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Make-to-order versus make-to-stock in aproduction—inventory system with general productiontimesANTONIO ARREOLA-RISA a & GREGORY A. DeCROIX ba Department of Information & Operations Management , Lowry Mays College & GraduateSchool of Business, Texas A&M University , College Station, Texas, 77843-4217, USA E-mail:b Fuqua School of Business, Duke University , Durham, NC, 27708-0120, USA E-mail:Published online: 30 May 2007.

To cite this article: ANTONIO ARREOLA-RISA & GREGORY A. DeCROIX (1998) Make-to-order versus make-to-stock in a production—inventory system with general production times, IIE Transactions, 30:8, 705-713, DOI:10.1080/07408179808966516

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IIE Transactions (1998) 30, 705-713

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

ANTONIO ARREOLA-RISA' and GREGORY A. D ~ C R O I X ~

'~eparlmenl of Informa~ion & Operations Management, Lowry Mays College & Graduate School of Business, Texas A&M University, College Station, Texas 77843-4217, USA E-mail: Tarreola@tamu.edu 2Fuqua School of Buriness, Duke University. Durham, NC 27708-0120, USA E-mail: decroix@mail.duke.edu

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 multiple heterogeneous products at a shared manufacturing facility. Manufacturing times are general i.i.d. random variables, and different products may have different manufacturing-time distributions. Demands for the products are independent Poisson processes with different arrival rates. The costs of managing the production-inventory system are stationary and include inventory holding and backordering costs. Backordering costs may be $ per unit or $ per unit per unit time. We derive optimality conditions for MTO and MTS policies. We also study the impact of several managerial considerations on the MTO versus MTS decision.

1. Introduction

Competitive pressures and continuous improvement efforts have led many firms to review their production- inventory practices. In particular, given the push toward a "zero-inventory" goal, world-class companies have begun a search for conditions that determine when it is optimal to hold a finished goods inventory and when it is not. If a finished goods inventory is held for a product, we say that the product is produced make-to-stockA(MTS); otherwise, we say that the product is produced make- to-order (MTO). Note that our designation of a product as MTO merely indicates stockless production, and does not mean that the product is made to order due to custom specifications.

To address the optimality of MTS and MTO policies, we consider a company that produces multiple products facing random demands. A single manufacturing facility produces the products one at a time, and successive manufacturing times for each product comprise a se- quence of general i.i.d. random variables. Different products may have different manufacturing-time.distri- butions. Demands for the products are independent Poisson processes with different arrival rates. The costs of managing the production-inventory system are station- ary and include inventory holding and backordering costs.

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 shown to be optimal in a similar system under periodic review [I]. Also, the base-stock inventory policy has commonly been used in previous work analyzing other aspects of systems similar to the one considered here [2-91. For these reasons we adopt a base-stock inventory policy for our system. The other element that defines the production- inventory policy is the scheduling rule - i.e., if multiple products are waiting to be manufactured, which product should be produced first? Some recent work [3-91 suggests that dynamic scheduling rules may outperform the first- come first-served (FCFS) rule. However, analysis of the MTOIMTS decision for dynamic scheduling rules be- comes intractable due to complex interactions among products. Furthermore, the additional information and coordination required at the production facility by dy- namic scheduling rules may counter the inventory savings over the FCFS rule. As a result and given that FCFS is common in practice, we will adopt FCFS as the sched- uling rule for our system. We believe that the insights regarding the MTOIMTS decision obtained under the FCFS scheduling rule should be suggestive of the be- havior of systems with more general scheduling rules, and in fact, we will show that this is the case for a number of our results.

The paper makes several contributions. Optimality conditions for MTO and MTS policies are established. These conditions are derived for two backorder-cost cases: $ per unit and $ per unit per unit time. The opti-

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mality conditions turn out to be independent of manu- facturing times in the first backorder-cost case but not in the second case. We identify special circumstances for the second backorder-cost case where the MTO versus MTS decision can be made entirely based on the first moments of the manufacturing-time distributions. We offer evi- dence that manufacturing-time diversity may or may not favor a MTO policy, depending on how diversity is conceptualized. Finally, these results are used to provide insights into the impact of manufacturing-time random- ness reductions efforts on the MTO versus MTS decision.

