optimal production and admission policies in make-to-stock/make-to-order manufacturing systems

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Optimal Production and Admission Policies in Make-to-Stock/Make-to-Order Manufacturing Systems Seyed M. R. Iravani Department of Industrial Engineering and Management Science, Northwestern University, Evanston, Illinois 60208, USA, [email protected] Tieming Liu School of Industrial Engineering and Management, Oklahoma State University, Stillwater, Oklahoma 74078, USA, [email protected] David Simchi-Levi Department of Civil and Environmental Engineering, MIT, Cambridge, Massachusetts 02139, USA, [email protected] I n this article, we study optimal production and admission control policies in manufacturing systems that produce two types of products: one type consists of identical items that are produced to stock, while the other has varying features and is produced to order. The model is motivated by applications from various industries, in particular, the automobile industry, where a part supplier receives orders from both an original equipment manufacturer and the aftermarket. The product for the original equipment manufacturer is produced to stock, it has higher priority, and its demands are fully accepted. The aftermarket product is produced to order, and its demands can be either accepted or rejected. We character- ize the optimal production and admission policies with a partial-linear structure, and using computational analysis, we provide insights into the benefits of the new policies. We also investigate the impact of production capacity, cost structure, and demand structure on system performance. Key words: production and inventory control; admission control; inventory rationing; make-to-stock and make-to-order History: Received: June 2009; Accepted: February 2011, after 1 revision. 1. Introduction Today, improved flexibility in manufacturing systems allows manufacturers to use the same production line to produce different products for different classes of customers. For example, in the automobile industry, a part supplier sells its products both to original equip- ment manufacturers (OEMs) for installation into new vehicles and to repair shops (or the so-called after- market) for replacement in old vehicles (Carr and Duenyas 2000). OEM and aftermarket demands are both important to the part supplier. OEM demands guarantee high utilization of the production capacity, whereas aftermarket demands bring high profit mar- gins to the supplier. Although its margin is lower than the aftermarket product, the OEM product typically has higher priority, because of its high penalty for delayed fulfillment (specified in the contract). After- market items are produced under the make-to-order (MTO) mode due to their large variety. OEM sales are based on long-term contracts, and they are produced under the make-to-stock (MTS) mode. Admission decisions that allow both backorders and lost sales are also important for manufacturing systems with multiple products or multiple classes of customers. When the system is out of stock, it would be profitable to backlog arriving orders if the number of existing backorders is not too high, or reject other- wise. Such admission decisions are consistent with human behavior, as no customers are infinitely patient although they are generally willing to wait (Benjaafar et al. 2010). The admission control also helps firms reduce potential complaints for long wait- ing times, and thus improves customer satisfaction. Although practically important, admission decisions are rarely considered in the literature on inventory problems, mainly due to the high complexity to ana- lyze systems with both backorders and lost sales. All these developments call for models that integrate pro- duction, sequencing, and admission decisions for manufacturing systems that produce under both MTS and MTO modes. There is a significant amount of literature address- ing multi-product production scheduling problems in 224 Vol. 21, No. 2, March–April 2012, pp. 224–235 DOI 10.1111/j.1937-5956.2011.01260.x ISSN 1059-1478|EISSN 1937-5956|12|2102|0224 © 2011 Production and Operations Management Society

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Page 1: Optimal Production and Admission Policies in Make-to-Stock/Make-to-Order Manufacturing Systems

Optimal Production and Admission Policies inMake-to-Stock/Make-to-Order Manufacturing Systems

Seyed M. R. IravaniDepartment of Industrial Engineering and Management Science, Northwestern University, Evanston, Illinois 60208, USA,

[email protected]

Tieming LiuSchool of Industrial Engineering and Management, Oklahoma State University, Stillwater, Oklahoma 74078, USA,

[email protected]

David Simchi-LeviDepartment of Civil and Environmental Engineering, MIT, Cambridge, Massachusetts 02139, USA,

[email protected]

I n this article, we study optimal production and admission control policies in manufacturing systems that produce twotypes of products: one type consists of identical items that are produced to stock, while the other has varying features

and is produced to order. The model is motivated by applications from various industries, in particular, the automobileindustry, where a part supplier receives orders from both an original equipment manufacturer and the aftermarket. Theproduct for the original equipment manufacturer is produced to stock, it has higher priority, and its demands are fullyaccepted. The aftermarket product is produced to order, and its demands can be either accepted or rejected. We character-ize the optimal production and admission policies with a partial-linear structure, and using computational analysis, weprovide insights into the benefits of the new policies. We also investigate the impact of production capacity, cost structure,and demand structure on system performance.

Key words: production and inventory control; admission control; inventory rationing; make-to-stock and make-to-orderHistory: Received: June 2009; Accepted: February 2011, after 1 revision.

1. Introduction

Today, improved flexibility in manufacturing systemsallows manufacturers to use the same production lineto produce different products for different classes ofcustomers. For example, in the automobile industry, apart supplier sells its products both to original equip-ment manufacturers (OEMs) for installation into newvehicles and to repair shops (or the so-called after-market) for replacement in old vehicles (Carr andDuenyas 2000). OEM and aftermarket demands areboth important to the part supplier. OEM demandsguarantee high utilization of the production capacity,whereas aftermarket demands bring high profit mar-gins to the supplier. Although its margin is lower thanthe aftermarket product, the OEM product typicallyhas higher priority, because of its high penalty fordelayed fulfillment (specified in the contract). After-market items are produced under the make-to-order(MTO) mode due to their large variety. OEM sales arebased on long-term contracts, and they are producedunder the make-to-stock (MTS) mode.

Admission decisions that allow both backordersand lost sales are also important for manufacturingsystems with multiple products or multiple classes ofcustomers. When the system is out of stock, it wouldbe profitable to backlog arriving orders if the numberof existing backorders is not too high, or reject other-wise. Such admission decisions are consistent withhuman behavior, as no customers are infinitelypatient although they are generally willing to wait(Benjaafar et al. 2010). The admission control alsohelps firms reduce potential complaints for long wait-ing times, and thus improves customer satisfaction.Although practically important, admission decisionsare rarely considered in the literature on inventoryproblems, mainly due to the high complexity to ana-lyze systems with both backorders and lost sales. Allthese developments call for models that integrate pro-duction, sequencing, and admission decisions formanufacturing systems that produce under both MTSand MTO modes.There is a significant amount of literature address-

ing multi-product production scheduling problems in

224

Vol. 21, No. 2, March–April 2012, pp. 224–235 DOI 10.1111/j.1937-5956.2011.01260.xISSN 1059-1478|EISSN 1937-5956|12|2102|0224 © 2011 Production and Operations Management Society

