Inven-tory Management and Production Planning, 2nd edn. John Wileyand Sons, New York, NY.

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Appendix

Proof of Proposition 1

Let DKi�Ri� � Ki�Ri � 1� ÿ Ki�Ri� and D2Ki�Ri� �DKi�Ri � 1� ÿ DKi�Ri�. From Equation (1), we have:

DK�p�i �Ri� � hiFOOi�Ri� ÿ pikifOOi�Ri�;

D2K�p�i �Ri� � �hi ÿ piki�fOOi�Ri � 1� � pikifOOi�Ri�:

Therefore, if DK�p�i �0� � �hi ÿ piki�fOOi�0� � 0, a base-stock level of Ri � 0 is preferred to Ri � 1. But this con-dition holds i� hi ÿ piki � 0, in which case D2K�p�i �Ri� � 0implying that Ri � 0 is preferred to any positive base-stock level. Conversely, DK�p�i �0� < 0 clearly impliesR�i � 1. Rearranging hi ÿ piki � 0 yields (3). j

Proof of Theorem 2

Recalling that DKi�Ri� � Ki�Ri � 1� ÿ Ki�Ri� andD2Ki�Ri� � DKi�Ri � 1� ÿ DKi�Ri�, we have from Equa-tion (2):

DK�p�i �Ri� � hiFOOi�Ri� ÿ pi�1ÿ FOOi�Ri��;D2K�p�i �Ri� � �hi � pi�fOOi�Ri � 1� � 0;

which implies that R�i � 0 i� DK�p�i �0� � 0, or equiva-lently, fOOi�0� � r�p�i . Let WA be the probability generatingfunction of random variable A. Note thatfOOi�0� � WOOi�0�. Since the production facility can berepresented as an M/G/1 queue, the probability gener-ating function for the outstanding orders of all products,OO �Pn

i�1 OOi, is:

WOO�z� � �1ÿ q��1ÿ z�A�z�A�z� ÿ z

; �A1�

where

A�z� �X1j�0

Z10

eÿkt�ktz�jj!

dFM�t�: �A2�

The combination of FCFS and Poisson arrivals implies[17, p. 287]

WOOi�z� � WOO��1ÿ di� � diz�:Using (A1) in this expression and fOOi�0� � WOOi�0�yields:

fOOi�0� ��1ÿ q�qi

q 1ÿ A�1ÿ di�� ÿ1n o

� qi A�1ÿ di�� ÿ1: �A3�

From (A2),

A�1ÿ di� �X1j�0

Z10

eÿkt�kt�1ÿ di��jj!

dFM�t�

�Z10

eÿktX1j�0

��kÿ ki�t�jj!

dFM�t�;

�Z10

eÿkte�kÿki�tdFM �t� � E�eÿkiM �; �A4�


where the interchange of the in®nite sum and integralis justi®ed by the monotone convergence theorem [18,p. 87]. Combining (A3), (A4) and algebra produces theresult. j

Proof of Corollary 3

Since fOOi�0� � 1ÿ q, if 1ÿ q � r�p�i then fOOi�0� � r�p�iwhich implies that R�i � 0. j

Proof of Corollary 4

Since 1ÿ qi � fOOi�0�, if 1ÿ qi � r�p�i , or equivalently,qi �1ÿ r�p�i , then R�i � 1 by the argument in the proof ofTheorem 2.

A straightforward application of Theorem 2 impliesthat if:

r�p�i qÿ �1ÿ q�qi

r�p�i �qÿ qi�> eqi �A5�

then R�i � 1. Equation (A5) can be rearranged as:

qiAi > qBi; �A6�where Ai � eqi r�p�i ÿ 1 and Bi � eqi r�p�i ÿ r�p�i ÿ qi. Clearly,if Ai < 0, Bi < 0, and q > qi�Ai=Bi�, then (A6) is satis®ed.

When r�p�i � qi < 1, we will prove that the condition in(7) implies that Ai < 0, Bi < 0, and q > qi�Ai=Bi�. Observethat when r�p�i � qi < 1, Ai < 0 and Bi < 0 become:

r�p�i �eqi ÿ 1� < qi:

Also, since if r�p�i � qi < 1 then:

r�p�i �eqi ÿ 1� < �1ÿ qi��eqi ÿ 1�;� �1ÿ qi� 1� qi � q2

i =2� q3i =6� � � � ÿ 1

� �;

� qi � q2i =2� q3

i =6� � � �� ÿ q2

i � q3i =2� q4

i =6� � � ��

;

� qi � q2i �1=2ÿ 1� � q3

i �1=6ÿ 1=2� � � � �� qk

i �1=k!ÿ 1=�k ÿ 1�!� � � � � ;< qi;

we have that r�p�i �eqi ÿ 1� < qi always holds, i.e., Ai < 0and Bi < 0. Condition (7) is simply the requirement thatq > qi�Ai=Bi�. j

Proof of Theorem 5

Using arguments similar to those in the proof of Theorem2, we arrive at:

fOOi�0� ��1ÿ q�ki

k 1ÿ 1=E�eÿkibM �� ki=E�eÿkibM � :

Observing that k � ki=di and:


j�1djE�eÿkiMj �;

plus algebraic manipulations, yields the desired result.j

Biographies

Dr. Tony Arreola-Risa is an Assistant Professor of Operations Man-agement in the Lowry Mays College & Graduate School of Business atTexas A&M University. He received his B.S. in Industrial and SystemsEngineering from the Monterrey Institute of Technology (ITESM) inMeÂ xico, his M.S. in Industrial Engineering from the Georgia Instituteof Technology, and his M.S. and Ph.D. in Operations Managementfrom Stanford University. His research interests include stochasticmodels of production and inventory systems as well as of service op-erations. Dr. Arreola-Risa is a member of DSI, IIE, and INFORMS.His research has appeared in European Journal of Operational Re-search, Management Science and Naval Research Logistics, and willappear in Decision Sciences.

Dr. Gregory A. DeCroix is a visiting Associate Professor of OperationsManagement in the Fugua School of Business at Duke University. Hereceived his B.S. in Mathematics and Statistics from Miami Universityand his Ph.D. in Operations Research from Stanford University. Hisresearch interests include inventory systems and environmental issuesin operations management. Dr. DeCroix is a member of INFORMS,and his research has appeared in European Journal of OperationalResearch, and Management Science, and will appear in Naval ResearchLogistics.

Contributed by Inventory and Supply Chain Management Department.


make-to-order versus make-to-stock in a production–inventory system with general production times

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