A search of the inventory literature reveals that ana- lytical treatments of the MTO versus MTS decision are limited to Li [lo] and Arreola-Risa [2]. Li has looked at the optimality of MTOIMTS in a multi-firm market where companies compete for customers based on deliv- ery time of orders. He considered the one-product case and assumed that unit manufacturing times are expo- nentially distributed. The inventory cost function is a present value where the interest rate is compounded continuously. Arreola-Risa has considered a production- inventory system similar to the one in this paper, but where manufacturing times for the different products are identical and gamma distributed, and backorder costs are incurred $ per unit per unit time. He determined the value of 'the squared coefficient of variation of manufacturing time a t which the firm is indifferent to produce a product MTO or MTS.

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

2. The cost niodel

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

E[.] = expected value operator; fA(d) = probability function of random variable

A; FA(') = distribution function of random variable

A; n = number of different products; Di = demand per unit time for product i;

li = E[Di] = average demand per unit time for product i;

I = z=, Ai = average arrival rate of orders at the manufacturing facility;

6; A;/I = fraction of orders a t the manufacturing facility coming from product i;

pi = liEIMi] = load offered by product i; p = C:=, pi = capacity utilization of the manufacturing

process; 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: = the value (or values) of Ri, i = 1,2,. . . , n,

which minimize K.

Inventory backordering costs may be incurred in a variety of ways. For a discussion of backorder costing the reader may consult, for example, Silver and Peterson [ll]. In this paper we will consider the following two types of backorder costs: $ per unit, and $ per unit per unit time. If inventory backordering costs are incurred $ per unit, for each product i, let pi denote the unit, backordering cost and BRi denote the backorders rate (number of backorders per unit time).Then from basic principles:

where @'I(-) is the average cost per unit time under $ per unit backorder costs.

If inventory backordering costs are incurred $ per unit per unit time, for each product i, let ni denote the unit backordering cost rate and BLi denote the backorder level. Again from basic principles:

where K ( ~ ) ( - ) is the average cost per unit time under % per unit per unit time backorder costs.

Under a FCFS scheduling rule, the quantities EIOHi], EIBR;], and E[BL;] are independent of R,, for j # i. This independence also holds for any scheduling rule that is independent of the base-stock levels. (The "lowest-in- ventory-level-first" scheduling rule for homogeneous products with equal base-stock levels [4,8,9] is an exam- ple of such a rule.) Thus we can minimize KO(.) or K(")(-) by minimizing K?(R;) or K,[~)(R~), respectively, for each i, where:

Let OOi denote the number of orders outstanding for product i. It is not difficult to show that EIOHi] =

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3. MTO V ~ M S MTS for e)O(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 for FCFS, but also for any scheduling rule that is indepen- dent of the base-stock levels. All proofs are included in the Appendix.

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

otherwise, it is optimal to produce product i MTS.

Proposition 1 shows that the factors that play a role in the optimality of MTO/MTS for product i are the ratio of its economic parameters (hi/pi) and its average demand per unit time (Ai). Surprisingly, the optimality of MTO/ MTS is independent of unit manufacturing times (Mi's), and hence of offered loads (pi's) and the capacity utili- zation of the manufacturing process (p). Before closing, notice that inspection of (3) indicateslhat small values of Ai or large values of hi/pi would favor a MTO policy for product i, and large values of Ai or small values of hi/pi would favor a MTS policy.

4. MTO versus MTS for * ) ( R ~ ) : homogeneous M:s

When backorder costs are incurred on a $ per unit per unit time basis, we will demonstrate that the optimal decision between MTO and MTS is fundamentally different and more complicated than when backorder costs are incurred on a $ per unit basis. In particular, the MTO/MTS opti- mality conditions for each product now depend on the unit manufacturing times of all products in the system.