Page 2: Optimal Production and Admission Policies in Make-to-Stock/Make-to-Order Manufacturing Systems

either a MTS or a MTO manufacturing system. Prob-lems with or without set-ups associated with switch-ing from producing one product to another have bothbeen extensively studied. We refer the reader toGraves (1980), Zheng and Zipkin (1990), Pena-Perezand Zipkin (1997), Wein (1992), Ha (1997c), and deVericourt et al. (2000) for scheduling policies in MTSenvironments, and to Varaiya et al. (1985), Gittins(1989), Duenyas and Van Oyen (1996), Reiman andWein (1998), and Xiao et al. (2010) and the referencestherein for scheduling policies in MTO environments.Single-product models with multi-class customers

in a MTS manufacturing system, the so-called stockrationing problems, have been studied in variouscontexts since the late 1960s. Some of the examplesof stock rationing in uncapacitated systems areTopkis (1968), Nahmias and Demmy (1981), Cohenet al. (1988), and Melchiors et al. (2000). For stockrationing in capacitated systems see Ha (1997a, 1997b,2000), de Vericourt et al. (2002), Huang and Iravani(2008), Iravani et al. (2007), Benjaafar et al. (2010), andXiao et al. (2010). Extensive reviews for the stockrationing problems can be found in Kleijn and Dekker(1998).All the papers above focused on the dynamic pro-

duction and sequencing problems for either MTO sys-tems or MTS systems; however, none of themconsidered a hybrid manufacturing system, andadmission control was rarely studied. Carr and Duen-yas (2000) considered both the production and admis-sion decisions for a MTS/MTO system, where theMTS orders have higher priorities. They found anoptimal policy characterized by monotonic switchingcurves. Gupta and Wang (2007) studied a MTS/MTOproblem in a periodic-review model with stochasticdemands and deterministic production capacities.Leadtime was quoted in their model. Due to the com-plexity and the large state space, they focused on thecase with single-period leadtime, and they developeda three-dimensional lookup table to find the optimaldecisions. For models with longer leadtime, theyacknowledged that “even if one could compute andstore look-up tables, the implementation of the result-ing policy can be problematic.” Dobson and Yano(2001) considered product offering, pricing, and MTOand MTS decisions with linear deterministic demandfunctions in strategic levels, whereas production andadmission controls were not their focus. Youssef et al.(2004) examined the manufacturing scheduling prob-lem in an MTS/MTO system under two static policies:the classical FIFO rule and a priority policy. In the lat-ter, preemptive priority was given to low volume(MTO) products, and approximations were used toobtain analytical and numerical results. They con-sidered the base-stock threshold, but rationing, admis-sion, and contingent outsourcing were not their focus.

In the literature on inventory systems (includingthe above papers, except for Carr and Duenyas 2000,and Gupta and Wang 2007), it is commonly assumedthat unsatisfied demands are either fully backorderedor lost. A relatively small number of papers consid-ered partial backordering, where in case of shortage,arriving customers may be either backordered or lost,each with a certain probability that is an outcomeof the customer behavior (see Moinzadeh 1989,Montgomery et al. 1981, Nahmias and Smith 1994,Smeitink 1990). In a few recent papers, partial back-ordering became part of the seller’s decisions. Theseller determines the amount of unsatisfied demandsto be backordered or rejected (see Chen and Kulkarni2007, Benjaafar et al. 2010, Rabinowitz et al. 1995).However, the admission decision increases the com-plexity of the model, and the current optimal policiesin the literature are characterized by state-dependent,non-linear switching curves. The complexity of thenon-linear structure makes the optimal policies diffi-cult to implement in the real world.In this article, we study policies to coordinate pro-

duction, sequencing, and admission controls for amanufacturing system with a high-priority MTSproduct and a low-priority MTO product. High-prior-ity customers order identical items from the manufac-turer, and these orders cannot be rejected. If theproduct is out of stock, high-priority orders will stillbe accepted and backlogging cost will be incurreduntil they are satisfied. Demands for low-priority,customized products can be rejected if the manufac-turer does not have enough production capacity, oraccepted otherwise. We characterize the structure ofthe optimal production and admission control poli-cies with a partial-linear structure, and we extend ourresults for the case where the manufacturer canoutsource low-priority orders.Our study is related to Carr and Duenyas (2000),

but it is different from it in the following sense.

1. In our article, unsatisfied high-priority demandsare fully backlogged when the high-priority prod-uct is out of stock, whereas in Carr and Duenyas(2000), unsatisfied demands were lost. Our back-logging assumption for the OEM product is rea-sonable as it is often difficult to immediately finda product matching all the required features fromother suppliers.

2. The optimal policies are characterized by a par-tial-linear structure in our article, whereas bynon-linear switching curves in Carr and Duenyas.

3. In addition, we examine the option of contingentoutsourcing in our model, which was not consid-ered in Carr and Duenyas.

The partial-linear structured optimal policy pre-sents a significant improvement over the current

Iravani, Liu, and Simchi-Levi: Optimal Production and Admission PoliciesProduction and Operations Management 21(2), pp. 224–235, © 2011 Production and Operations Management Society 225

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non-linear, switching-curve policies in the literatureon inventory rationing with partial backorders, asthey are easier to implement and less computa-tionally intensive. Furthermore, the partial-linearstructured optimal policy presented in this articleprovides a solid theoretical support for the findingsin Benjaafar et al. (2010), Carr and Duenyas (2000),and Chen and Kulkarni (2007), where linear-struc-tured heuristic policies were only supported bynumerical tests. To prove the partial-linear structure,we develop a unique technique, where the initialthreshold positions and the threshold movementdirections are restricted. We explain this technique indetail later in section 2.2.The article is organized as follows. In section 2,

we study the model with both OEM and aftermar-ket demands, and we extend the model with theoption to outsource the aftermarket orders when thesystem does not have sufficient capacity. In section3, we use computational analysis to obtain insightsinto the benefits of the new policies and the impactsof production capacity, demand structure, and pricestructure on system performance. Finally, we con-clude in section 4.

2. The Model with OEM andAftermarket Products

2.1. Problem FormulationWe consider a manufacturing system with two typesof products over an infinite horizon. Type 1 is forhigh-priority orders and has a standard configuration,so it is produced to stock. Demands for type 1 are sat-isfied if inventory is available, or fully backloggedwith high backlogging cost b1 per item per unit timeotherwise. Backlogging cost serves as a proxy for themanufacturer’s loss of good will, as well as extra ship-ping and handling cost due to the tardiness. Type 2has varying features and is produced to order.Demands for type 2 can be either accepted or rejected.If an order for type 2 is rejected, a rejection penalty, r2,associated with lost sales and loss of goodwill, isincurred. If an order is accepted, we assume that abacklogging cost b2 per item per unit is charged as theorder is accepted, as in the literature on MTO systems(see Carr and Duenyas 2000, Youssef et al. 2004). Weassume b2 < b1, since suppliers usually have long-term contracts with OEMs, and delayed fulfillmentsare heavily penalized. The contribution margin forproduct i is pi, and the inventory holding cost forproduct 1 is h per unit time.We assume that customers for each type of product

arrive according to a Poisson Process, and let ki bethe demand arrival rate of Class i, for i = 1,2. Wealso assume that the production time of each type of

product follows the same exponential distributionwith production rate μ. The same-production-timeassumption is reasonable if the production processconsists of similar operations. For example, the timedifference between making an OEM product (e.g., abattery for 2010 Chevrolet Malibu, General MotorsInc., Detroit, MI) and an aftermarket product (e.g., abattery for 2005 Chevrolet Malibu) is insignificant.We further assume that preemptions are allowed, andno set-up time is needed when the manufacturingsystem switches from one type of job to the other.This assumption is also reasonable for assembly pro-duction systems where set-up times are negligiblecompared to production times. The assumptions ofPoisson arrival demand and exponential productiontimes allow us to formulate the problem and charac-terize the structure of the optimal policy. Carr andDuenyas (2000) studied a similar exponential modeland then extended it with Erlang distributed interarri-val and production times, and they showed that mostof the insights obtained with the exponential modelwere not influenced by the assumption on demandand production processes.As the system cost (e.g., backlogging or holding

cost) is the same for the same type of product, it isclear that there is no need for rationing amongorders for the same type of product. Therefore, thesystem state can be described by a vector of twovariables, y(t) = (y1(t),y2(t)), where yi(t) is the netinventory level of product i at time t, with y1(t) ∈ Zand y2(t) ∈ Z�, where Z is the set of all integers,whereas Z� only includes non-positive ones. We useyþ1 ðtÞ ¼ maxf0; y1ðtÞg to show the amount of inven-tory of type 1 at time t, and y�1 ðtÞ ¼ maxf0; � y1ðtÞgto show the number of backorders at time t. Simi-larly, �y2(t) is the number of backorders for product2 at time t. The system state space is Ω = Z 9 Z�.In state y = (y1,y2), the system incurs a cost at rate,

cðyÞ ¼ �hyþ1 � b1y�1 þ b2y2:

Let a be the time discount rate, let Nai ðtÞ be the num-

ber of accepted orders over interval [0,t] for product i(i = 1,2), and let Nr

2ðtÞ be the number of rejectedorders for type 2 over the same period of time. Weseek an optimal control policy π so as to maximizeeither the discounted system profit over an infinitehorizon,

maxp

Jpðyð0ÞÞ¼Epyð0Þ

X2i¼1

Z 1

0

e�atpidNai ðtÞ

"

�Z 1

0

e�atr2dNr2ðtÞþ

Z 1

0

e�atcðyðtÞÞdt�;

ð1Þ

Iravani, Liu, and Simchi-Levi: Optimal Production and Admission Policies226 Production and Operations Management 21(2), pp. 224–235, © 2011 Production and Operations Management Society

Page 4: Optimal Production and Admission Policies in Make-to-Stock/Make-to-Order Manufacturing Systems

or the average profit over an infinite horizon,

maxp

Jpa¼ limT!1

1

TEp

X2i¼1

piNai ðTÞ�r2N

r2ðTÞþ

Z T

0

cðyðtÞÞdt" #

:

ð2ÞIn Equation (1), Jπ(y(0)) is the expected profit func-

tion under policy π starting from initial state y(0) =(y1(0),y2(0)). In the rest of the article, we will mainlyfocus on the discounted-profit problem. However, asshown in de Vericourt et al. (2002), the theoreticalresults for the discounted-profit model also apply tothe average-profit problem.Without loss of generality, we redefine the time

scale such that a + μ + k1 + k2 = 1. Then, all thedenominators in the transition probability functionsbecome 1, and thus disappear. The time discount fac-tor a, only appearing as a term in the denominators,disappears as well. According to Bertsekas (1995), theoptimality equation J*(y1,y2) under the time-discountcriterion satisfies the following Bellman equation:

Jðy1; y2Þ ¼ cðy1; y2Þ þ lH0Jðy1; y2Þ þ k1H1Jðy1; y2Þþ k2H2Jðy1; y2Þ

:¼ HJðy1; y2Þ ð3Þ

where H0, H1, and H2 are functions defined by,

H0Jðy1;y2Þ¼max Jðy1;y2Þ;Jðy1þ1;y2Þ;Jðy1;y2þ1jy2\0Þf gH1Jðy1;y2Þ¼Jðy1�1;y2Þþp1

H2Jðy1;y2Þ¼max Jðy1;y2�1Þþp2;Jðy1;y2Þ�r2f g:

The model includes production and admissiondecisions. H0 corresponds to the production decision:when the machine becomes available, the manufac-turer needs to decide which product to produce orstop production. J(T|C) is a conditional value function,which indicates that the transition to state T is onlyvalid when condition C holds. Therefore, the thirdterm means producing product 2 when there arebackorders for it. H1 indicates that demands for type 1will always be accepted. H2 is associated with theadmission control for an arriving order for type 2. Theadmission decision is for MTO orders only, as OEMorders are fully accepted. The manufacturer can eitheraccept (backlog) or reject the order. We denote theoverall Bellman function as HJ(y1,y2).The optimality equation under the average-profit

criterion is:

Jðy1;y2Þþg¼ cðy1;y2ÞþlH0Jðy1;y2Þþk1H1Jðy1;y2Þþk2H2Jðy1;y2Þ; ð4Þ

where g is the optimal average profit per unit time.

2.2. The Optimal PolicyIn this section, we prove that the optimal policy hasa partial-linear structure. To prove the partial-linearstructure, we develop a unique technique, wherethe initial positions and the movement directions ofthe thresholds are restricted. In a general value itera-tion process, the initial threshold positions are deter-mined by the initial value function, and then thethresholds move toward their final (optimal) posi-tions. The movement directions are determined bythe initial threshold positions, and thus by the initialvalue function. In the existing literature, there havebeen no restrictions on the initial value functionother than satisfying all the optimality conditions.Thus, the initial thresholds could be in any possiblepositions, depending on the initial value function,and the threshold movement directions are notrestricted either. This standard approach works ifthe structure of the optimal policy is completely lin-ear (Ha 1997a, 2000, de Vericourt et al. 2002), orcompletely non-linear (Benjaafar et al. 2010, Carrand Duenyas 2000, Chen and Kulkarni 2007, Ha1997b).When the optimal policy has a partial-linear struc-

ture such as the one in our problem, the standardapproach does not work. We need to restrict the initialpositions and the movement directions of the thresholds.The objective is to keep the linear conditions valid inthe final linear region throughout the value iterationprocess. The linear conditions are more restrictivethan the non-linear ones. Once the linear conditionsare not satisfied in a state, it would be extremely diffi-cult to prove that the linear conditions will hold inthat state in later iterations. To avoid this difficult sit-uation in the proof, we restrict the initial rationingthreshold to be higher than its final position. Therationing threshold keeps moving downward towardthe final position during the value iteration process.In this way, the linear region starts at its largest andshrinks during the value iteration process, and the lin-ear conditions are kept valid in the region below thefinal rationing threshold throughout the value itera-tion process.To restrict the initial rationing threshold position,

we select an initial function to let the initial ration-ing threshold equal to N, where N is a number thatis guaranteed to be larger than the optimal ration-ing threshold. To show the existence of such a num-ber, we take two steps to prove the optimal policy.In the first step, we prove the optimal policy ischaracterized by a base-stock level and two switch-ing curves. The purpose of this step is to show theexistence of a base-stock level, which is guaranteedto be larger than the optimal rationing level. In thesecond step, we prove the partial-linear structure of

Iravani, Liu, and Simchi-Levi: Optimal Production and Admission PoliciesProduction and Operations Management 21(2), pp. 224–235, © 2011 Production and Operations Management Society 227