In this section we will consider homogeneous manu- facturing times - i.e., manufacturing times for the various products are identically distributed, where Mi = M for all i. Homogeneous manufacturing times may arise in a system where the manufacturing process is similar for all products. Note, howevkr, that the assumption of homo-

genenous manufacturing times does not imply that all product parameters are the same across products. Dif- ferences in price, component quality, configuration, etc., may result in different values of Ri, q, and hi even though manufacturing-time distributions are the same.

Consideration of homogeneous manufacturing times will allow us not only to pave the way for heterogenous manufacturing times, but also to isolate the impact on the MTO versus MTS decision of efforts to reduce manu- facturing-time randomness. More general systems con- sisting of products with different manufacturing-time distributions will be considered in Section 5.

4.1. M TOIMTS 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 y ) ni/(hi-+ xi ) . (Note that ri(') is the well-known newsperson fractile.)

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

otherwise, it is optimal to produce product i MTS.

Theorem 2 reveals that the factors that play a role in the optimality of MTO/MTS for product i are its average demand per unit time (Ai), its offered load (pi), its eco- nomic ratio (r,(")), the unit manufacturing time (M), and the capacity utilization of the manufacturing process (p). Contrasting Theorem 2 with Proposition 1 we note that more factors influence the MTO versus MTS decision when backorder costs are $ per unit per unit time rather than $ per unit. Notably, these factors not only include first-moment information of M (via pi and p) but also the probability distribution of M (via ~ [ e - ~ i ~ ] ) . Applying calculus to (4) verifies that small values of p, pi, or r!') would favor a MTO policy for product i, while large values 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, application of Theorem 2 depends on being able to evaluate ~ l e - ' ~ ~ ] . The process of finding ~ [ e - ~ ; ~ ] can be simplified by ob- serving that if M is modeled as a nonnegative random variable (e.g., exponential) or by an unrestricted random variable (e.g., normal), ~ [ e - " ~ ] is equivalent to the Laplace-Stieltjes transform of M evaluated at Ai or the moment generating function of M evaluated a t -Ai, re- spectively. Closed-form expressions for these transfoms and generating functions are available for common distributions.

If the behavior of M cannot be captured by a common distribution, assuming that M is a "reasonable" random

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variable (meaning that it has a rational Laplace-Stieltjes transform) FM(.) can always be expressed as a Coxian distribution [12]. Cox's method of stages representation is the most general way of constructing a nonnegative random variable from independent exponential stages. Let k denote the number of stages, Cj the parameter of the jth stage, p,-1 the probability of entering the jth stage for service, and qj = 1 - p, the probability of terminating service after the jth stage. It can be demonstrated that when & ( a ) is modeled as a Coxian distribution the left- hand side of (4) becomes:

Manufacturing-time randomness and the MTO versus MTS decision

In addition to providing practical conditions for identi- fying whether a product should be produced MTO or MTS, the' preceding results provide a foundation for further insights into questions currently of interest to manufacturers. For example, based on the work of Hall 1131 and Schonberger [14], many manufacturers are Fo- cusing on reducing randomness in their manufacturing processes in order to move towards MTO. Proposition 1 .clearly implies that if backorders incur costs on a $ per unit basis, while randomness-reduction efforts may help reduce 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/MTS decision. In fact, the condition in (4) can be used to es- tablish the general impact of manufacturing randomness reductions on this decision. For this purpose, consider two manufacturing-time random variables M and M'. Suppose that MI is a manufacturing time with reduced randomness, which we will represent by saying that M is stochastically more variable than M' - i.e., E[g(M)] > E[g(M1)] for all increasing, convex functions g(.). (This 'point is discussed by Ross [15, p. 2701.) Since M and MI are nonnegative and E[M] =EIM1], it follows that E[h(M)] 2 E[h(M1)] for all convex functions h( - ) (see Ross [ I 51, Corollary 8.5.2), so in particular:

As a result, Theorem 2 implies that, when the backorder costs are $ per unit per unit time, a reduction in manu- facturing-time randomness increases the likelihood that each product should be MTO.