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the optimal policy, by setting N equal to the opti-mal base-stock level. Thus, the initial rationingthreshold position is at the base-stock point. Inter-ested readers may refer to the beginning of theproof for Theorem 1 in the Appendix for details.Here, we only present the optimality conditions andthe properties of the partial-linear structure of theoptimal policy.For any function f defined on Ω, let Δ1f(y1,y2) = f

(y1 + 1,y2) � f(y1,y2),Δ2f(y1,y2) = f(y1,y2 + 1) � f(y1,y2),and Δ12f(y) = f(y1 + 1,y2) � f(y1,y2 + 1). We definethe set of functions as C1, such that if fðy1; y2Þ 2 C1,then,

CONDITION SET C1. For (y1,y2) ∈ Ω,

• C.1.1: Δif(y1,y2) � 0, if yi < 0, i = 1,2;• C.1.2: Δ12f(y1,y2) � 0, if y1 < 0;• C.1.3: Δ1f(y1,y2) and Δ2f(y1,y2) are non-increasing

in y1 and y2;• C.1.4: Δ12f(y1,y2) is non-increasing in y1 and

non-decreasing in y2.

To have some intuition about the above condi-tions, we apply them to the expected profit functionJ(y1,y2). Condition C.1.1 implies that, if there arebackorders for a product, it is better to produce theproduct rather than idle the facility. Condition C.1.2implies that, if there are backorders for both prod-ucts, type 1 has higher priority than type 2. Condi-tions C.1.3 and C.1.4 are the concavity andsubmodularity conditions.We first show that any functions satisfying the

above conditions have the following properties.

PROPERTY 1. If fðy1; y2Þ 2 C1, then

(i) Δ1f(y1,y2) � 0 for y1 < Sf, whereSf ¼ minfz j�1fðz; 0Þ\0g;

(ii) Δ12f(y1,y2) � 0 for y2 < 0, y1 < Qf(y2), whereQfðy2Þ ¼ minfz j�12fðz; y2Þ\0g;

(iii) Δ2f(y1,y2 � 1) � p2 + r2 for y2 > Af(y1), whereAfðy1Þ ¼ maxfz j�2fðy1; z � 1Þ[p2 þ r2g;

(iv) 0 � Af(y1) � Af(y1 + 1) � 1;(v) Sf � Qf(�1) � Qf(y2), for y2 < �1.

For the proof of all the properties, propositions,lemmas, and theorems in 2.2, please see Appendix S1.Property (i) of the proposition implies that for thresh-old level Sf, if y1 < Sf, then producing type 1 is betterthan idling the facility. Property (ii) implies that type1 has higher priority if y1 < Qf(y2), and type 2 hashigher priority otherwise. Property (iii) implies thatfor threshold level Af(y1), if y2 > Af(y1), then acceptinga type 2 order is better than rejecting. Property (iv)implies that the admission threshold level Af(y1) goesup either vertically (when Af(y1) � Af(y1 + 1) = 0) orin a 45� line (when Af(y1) � Af(y1 + 1) = 1). Property

(v) implies that the production threshold level Sf is nolower than the rationing threshold level.In the following, we show that the optimal profit

function J(y1,y2) satisfies all the conditions in set C1,i.e., Jðy1; y2Þ 2 C1. Lemma 1 indicates that the struc-ture of function f in C1 is preserved under function H.

LEMMA 1. If fðy1; y2Þ 2 C1, then Hfðy1; y2Þ 2 C1.

In the next proposition, we use the lemma to showthat there exists an optimal policy satisfying all theconditions in C1. It will be a switching-curve policycharacterized by a base-stock level (S), a rationingthreshold curve (Q(y2)), and an admission thresholdcurve (A(y1)).

PROPOSITION 1. The optimal policy is characterized bythree thresholds: the base-stock level S, the rationing levelQ(y2), and the admission level A(y1), such that at state(y1,y2),

• Production control: when there are no backorders forproduct 2 (i.e., y2 = 0), it is optimal to produce type1 if y1 < S and to stop production if y1 � S.

• Rationing control: when there are backorders for pro-duct 2 (i.e., y2 < 0), it is optimal to produce type 1 ify1 < Q(y2) and to produce type 2 if y1 � Q(y2),where Q(y2) decreases as y2 decreases (the number ofbackorders increases).

• Admission control: when y1 < Q(y2), it is optimal toaccept an arriving demand for type 2 if y2 > A(y1)and to reject otherwise, where A(y1) decreases as y1increases.

The proof for the above policy is omitted becauselater we will prove a more restrictive policy using thesame approach. This policy is similar to the switch-ing-curve policy in Carr and Duenyas (2000), and thusnot surprising. However, it shows the existence of theoptimal base-stock level S, which we will use as theinitial position of the rationing threshold in the valueiteration process in the proof for Theorem 2. Wedenote f*(y1,y2) as the optimal function that satisfiesEquation (3).In the following, we will show that the optimal

policy could be characterized more restrictively by apartial-linear structure, which is a special case of theswitching-curve structure. For this purpose, we intro-duce C2, a set of functions that satisfy all the condi-tions in C1 and the following additional conditions.Denote Rf = Qf(�1) and Bf = Af(0), we define C2, suchthat, if fðy1; y2Þ 2 C2 � C1, then,

CONDITION SET C2. For y1 < Rf and y2 < 0,

• C.2.1: Δ12f(y1,y2) is independent of y2;• C.2.2: Δ12f(y1,y2) � 0;

Iravani, Liu, and Simchi-Levi: Optimal Production and Admission Policies228 Production and Operations Management 21(2), pp. 224–235, © 2011 Production and Operations Management Society

Page 6: Optimal Production and Admission Policies in Make-to-Stock/Make-to-Order Manufacturing Systems

• C.2.3: Δ12f(y1,y2) � Δ12Hf(y1,y2)�Δ12f*(y1,y2);• C.2.4: Δ2f(y1,y2) is independent of y1 and y2 as

long as y1 + y2 is fixed.

Condition C.2.1 implies that the marginal differencebetween producing type 1 or type 2 is independent ofthe number of backorders for type 2, when y1 < Rf.Condition C.2.2 implies that type 1 has higher priorityif y1 < Rf, or type 2 has higher priority otherwise.Condition C.2.3 implies that the value of Δ12f(y1,y2)monotonically decreases during the value iterationprocess, until it converges. The sign of p2 + r2 � Δ2J(y1,y2 � 1) determines whether to reject an order fortype 2. Condition C.2.4 suggests that the admissiondecision for orders of product 2 depend on the totalinventory level, but not on the inventory level of eachproduct.We next show that under the above additional

conditions, the threshold levels have the followingadditional properties.

PROPERTY 2. If fðy1; y2Þ 2 C2 � C1, then

(vi) Δ2f(y1,y2 � 1) � p2 + r2 for y1 < Rf

and y1 + y2 > Bf;(vii) y1 + Af(y1) = Bf when y1 < Rf;(viii) RHf � Rf.

Property (vi) implies that accepting a new orderfor type 2 is profitable, as long as the the totalinventory of products 1 and 2 is larger than athreshold level Bf (i.e., y1 + y2 > Bf). Property (vii)implies that the admission threshold Af(y1) presentsa 45� line when y1 < Rf. Property (viii) of the propo-sition suggests that the rationing threshold levelmonotonically decreases during the value iterationprocess (until it converges).In the following, we show that the optimal profit

function J(y1,y2) satisfies all the conditions in C2,i.e., Jðy1; y2Þ 2 C2. Lemma 2 indicates that the struc-ture of function f in C2 is preserved under function H.

LEMMA 2. If fðy1; y2Þ 2 C2 � C1, then Hfðy1; y2Þ 2C2 � C1.

In the next theorem, we use the lemma to show thata modified base-stock policy is optimal.

THEOREM 1. The optimal policy is characterized by threeparameters: the base-stock level, S, the rationing level, R(R � S), and the admission level, B, such that at state(y1,y2),

• Production control: when there are no orders back-orders for product 2 (i.e., y2 = 0), it is optimal to pro-duce type 1 if y1 < S and to stop production if y1 � S.

• Rationing control: when there are backorders for pro-duct 2 (i.e., y2 < 0), it is optimal to produce type 1 ify1 < R and to produce type 2 if y1 � R.

• Admission control: when y1 < R, it is optimal toaccept an arriving demand for type 2 if y1 + y2 > Band to reject otherwise; when y1 � R, it is optimal toaccept an arriving demand for type 2 if y2 > A(y1)and to reject otherwise.

The optimal policy is illustrated in Figure 1 (left).Under the optimal policy, the manufacturer will stopproduction if the inventory level of type 1 is greateror equal to S and there are no backorders of type 2.When the inventory level of type 1 is higher than Rand there are backorders for type 2 in the system, themanufacturer will give higher priority to type 2orders. If the inventory level of type 1 is < R or whenthere are no backorders for type 2, the manufacturerwill produce type 1 to increase its inventory level.When the inventory level of type 1 is lower than R,the manufacturer will accept an arriving order fortype 2 if the total net inventory y1 + y2 is higher thanB or reject it otherwise; when y1 � R, the manufac-turer will accept a type 2 order if y2 > A(y1).The optimal policy has a partial-linear structure.

The rationing threshold is a horizontal line, and theadmission threshold below y1 = R represents a 45�

Figure 1 Left: The Optimal and the Linear (S,R,B) Heuristic Policy; Right: The Optimal Policy with Contingent Outsourcing

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line. Only the admission threshold above R may notbe linear. If we use 45� to approximate the admissionthreshold, as shown by the dotted line in Figure 1(left), we come to a linear-structured policy. We referto this linear heuristic policy as the (S,R,B) policy.Due to its simple structure, the computation complex-ity of the (S,R,B) policy is much lower than that of theoptimal policy. As we will later show in section 3.1,the performance of the (S,R,B) policy is very close tothat of the optimal policy.

2.3. Contingent Outsourcing for AftermarketOrdersIn this section, we consider a manufacturer who hasthe option to outsource the production of some ordersto other suppliers, and the manufacturer pays by thenumber of units outsourced. In this model, weassume that the manufacturer only outsources theproduction of aftermarket products but not the OEMorders. Such an assumption is reasonable for manu-facturers who want to provide consistent quality forOEM products. The quality of outsourced products isnot guaranteed, and it may hurt the relationshipbetween the manufacture and the OEM, if the qualityof the outsourced product is low. In some industries,outsourcing may simply be prohibited by the contractbetween a manufacturer and an OEM customer.We focus on the case where contingent outsourcing

of an order for product 2 incurs a cost of l2, higherthan the rejection penalty r2. (The case with l2 � r2 istrivial, as the manufacturer will never reject anydemand—s/he will accept and then outsource theproduction immediately.) For the detailed model andthe optimality conditions, see Appendix S2. The fol-lowing theorem characterizes the structure of theoptimal policy for the two-product problem with theoutsourcing option:

THEOREM 2. The optimal policy is characterized by fourparameters: the base-stock level, S, the rationing level, R,the admission level, B, and the outsourcing level, L(L � B), such that at state (y1,y2),

• Production control: when there are no backorders forproduct 2 (i.e., y2 = 0), it is optimal to produce type1 if y1 < S, and to stop production if y1 � S.

• Rationing control: when there are orders backorders forproduct 2 (i.e., y2 < 0), it is optimal to produce type 1if y1 < R, and to produce type 2 if y1 � R.

• Admission control: when y1 < R, it is optimal toaccept an arriving demand for type 2 if y1 + y2 > B,and to reject otherwise; when y1 � R, it is optimal toaccept an arriving demand for type 2 if y2 > A(y1),and to reject otherwise.

• Outsourcing control: when y1 < R, it is optimal tooutsource an order for type 2 upon arrival of an order

for type 1, if y1 + y2�L; when y1 � R, it is optimalto outsource an order for type 2 if y2 � K(y1).

The optimal policy for a two-product problem withoutsourcing options is described in Figure 1 (right).Under the optimal policy, when y1 < R, if the totalinventory y1 + y2 is lower or equal to L, the manufac-turer will outsource an order for type 2 when an orderfor type 1 arrives; when y1 � R, the manufacturer willoutsource an order for type 2 if y2 � K(y1). The opti-mal policy entails simultaneously accepting a high-priority OEM order while sending a previouslyaccepted aftermarket order to the outsourced sup-plier. In Figure 1 (right), this corresponds to movingdiagonally south-east in the state space, and thus anystates below the outsourcing threshold are transient.Except for the admission and outsourcing thresholdsabove R, the optimal policy has linear thresholds any-where else.

3. Computational Analysis

The goal of our numerical analysis is to investigatethe following questions:

1. How much profit will a manufacturer lose if s/heuses the linear-structured (S,R,B) policy instead ofthe optimal policy?

2. Outsourcing orders prevents large backloggingcosts for the manufacturer. However, it alsodecreases customer satisfaction. How much willthe profit decrease if the manufacturer does notutilize the flexibility of outsourcing?

3. What is the profit improvement of the (S,R,B) pol-icy relative to other commonly used policies?How is the profit improvement affected by theprices and system capacity?

4. When the objective is to maximize profit, it issometimes inevitable to sacrifice the service levelof some customers. Therefore, an important ques-tion is: What is the impact of the (S,R,B) policyon the service level of both the aftermarket andthe OEM customers?

Our numerical study consists of 320 cases gener-ated by varying the following parameters:

• ρ = (k1 + k2)/μ: This parameter is an indicationof the manufacturer’s relative capacity comparedto the market size. We refer to ρ as potentialfacility utilization in the rest of the article. Pleasenotice that ρ is not the real facility utilizationsince aftermarket demands may be rejected.We considered five values for ρ, specifically,ρ ∈ {0.4,0.6,0.8,1.0,1.2}.

• k1/(k1 + k2): This demand ratio represents the sizeof the OEM demand compared to the aftermarket

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demand. In our numerical study we consideredfour cases, namely k1/(k1 + k2) ∈ {0.1,0.3,0.5,0.7}.