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

cific reduction in manufacturing-time randomness. For example, if manufacturing time follows a normal distri- bution with parameters (19, $), we know that:

E[~-W] = e-(M-($d/2)) .

If we fix B and decrease a2, the right-hand side of (4) remains unchanged, so we can determine how much of a reduction in the randomness of M is necessary for MTO to become optimal.

4.3. Extensions and other managerial implications of the MTOI MTS optimality conditions

Although Theorem 2 establishes simple and powerful conditions for deciding whether a product should be MTO or MTS, using these conditions requires knowledge of the entire distribution FM(.). There may be cases however, where complete information on FM(.) is not available - e.g., in the design or reengineering of a pro- duction-inventory system. Consequently, in such cases it would be useful to have alternative conditions implying the optimality of MTO or MTS that require only minimal information about the manufacturing distribution, namely its expected value E[M]. Corollaries 3 and 4 provide such conditions.

Corollary 3. I f : .

then it is optimal to produce product i MTO.

Corollary 4. Ifeither of the following conditions holak, then it is optimal to produce product i MTS:

In addition to being useful as simple screens for the optimality of MTO or MTS, Corollaries 3 and 4 provide insight into the capacity and economic trade offs that determine the optimal inventory strategy for each prod- uct. Corollary 3 states that if the capacity utilization of the manufacturing process' p is sufficiently low, then product i should be MTO. Exactly how low the utiliza- tion must be is determined by the economic ratio r,!") for each 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 utilization of the manufacturing process or the ratio of backorder costs to holding costs is high, we are more likely to want some inventory as a buffer.

Corollary 4 states that if the load pi that product i alone places on the system is sufficiently high, then that

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Production-inven tory system 709

product s'hould be MTS. Again, the critical threshold for pi is determined by the economic ratio for that product - the larger the ratio, the more likely the product should be MTS. Corollary 4 also provides a second condition leading to the optimality of MTS: Even when the load pi due to product i is not sufficiently high to imply MTS, if the capacity utilization of the manufacturing process p is sufficiently high, then we can still conclude that product i should be MTS.

By considering the proofs of the above corollaries, one notices that the conditions in (5) and (6) imply the opti- mality of MTO and MTS, respectively, for systems using any scheduling rule that is independent of the base-stock levels. As a result, these conditions provide assistance in making the MTO/MTS decision not only in the absence of complete information on FM(.), but also in systems with more general scheduling rules.

T o estimate the likelihood that the conditions in Cor- ollaries 3 and 4 are sufficient for making the MTO versus MTS decision, we generated 540 cases by combining the following parameter values:

For example, the combination (pi/p = 0.10, nilhi = 5, p = 0.7) is equivalent to a product whose offered load represents 10% of the capacity utilization of the manu- facturing process, whose unit backordering cost rate is five times its unit inventory holding cost rate, and where the product is being produced in a manufacturing process with a capacity utilization of 70%. We think that these 540 cases represent products under a wide variety of economic and production characterisitics, and thus should be repres'entative of most situations found in practice.

By applying Corollaries 3 and 4 to the 540 cases, we determined that in 76.3% of the cases, first-moment in- formation of M would suffice to make the MTO versus MTS decision. This suggests that it should be more likely than not to find cases in practice where the importance of higher moments of M is small relative to E[M] when it comes to deciding between MTO and MTS. By solely applying (5) and (6) to the 540 cases, we determined that knowledge of E[M] was sufficient to make the MTO versus MTS decision in 69.7% of the cases even if the scheduling rule was of the more general type indicated above.

One final use of the results in Corollaries 3 and 4 is to mitigate the problem of measuring the stockout-cost rate xi. Hadley and Whitin [16, p. 4201 have written: ". . . stockout costs are often the most difficult to determine. They cannot normally be measured directly, since they usually include such intangibles as good will losses.

Consequently, the usual procedure, if stockout costs are specified at all, is for someone to make a guess as to what they are." For example, for pi/p = 0.1 and p = 0.8, i.e., a product i produced in a manufacturing process with a capacity utilization of 80% and offering a load of 8%, using Corollaries 3 and 4 we conclude that regardless of FM(-), product i should be produced MTO when nilhi 5 0.25 and should be produced MTS when nilhi 2 3.97. It is only when the decision maker has reason to believe that 0.25 < nilhi < 3.97, that complete information on FM(.) would be needed to decide if it is optimal to produce product i MTO or MTS.

5. MTO versus MTS for @I(&): heterogeneous M.'s

The prior section examined production-inventory sys- tems with homogeneous manufacuring times. Some sys- tems, however, exhibit heterogeneous manufacturing times - i-e., manufacturing times for different products follow different distributions. In this section we extend most of the results in Section 4 to such systems. We also explore the impact of manufacturing-time diversity across products on the MTO versus MTS decision.

5.1. M TOIMTS optimality conditions

When manufacturing times are heterogeneous, the time required for the manufacturing process to make one unit of product i is the random variable Mi. Recalling that Ji represents the fraction of overall demand coming from product i, the composite man~facturing time a t the pro- duction facility, denoted by M, is a mixture'of the indi- vidual manufacturing times, i.e.:

The following theorem uses % to provide optimality conditions for the MTO' versus MTS decision when manufacturing times are heterogeneous.

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

otherwise, it is optimal to produce product i MTS.

Theorem 5 makes it clear that the factors influencing the MTO versus MTS decision given homogeneous manufacturing times also influence the decision when manufacturing times are heterogeneous. But in contrast to Theorem 2, the optimality conditions in (8) include information about each manufacturing time (Mi) as well

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as the fraction of orders coming from each product j (~3~). However, the optimality conditions for heterogeneous manufacturing times are not significantly more difficult to evaluate than their homogeneous counterparts. The work involved in evaluating the left-hand side of (8) is just computing a weighted sum of Laplace-Stieltjes trans- forms and/or moment generating functions, evaluated at

or a t -Ai, as appropriate. When the homogeneous FM(.) is unavailable, in Section

4 we developed several results that would allow managers to make the MTO versus MTS decision based solely on E[M]. For those situations in practice where manufac- turing times are heterogeneous and F - ( a ) is unknown, it is

M again desirable and perhaps now more important to ob- tain conditions under which the MTO versus MTS deci- sion can be made based only on first-moment information about manufacturing times. This is so because in the heterogeneous manufacturing times case, determining F;(.) requires complete information of FMi(.), r = I , . . . , n. Fortunately, with minor modifications, most of the results of this type from Section 4 can be extended to apply here as well. For example, it is easy to see that Corollary 3 also holds for heterogeneous manufacturing times, and letting:

it can be demonstrated that Corollary 4 becomes Corol- lary 6 for heterogeneous manufacturing times.

Corollary 6. Ifeither of the following conditions ho lh , then it is optimal to produce product i MTS:

A moment's reflection 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 any scheduling rule that is independent of the base-stock levels.

5.2. Manufacturing-time diversity and the MTO versus MTS decision

The results of the preceding sections can be used, as in Section 4, to address the impact on the MTO/MTS decision of variability in the unit manufacturing time for an individual product. When manufacturing times are heterogeneous, however, there is an additional source of variation in the system - the diversity of manufacturing 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 different products take different amounts of time to produce. The impact of this source of variation on the MTO versus MTS decision is the focus of this section.