• p2/p1: The price ratio represents the price differ-ence between the two products. As the price ofaftermarket product is higher than the priceoffered to the OEM, we consider four values forp2/p1 that are all >1, namely p2/p1 ∈ {1.0,1.3,1.6,2.0}. We did not consider ratios >2, as it isvery uncommon in practice to find an aftermar-ket item sold at twice its price for OEM.

• b2/b1: The penalty ratio represents the backlog-ging penalty difference between the two prod-ucts. In the numerical study, we assume that thebacklogging cost of each product is proportionalto its price, and we consider four scenarios:(b1 = 20%p1,b2 = 5%p2), (b1 = 40%p1,b2 = 5%p2),(b1 = 100%p1,b2 = 5%p2), and (b1 = 200%p1,b2 = 5%p2). Note that, although p1 < p2, in all the scenar-ios, backlogging of type 1 customers is costlierthan that of type 2 customers, which follows theassumption for the model with OEM and after-market products.

We also considered other selected demand andprice ratios to obtain more data points for the figures,but they are not included in the 320 cases. To betterpresent the effects of the parameters on the systemperformance (and omit the effects of discount factor),our numerical study uses total expected profit perunit time as the performance measure.

3.1. (S,R,B) Policy vs. Optimal PolicyTo evaluate the performance of the (S,R,B) policy, wefirst compare it with the optimal policy in ournumerical study. To obtain the expected profit underthe (S,R,B) policy, we revise the admission decisionin the MDP model to follow the linear thresholdlevel B. This changes our MDP model from an opti-mization model to a performance evaluation modelthat obtains the optimal expected profits for the(S,R,B) policy.By examining the 320 cases in our numerical study,

we found that, on average, the (S,R,B) policy resultsin 0.6% less profit than the optimal policy. The differ-ence was < 2% in most of the cases we tested. Besidesthe similarities between the two policies, another keyreason that the (S,R,B) policy performs so close to theoptimal policy is that the probability that the systemvisits a state close to the non-linear threshold (in theupper left area where y1 � R and y2 � B � R in Fig-ure 1) is very small. Intuitively, on one hand, whenthe production capacity is tight, the manufacturermay have low inventory or high backorders for bothproducts most of the time, and the system stays in thelower area (where y1 < R) most of the time; on theother hand, when the production capacity is suffi-

cient, the manufacturer may have high inventorylevel for product 1 and very few backorders for prod-uct 2, and the probability that the system stays in theupper right area (where y1 � R and y2 > B � R) ishigh. So although the admission threshold may not belinear when y1 � R, the frequency that we need torefer to it is very small.The close performance of the linear (S,R,B) policy to

that of the optimal policy is an indication that chang-ing the partial-linear structured optimal policy to alinear heuristic does not have a significant impact onthe expected profit. The (S,R,B) policy approximatesthe optimal policy very well.

3.2. Effectiveness of Contingent OutsourcingWe also studied the performance of the outsourcingpolicy analyzed in section 2.3. Interestingly, we foundthat in all the scenarios in our numerical study, thecontingent outsourcing option only provides slightprofit increase (i.e., 1.8% on average and < 3% in mostof the cases we tested), compared with the (S,R,B) pol-icy in which outsourcing is not applied. Intuitively,under the (S,R,B) policy, an aftermarket demand willbe accepted only when the system has enough capac-ity, so the probability that an accepted order is lateroutsourced is very small. In our model settings (withsingle-unit Poisson arrivals), for a manufacturer withthe flexibility to accept or reject an order, having addi-tional flexibility of outsourcing an order later does nothave much value.

3.3. (S,R,B) Policy vs. Simple Base-Stock PolicyBecause of its non-linear component, the optimal pol-icy is not easy to implement in practice. Fortunately,the (S,R,B) policy has a simple structure and wellapproximates the optimal policy. In this section, wecompare the performance of the (S,R,B) policy with apolicy that is commonly used in practice (i.e., thebase-stock policy). The objective is to investigate howmuch the expected profit increases if a manufacturingsystem switches from other commonly used policiesto the (S,R,B) policy.The simple base-stock policy is often used in prac-

tice, and it is characterized by two threshold levels:base-stock level, S’, for type 1 products, and admis-sion level, B’, for type 2 products. Under this policy,the system produces type 1 products as long as theinventory is < S’. When the inventory of type 1 prod-uct reaches S’, the system produces type 2 products ifthere is an order of type 2 in the system, or idles other-wise. The orders for type 2 product will only beaccepted if the number of orders of type 2 in the sys-tem is < B’. We call this policy the (S′,B′) policy. Theoptimal values for S’ and B’ are obtained by using anexhaustive grid search.

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3.3.1. Impact on Expected Profit. Denoting JSB asthe system’s expected profit per unit time under theoptimal (S′,B′) policy, we evaluate the profit improve-ment under the (S,R,B) policy using the followingmeasure:

Profit PotentialSRB ¼ JSRB � JSBJSB

� 100%; ð5Þ

where JSRB is the expected profit per unit time underthe (S,R,B) policy.Based on our numerical study, we found that, on

average, the profit improvement obtained by usingthe (S,R,B) policy is 8.1%. The maximum profitimprovement can be up to 40%.We also examined the effects of price ratio and

potential facility utilization on the profit improve-ment. Figure 2 (left), which is for one set of examplesamong several that we studied, depicts the typicalbehavior of the system. In Figure 2 (left), we fixk1/(k1 + k2) = 0.7 and b1 = 20%p1, and we change theprice ratio and the potential facility utilization rate.As the figure shows, when the price ratio increases,implementing the (S,R,B) policy results in more profitimprovement. The reason is as follows. Both the (S,R,B) and the base-stock policies do not reject the type 1orders, and thus in the long run, the number of satis-fied type 1 orders is the same under both policies.However, the number of type 2 orders satisfied underthese two polices are different. The (S,R,B) policy sat-isfies more type 2 orders than the base-stock policy,because as shown in section 3.1, the (S,R,B) policy isalmost as efficient as the optimal policy (which hasthe highest number of accepted orders of type 1). As aresult, the gap between the performance of these twopolicies (i.e., the profit potential) increases as theorders of type 2 become more profitable (i.e., the priceration increases).The figure also suggests that, when the capacity is

sufficient (ρ � 1.0), profit potential increases as ρincreases. This is true because as potential facility uti-lization rate increases, i.e., capacity becomes tighter, itis more and more important to allocate capacity effec-

tively between the two classes of products, which isexactly what the (S,R,B) policy achieves. However,when ρ > 1.0 (see Figure S1 in Appendix S3), theexpected profit under both the (S,R,B) policy and thesimple base-stock policy turns negative as ρ furtherincreases, and the profit difference between the twopolicies becomes smaller. In this case, as the produc-tion capacity is very tight, both policies behave thesame, keeping the facility running almost all the timeand rejecting aftermarket demand if capacity is notenough. So manufacturers benefit most from imple-menting the (S,R,B) policy when the capacity is nei-ther too tight nor too loose.We also examined the effects of backlogging pen-