There is a vast number of ways that manufacturing- time diversity could be interpreted. For example, it could mean that manufacturing times for different products have different means, different variances, different coef- ficients of variation, or different distributional forms. As a result, achieving general statements about the effect of manufacturing-time diversity on the MTO versus MTS decision seems unlikely. Instead, we explore different possible effects by examining three manufacturing-time diversity scenarios. Since manufacturing times only affect the left-hand side of (8), for each scenario below we an- alyze the effects of manufacturing-time diversity by comparing the left-hand side of (8) for the system with manufacturing-time diversity to the corresponding quantity under an "equivalent" system that has had the manufacturing-time diversity removed.

Scenario 1. Deterministic manufacturing times, different means Suppose the time to manufacture product i is a deter- ministic value l / p l . Then the left-hand side of (8) for product i in this system with manufacturing-time diver- sity is:

If we strip out manufacturing- time diversity but preserve the other key attributes of the system (most notably the capacity utilization of the manufacturing process p), we obtain a system where all products have a deterministic manufacturing time 1/p = EL, &/pi . For this system without manufacturing-time diversity, the left-hand side of (8) is

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

Scenario 2. Gamma manufacturing times, different. means, same scv's Suppose the time to manufacture product i is gamma distributed with parameters a; and /I, where the parame- ter p is shared across products so that all products have the same squared coefficient of variation. The left-hand side of (8) for product i in this system with manufactur- ing-time diversity is:

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Production-in ventory system 71 1

If we strip out manufacturing-time diversity, and replace each manufacturing time with a common gamma distri- bution with the same squared coefficient of variation 1/Q and the same system utilization p, .we obtain a system where all products have a gamma-distributed manufac- turing time with parameters a = A/ &, (Aj/uj) and j3. For this system without manufacturing-time diversity, the left-hand side of (8) is:

Define a discrete random variable J that takes on the value 1 + L i / q with probability 6, and a function g(z) = z-8. Since g( . ) is convex, Jensen's inequality implies that

E[dJ ) I 1 g(E[Jl). (15) By noting that the left-hand side of (15) is equal to the quantity in (1 3) and the right-hand side of (1 5) is equal to the value in (14), we again observe that, in this scenario, diversity among manufacturing times makes MTO less likely to be optimal for every product in the system.

Scenario 3. Gamma manufacturing times, identical means, different variances Suppose the time to manufacture product i is gamma distributed with parameters ai and Pi, where the products share a common mean manufacturing time fli/ai = l / p . Then the left-handside of (8) for product i in this system with manufacturing-time diversity is:

We now strip out manufacturing-time diversity, and re- place each manufacturing time with a common gamma distribution with parameters a and Q such that the new distribution preserves the common mean and has a variance equal to the weighted average of the variances of the individual manufacturing times, i.e.:

R

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

Now define a discrete random variable J that takes on the value 1 / j . with probability 6, and a function g(z) = (1 + piz)"lz. Since g ( - ) is concave, Jensen's inequality implies that:

By noting that the left-hand side of (18) is equal to the quantity in (16) and the right-hand side of (18) is equal to the value in (17), we observe that, for this scenario, diversity among manufacturing times makes MTO more likely to be optimal for every product in the system.

The preceding three scenarios illustrate that diversity of manufacturing times among products can certainly have an impact on the MTO versus MTS decision. However, although Scenarios I and 2 seem to reinforce the natural intuition that more manufacturing-time diversity would tend to make MTO less attractive, Scenario 3 provides a counter-example to this notion. Clearly the relationship between manufacturing-time diversity and the MTO versus MTS decision is a complex one. Although the work presented here provides tools and a framework for studying manufacturing-time diversity, further analysis of this issue is beyond the scope of the paper.