alty ratio for OEM products. For this purpose, wekeep b2 = 5%p2 unchanged, and increase b1 from20%p1 to 40%p1, 100%p1, and 200%p1. We find thatthe profit potential of the (S,R,B) policy decreasesand finally approaches zero. Intuitively, as the OEMbacklogging penalty increases, giving priority toOEM product becomes more critical, and thus anypolicy such as our base-stock policy that giveshigher priority to the OEM also performs relativelywell. Therefore, the gap between the performance ofthe base-stock policy and the (S,R,B) policy (i.e., theprofit potential) decreases as the OEM backloggingpenalty increases.Figure 2 (right) shows the typical system behavior

of how changes in demand ratio and potential facilityutilization affect the profit improvement. Figure 2(right) corresponds to one set of our numerical studyin which p2/p1 = 1.6 and b1 = 20%p1. As the figuresuggests, when the demand ratio, k1/(k1 + k2), equals0 or 1, there is only one type of products in the sys-tem, and therefore there is no difference between the(S,R,B) policy and the simple base-stock policy. How-ever, as the demand ratio increases from 0 to 1, therelative performance of the (S,R,B) policy increasesand then decreases. The reason is the same as whatwas described for Figure 2 (left). When the two prod-ucts are intensively competing for the capacity, thebenefit of the (S,R,B) policy becomes prominent,

Figure 2 Left: Impact of Price Ratio (p2/p1); Right: Impact of Demand Ratio k1=ðk1 þ k2Þ

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because the (S,R,B) policy can manage the productionand the inventory more effectively than the simplebase-stock policy.Figure 2 (right) also depicts that, when ρ � 1.0, the

profit improvement of the (S,R,B) policy increases asthe total demand relative to the production capacityincreases. However, when ρ > 1.0 (see Figure S1 inAppendix S3), the expected profit under both the (S,R,B) policy and the simple base-stock policy turnsnegative as ρ further increases, and the profit differ-ence between the two policies becomes smaller. Thereason is the same as for Figure 2 (left).

3.3.2. Impact on Service Levels. When the objec-tive is to maximize profit, it is sometimes inevitable tosacrifice the service level of some customers. There-fore, it is important to understand what the impact ofthe (S,R,B) policy is on the service levels for both theaftermarket and the OEM customers. Thus, in thissection we study the impact of the (S,R,B) policy oncustomers’ service levels.We measure the service level for aftermarket by the

fraction of aftermarket orders that are accepted bythe manufacturer. On the other hand, we measurethe service level for OEM demand by the fraction ofOEM orders that are immediately satisfied frominventory.Figure 3 (left) shows the service levels for aftermar-

ket demands under the (S,R,B) policy and the simple(S′,B′) policy, respectively. As the figure shows, whenthe price ratio increases, the service level for aftermar-ket demands increases slightly under both policies.This is very intuitive, as the manufacturer will acceptmore aftermarket demands when they contributemore profits.Figure 3 (left) also shows that when the production

capacity is sufficient (ρ = 0.8), both policies maintainhigh service levels for aftermarket demands, and theservice level under the (S,R,B) policy is slightly higherthan that under the simple base-stock policy. Ascapacity becomes tight (ρ = 1.0), the (S,R,B) policyprovides higher service level than the simple base-

stock policy. Since the (S,R,B) policy can manage thelimited capacity more effectively than the simple pol-icy, more aftermarket demand could be satisfiedunder the (S,R,B) policy. However, as ρ furtherincreases (ρ = 1.2), the difference in service levels ofthe two policies becomes smaller. In this case, as theproduction capacity is very tight, both policies behavesimilarly, keeping the facility running almost all thetime and rejecting aftermarket demands if the numberof orders exceeds the threshold.Figure 3 (right) illustrates the service level for OEM

demand, i.e., the percentage of OEM orders that areimmediately satisfied upon their arrivals. When theprice ratio increases, the service level for OEMdemand under both the (S,R,B) policy and the simplebase-stock policy decreases.In Figure 3 (right), we find that the service level for

OEM is lower under the (S,R,B) policy than under thesimple base-stock policy. This can be explained in twoways. One reason is that the (S,R,B) policy sacrificesthe service level for OEM demands to accept moredemands from the aftermarket (which has a higherprofit margin) to improve the overall profit. Anotherreason is that the service level for OEM demand ismainly determined by R in the (S,R,B) policy, and byS’ in the simple policy. In the numerical results, wefind S′ � R, so the service level for OEM demand ishigher under the simple policy. The reason forS′ � R is also intuitive. In the simple policy, S’ acts asboth the base-stock level for product 1 and the ration-ing level for product 2, so the value of S′ is usuallybetween S and R.Figure 3 (right) also shows that when the produc-

tion capacity is sufficient (ρ = 0.8), both policies main-tain high service levels for OEM, and the service leveldifference between the two policies is very small. Ascapacity becomes tight (ρ = 1.0), the service levels ofboth policies decrease, and the difference betweenthem increases. As ρ further increases (ρ = 1.2), theservice level difference between the two policiesbecomes smaller, which could be explained similarlyas for Figure 3 (left).

Figure 3 Left: Service Level for Aftermarket Demands under the (S,R,B) Policy and the (S′,B′) Policy; Right: OEM Service Level under the (S,R,B) Policy and the (S′,B′) Policy

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4. Conclusions

In this article, we study the production and inven-tory policies in a manufacturing system with ahigh-priority MTS product and a low-priority MTOproduct. We show that the optimal policy has apartial-linear structure characterized by three para-meters, the base-stock level, the rationing level, andthe admission level. The partial-linear structuredoptimal policy presents a significant improvementover the current non-linear, switching-curve poli-cies in the literature on inventory rationing prob-lems with partial backorders. Finding the optimalpolicy is much less computationally intensive, andthe policy is easier to implement. The results inthis article also provide a strong theoretical supportfor the findings in (Benjaafar et al. 2010, Carr andDuenyas 2000, Chen and Kulkarni 2007), where lin-ear-structured heuristic policies are suggestedbased on numerical tests. The article also contrib-utes to the value iteration approach for provingstructural results when the aimed optimal policieshave partial-linear structures. To prove the partial-linear structure, we select an initial value functionto restrict the initial position of the rationingthreshold and its movement direction in the valueiteration process. Using this technique, we can alsoprove that a linear-structured policy is optimal fora manufacturing system with a high-priority MTOproduct and a low-priority MTS product (Liu2010).Our numerical study shows that a linear-structured

heuristic approximates the optimal policy very well.In the computational analysis, we also observed thatthe (S,R,B) policy can result in, on average, 8% (andsometimes up to 40%) more profit than that under thecommonly used base-stock policy. The profit differ-ence is high when production capacity is tight, whenthe backlogging costs for the MTS product and theMTO product are close, or when demand arrival ratesfor the two products are close. The (S,R,B) policyimproves the profit by allocating the productioncapacity more effectively, and therefore having thecapability to accept more low-priority demands. As atrade-off, when the production capacity is very tight,the service level for the high-priority demands underthe optimal policy is lower than that under the simplebase-stock policy.

Acknowledgments

The authors thank the department editor, the associ-ate editor, and two referees for their thorough reviewsand constructive comments. The research is supportedin part by the Center for Excellence in Logistics andDistribution, Center of eBusiness at MIT, ONR Con-

tracts N00014-95-1-0232 and N00014-01-1-0146, andNSF Contracts DMI-0245352, IIP-0214416, andCMMI-0758069.

ReferencesBenjaafar, S., M. ElHafsi, T. Huang. 2010. Optimal control of a

production-inventory system with both backorders and lostsales. Naval Res Logist. 57(3): 252–265.

Bertsekas, D. 1995. Dynamic Programming and Optimal Control, Vol.2. Athena Scientific, Nashua, NH.

Carr, S., I. Duenyas. 2000. Optimal admission control and sequenc-ing in a make-to-stock/make-to-order production system. Oper.Res. 48(5): 709–720.

Chen, F., V. G. Kulkarni. 2007. Individual, class-based, and socialoptimal admission policies in two-priority queues. Stochast.Models 23(1): 97–127.

Cohen, M. A., P. R. Kleindorfer, H. L. Lee. 1988. Service con-strained (s,S) inventory systems with priority demand classesand lost sales. Manage. Sci. 34(4): 482–499.

de Vericourt, F., F. Karaesmen, Y. Dallery. 2000. Dynamic sched-uling in a make-to-stock system: A partial characterization ofoptimal policies. Oper. Res. 48(5): 811–819.

de Vericourt, F., F. Karaesmen, Y. Dallery. 2002. Optimal stockallocation for a capacitated supply system. Manage. Sci. 48(11): 1486–1501.

Dobson, G., C. A. Yano. 2001. Production offering, pricing, andmake-to-stock/make-to-order decisions with shared capacity.Prod. Oper. Manag. 11(3): 293–312.

Duenyas, I., M. Vanoyen. 1996. Heuristic scheduling of parallelheterogeneous queues with set-ups. Manage. Sci. 42(6): 814–829.

Gittins, J. C. 1989. Multi-armed Bandit Allocation Indices. Wiley,New York.

Graves, S. C. 1980. The multi-product production cycling prob-lem. AIIE. Trans. 12(3): 233–240.

Gupta, D., L. Wang. 2007. Capacity management for contractmanufacturing. Oper. Res. 55(2): 367–377.

Ha, A. 1997a. Inventory rationing in a make-to-stock productionsystem with several demand classes and lost sales. Manage.Sci. 43(8): 1093–1103.

Ha, A. 1997b. Stock-rationing policy for a make-to-stock produc-tion system with two priority classes and backordering. NavalRes. Logist. 44(5): 458–472.

Ha, A. 1997c. Optimal dynamic scheduling policy for a make-to-stock production system. Oper. Res. 45(1): 42–53.

Ha, A. 2000. Stock-rationing in an M/Ek/1 make-to-stock queue.Manage. Sci. 46(1): 77–87.

Huang, B., S. M. R. Iravani. 2008. Technical note—A make-to-stock system with multiple customer classes and batch order-ing. Oper. Res. 56(5): 1312–1320.

Iravani, S. M. R., T. Liu, K. L. Luangkesorn, D. Simchi-Levi.2007. A produce-to-stock system with advance demandinformation and secondary customers. Naval Res Logist. 54(3): 331–345.

Kleijn, M., R. Dekker. 1998. An overview of inventory systemswith several demand classes. Report EI9838/A, EconometricInstitute, Erasmus University, Rotterdam, The Netherlands.

Liu, T. 2010. Optimal production and admission policies withboth customized and pre-configured products. Workingpaper, Oklahoma State University, Stillwater, OK.

Melchiors, P., R. Dekker, M. J. Kleijn. 2000. Inventory rationing inan (s,Q) inventory model with lost sales and two demandclasses. J. Oper. Res. Soc. 51(1): 111–122.

Iravani, Liu, and Simchi-Levi: Optimal Production and Admission Policies234 Production and Operations Management 21(2), pp. 224–235, © 2011 Production and Operations Management Society

Page 12: Optimal Production and Admission Policies in Make-to-Stock/Make-to-Order Manufacturing Systems

Moinzadeh, K. 1989. Operating characteristics for the (S�1,S)inventory system with partial backorders and constant resup-ply time. Manage. Sci. 35(4): 472–477.

Montgomery, D. C., M. S. Bazaraa, A. K. Keswani. 1981. Inventorymodels with a mixture of backorders and lost sales. NavalRes. Logist. 20(2): 255–263.

Nahmias, S., W. S. Demmy. 1981. Operating characteristics of aninventory system with rationing. Manage. Sci. 27(11): 1236–1245.

Nahmias, S., S. A. Smith. 1994. Optimizing inventory levels in atwo-echelon retailer system with partial lost sales. Manage.Sci. 40(5): 582–596.

Pena-Perez, A., P. Zipkin. 1997. Dynamic scheduling rules for amulti-product make-to-stock queue. Oper. Res. 45(6): 919–930.

Reiman, M. I., L. M. Wein. 1998. Dynamic scheduling of a two-class queue with setups. Oper. Res. 46(4): 532–547.

Rabinowitz, G. A., A. Mehrez, C. Chull, B. E. Patuwo. 1995. Apartial backorder control for continuous review (r,Q) inven-tory system with poisson demand and constant leadtime.Comp. Oper. Res. 22(7): 689–700.

Smeitink, E. 1990. A note on the operating characteristics of the(S�1,S) inventory systems with partial backorders and con-stant resupply times. Manage. Sci. 36(11): 1413–1414.

Topkis, D. M. 1968. Optimal ordering and rationing policies in anonstationary dynamic inventory model with n demand clas-ses. Manage. Sci. 15: 160–176.

Varaiya, P., J. Walrand, C. Buyukkoc. 1985. Extensions of themulti-armed bandit problem: The discounted case. IEEETrans. Automat. Contr. AC-30(5): 426–439.

Wein, L. M. 1992. Dynamic scheduling of a multiclass make-to-stock queue. Oper. Res. 40(4): 724–735.

Xiao, Y., J. Chen, C. Y. Lee. 2010. Single-period two-productassemble-to-order systems with a common component anduncertain demand patterns. Prod. Oper. Manag. 19(2): 216–232.

Youssef, K. H., C. van Delft, Y. Dallery. 2004. Efficient schedulingrules in a combined make-to-stock and make-to-order manu-facturing system. Ann. Oper. Res. 126(1-4): 103–134.

Zheng, Y., P. Zipkin. 1990. A queueing model to analyze thevalue of centralized inventory information. Oper. Res. 38(2):296–307.

Supporting InformationAdditional supporting information may be found in theonline version of this article:

Appendix S1. Proof of Analytical ResultsAppendix S2. Analytical Supporting Results for the OEM/

Aftermarket Model with Contingent OutsourcingAppendix S3. Impact on Expected Profit when Capacity

is Tight

Please note: Wiley-Blackwell is not responsible for thecontent or functionality of any supporting materials sup-plied by the authors. Any queries (other than missingmaterial) should be directed to the corresponding authorfor the article.

Iravani, Liu, and Simchi-Levi: Optimal Production and Admission PoliciesProduction and Operations Management 21(2), pp. 224–235, © 2011 Production and Operations Management Society 235