6. Conclusions

We studied the optimality of MTOIMTS policies for a multi-product company with generally distributed man- ufacturing times. Optimality conditions were derived for the backorder-cost cases of $ per unit and $ per unit per unit time. While in the first backorder-cost case the optimality conditions are independent of manufacturing times, in the second case they are not. When the dis- tributions of manufacturing times play a role in the optimality of MTOIMTS, we identified circumstances where the MTO versus MTS decision can be made en- tirely on first-moment information. Our results also provide a framework to assess the impact of several managerial considerations on the MTO versus MTS decision. We discussed to what extent reducing manu- facturing-time randomness helps ' a product to be pro- duced MTO. We also discovered that depending on how diversity is defined, manufacturing-time diversity across products 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 be pursued along several fronts. First, a generalization of the Poisson process to a renewal process as a model of de- mands would be worthwhile to pursue because of its practical relevance. A second research direction would be to 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 the MTO/MTS optimality conditions. Finally, simulation studies may be conducted to formally study the effects on the MTO versus MTS decision of various dynamic scheduling rules.

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Acknowledgements

We want to thank Professor Yigal Gerchak, Inventory Department Editor, and the anonymous referees for their comments which led to a much improved paper.

References

(I] DeCroix, G.A. and Arreola-Risa, A. (1998) Optimal production and inventory policy for multiple products under resource con- straints. Management Science, 44(7), 950-961.

121 Arreola-Risa, A. (1996) On the existence of a manufacturing randomness threshold. Working Paper. Department of Informa- tion & Operations Management, Lowry Mays College & Gradu- ate School of Business, Texas A&M Unwers~ty.

131 Federgruen, A. and Katalan, Z. (1995) Make-to-stock or make- to-order: that is the question - novel answers to an ancient debate. Workmg Paper, Department of Operat~ons and Information Management. The Wharton School, University of Pennsylvania.

[4] I-la, A. (1992) Optimal scheduling and stock rationing in make-to- stock production systems. Ph D dissertation, Stanford University

(51 Perez, A.P. and Zipkin, P. (1997) Dynamic scheduling rules for a multiproduct make-to-stock queue. Operatlons Research, 45(6). 919-930.

[6] Veatch, M.H. and Wein, L.M. (1996) Scheduling a make-to-stock queue: index policies and hedging points. Operations Research, 44(4), 634-647.

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

[8] Zheng, Y. and Z ~ p k ~ n , P. (1990) A queuelng model to analyze the value of centralized inventory information. Operations Research, 38(2). 296-307.

191 Zipkin, P. (1995) Performance analysis of a multi-~tem produc- tion-inventory system under alternative policies. Management Science, 41 (4), 690-703.

[lo] Li, L. (1992) The role of inventory in delivery-time competition. Management Science, 38(2), 182-1 97.

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

1121 Cox. D.R. (1955) A use of complex probab~lrt~es in the theory of stochastic processes. Proceedmgs o/ rhe Cambridge Phrlosoph~cal Society, 51, 3 13-3 19.

1131 Hall, R. (1983) Zcro Inventories, Dow-Jones Irwin, Homewood, IL.

[I41 Schonberger, R. (1982) Japanese Manufacturing Techniques: Nine Hrdden LEssons m Slmpllcrty. Free Press, New York, NY.

[15] Ross, S.M. (1983) Stochastic Processes. John Wiley and Sons, New York.

[I61 Hadley. G. and Whitin, T.M. (1963) Analysis of Inventory Sys- tems, Prentice-Hall, Englewood Cliffs, NJ.

117) Feller, W. (1968) An Inrroductron to Probabdrty Theory and rts Applications, Vol. 1. Wiley, New York, NY.

[I81 Royden, H.L. (1988) Real Anal-vsis. Macmillan, New York, NY.

Appendix

Proof of Proposition I

Let M i ( & ) = Kt (R, + 1 ) - K,(R,) and A~K, ( R ~ ) = AKi(Ri + 1) - AK,(R,). From Equation (I), we have:

Therefore, if M ~ ) ( o ) = (h, - pJ1) foot (0) 2 0, a base- stock level of Ri = 0 is preferred to Ri = 1. But this con-

2 (PI dition holds iff h, - piRi 2 0, in which case A Ki (R,) >_ 0 implying that Ri = 0 is preferred to any positive base- stock level. Conversely, AK(P'(0) < 0 clearly implies R,* 2 1. Rearranging h, - p,Al 2 0 yields (3).

Proof of Theorem 2

Recalling that AK,(R,) = K,(R, + 1) - Kl(R,) and A~K,(R,) = AK1(Ri + 1) - AK,(R,), we have from Equa- tion (2):

which implies that R: = 0 iff AK,'")(0) 2 0, or equiva- lently, fooi(0) 1 rjn)- Let !PA be the probability generating function of random variable A. Note that fooi(0) = Yoo,(0)- Since the production facility can be represented as an M/G/l queue, the probability gener- ating function for the outstanding orders of all products, 00 r OOi, is:

where

The combination of FCFS and Poisson arrivals implies [17, p. 2871

YOO, (2) = YOO [(I - Ji) + 421. Using (Al) in this expression and fooi(0) = Yoo,(0) yields:

From (A?),

00

[ ( A - li)t]j = / e - " c

j ! ~ F M (4, j=o

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Product ion- inven t o r y s y s t e m

where the interchange of the infinite sum and integral is justified by the monotone convergence theorem [18, p. 871. Combining (A3), (A4) and algebra produces the result. H

Proof of Corollary 3

Since foo,(0) 1 1 - p, if 1 - p 2 rp) then /oo,(O) 1 rp) which implies that R: = 0 .

Proof of Corollary 4

Since 1 - pi 2 foo,(0), if 1 - pi 5 rfX), or equivalently, pi 2 1 - rin), then R; 2 1 by the argument in the proof of Theorem 2.

A straightforward application of Theorem 2 implies that if:

then R,' 2 1. Equation (A5) can be rearranged as:

piAi > pBi, ( A 6 )

where Ai = epi$) - I and Bi = epic) - rp) - pi. Clearly, if Ai < 0, Bi < 0, and p > pi (Ai /Bi ) , then (A6) is satisfied.

'When rjn) + pi < 1 , we will prove that the condition in (7) implies that Ai < 0, Bi < 0 , and p > pi (Ai /Bi ) . Observe that when 6') + pi < 1 , Ai < 0 and Bi < 0 become:

Also, since if ri(") + pi < 1 then:

we have that ri(nl(epi - 1) < pi always holds, ix., Ai < 0 and Bi < 0. Condition (7) is simply the requirement that P > pi(Ai/Bi). w

Proof of Theorem 5

Using arguments similar to those in the proof of Theorem 2, we arrive at:

Observing that L = &/ai and:

plus algebraic manipulations, yields the desired result. w

Biographies

Dr. Tony Arreola-Risa is an Assistant Professor of Operations Man- agement in the Lowry Mays College & Gradua~e School of Business at Texas A&M University. He received his B.S. in Industrial and Systems Engineering from the Monterrey Institute of Technology (ITESM) in Mixico. his MS. in Industrial Engineering from the Georgia Institute of Technology, and his M.S. and Ph.D. in Operations Management from Stanford University. His research interests include stochastic models of production and inventory systems as well as of service op- erations. Dr. Arreola-Risa is a member of DSI, UE, and INFORMS. His research has appeared in European Journal o j Operational Re- search, Management Science and Naval Research Logistics, and will appear in Decision Sciences. .

Dr. Gregory A. DeCroix is a visiting Associate Professor of Operations Management in the Fugua School of Business at Duke University. He received his B.S. in Mathematics and Statistics from Miami University and his Ph.D. in Operations Research from Stanford University. His research interests include inventory systems,and environmental issues in operations management. Dr. DeCroix is a member of INFORMS, and his research has appeared in European Journal o j Operational Research, and Management Science, and will appear in Naval Research Logistics.

Contributed by Inventory and Supply Chain Management Department.

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