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Optimal Admission Control and Sequencing in a Make-to-Stock/Make-to-Order Production SystemScott Carr, Izak Duenyas,

To cite this article:Scott Carr, Izak Duenyas, (2000) Optimal Admission Control and Sequencing in a Make-to-Stock/Make-to-Order ProductionSystem. Operations Research 48(5):709-720. http://dx.doi.org/10.1287/opre.48.5.709.12401

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OPTIMAL ADMISSION CONTROL ANDSEQUENCING IN AMAKE-TO-STOCK/MAKE-TO-ORDER PRODUCTION SYSTEM

SCOTT CARR and IZAK DUENYASAnderson Graduate School of Management; University of California at Los Angeles; Los Angeles; California, scarr@anderson.ucla.edu

University of Michigan; School of Business; Ann Arbor; Michigan 48109, duenyas@umich.edu

(Received February 1997; revision received August 1998; accepted January 1999)

In this paper, we address the problem of admission control and sequencing in a production system that produces two classes of products.The &rst class of products is made-to-stock, and the &rm is contractually obliged to meet demand for this class of products. The secondclass of products is made-to-order, and the &rm has the option to accept (admit) or reject a particular order. The problem is motivatedby suppliers in many industries who sign contracts with large manufacturers to supply them with a given product and also can take onadditional orders from other sources on a make-to-order basis.

We model the joint admission control=sequencing decision in the context of a simple two-class M=M=1 queue to gain insight into thefollowing problems: 1. How should a &rm decide (a) when to accept or reject an additional order, and (b) which type of product to producenext? 2. How should a &rm decide what annual quantity of orders to commit to when signing a contract to produce the make-to-stockproducts?

1. INTRODUCTION

Rarely does a manufacturing line produce a single prod-uct for sale to a single customer. Rather, a typical &rm islikely to use its manufacturing facilities to produce a vari-ety of di0erent products for a variety of di0erent customersand market segments. The &rm’s customers in di0erent mar-ket segments may have di0erent requirements, and the rela-tionship between producers and customers may vary acrossdi0erent segments.

This work is motivated by a major glass manufacturerthat fabricates laminated and tempered windows for instal-lation into automobiles, trucks, and farm implements. This&rm’s products are sold to automotive assembly plants (theoriginal equipment manufacturer (OEM) market segment)for installation into new vehicles and to the automotiveglass replacement (aftermarket) segment for installation intoolder vehicles. The two segments consume nearly identicalproducts but are otherwise of very di0erent character.

Sales to OEMs are based on long-term contracts thatinclude large contractually speci&ed penalties if productsare not available when required by the OEM’s just-in-timeproduction. These long-term contracts are appealing toautomotive suppliers because they result in high guaranteedcapacity utilization. The trade-o0 is that the unit pro&ts gen-erated by these contracts typically are low, and the potentialfor large penalties dictate that the manufacturer maintaina suitable &nished goods inventory to bu0er against pro-duction delays and to allow production runs of aftermarketitems. As noted in Duenyas et al. (1997), although the JITliterature widely espouses the virtues of providing supplierplants with visibility to production schedules, in realitymany suppliers have very poor information about futuredemand. Duenyas et al. discuss this issue in detail and give

examples from several industries. One result of this poorvisibility to demand is that suppliers often have to carrysigni&cant amounts of inventory to meet the demands oftheir customers “just-in-time.” In this paper, we consider anadditional reason for the supplier to choose to carry inven-tory: namely, to allow acceptance and production of morepro&table orders.

In contrast to the OEM demand, the aftermarket is charac-terized by higher pro&t margins but more occasional and lim-ited demand. A manufacturing facility may produce itemsfor a single OEM contract but many di0erent aftermarketitems. The OEM contract is served in a make-to-stock fash-ion, while the uncertain arrival pattern and the very largevariety of aftermarket items necessitates make-to-order ser-vice. Prices are fairly standard in the aftermarket; quickservice is the primary dimension of competition. Notablythe manufacturer is not contractually required to accept anaftermarket order; an order can be either refused or sourcedfrom another &rm.

Several interesting topics emerge from this situation.A long-term strategic question is the optimal proportionof production capacity that the &rm should dedicate to theOEM segment. Once the &rm commits to a certain volumeof OEM production, how should it decide whether to takeon additional work? Finally, how should a &rm decide tosequence work? In this paper, we provide simple models togain insight into the nature of these problems.

We &rst address these problems in the context of a simpletwo-class M=M=1 queue. The &rst class of jobs (representingOEM jobs), are made-to-stock. Demands arrive randomly,and there is a penalty if demand is not satis&ed immedi-ately. The second class of jobs represents possible aftermar-ket orders. These can be accepted or rejected by the &rm

Subject classi%cations: Inventory=production: inventory policies for make-to-stock=make-to-order systems. Sequencing: stochastic sequencing.Area of review: MANUFACTURING OPERATIONS.

Operations Research, ? 2000 INFORMS 0030-364X/00/4805-0709 $05.00Vol. 48, No. 5, September–October 2000, pp. 709–720 709 1526-5463 electronic ISSN

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depending on the state of the system. Rejected orders arelost. The &rm also has to decide whether to idle, or to pro-cess a job of the &rst or of the second class at a given pointin time. We assume that no set-ups are required to switchfrom one type of job to another and that the &rm faces hold-ing costs for the amount of OEM inventory it holds in stock.We are therefore addressing the joint problem of admis-sion control=sequencing in a make-to-order=make-to-stockM=M=1 queue. (To keep the presentation simple, we initiallyfocus on M=M=1 queues as in other recent papers focusingon structural insight including Veatch and Wein 1996 andHa 1997a–1997c. However, in §5, we numerically explorethe e0ects of lower variability in processing or demand ar-rival times by allowing Erlang distributions.)

There is a rich literature on admission control and se-quencing, although most papers consider these problemsin isolation and not jointly. Stidham (1985) reviews the lit-erature on customer admission to single class make-to-orderqueues. There has been a signi&cant amount of recent inter-est in control and performance evaluation of single serversystems that produce multiple classes of products. Whilepast work has focused on the optimality of simple index rulesfor make-to-order systems where set-ups are not required toswitch from one class to another (e.g., Varaiya et al. 1985,Walrand 1988, and the references therein), more recent workhas focused on systems where a set-up is required to switchfrom one product to another and the dynamic stochasticeconomic lot sizing problem (SELSP) arising from the pro-duction of multiple make-to-stock products. Duenyas andVan Oyen (1996) and Reiman and Wein (1994) focus oncontrol of make-to-order systems with set-ups. Gallego(1990) and Bourland and Yano (1995) develop heuristicpolicies for SELSP. Anupindi and Tayur (1994) developa simulation-based approach to obtain e0ective base-stockpolicies, and Federgruen and Katalan (1996) analyze theperformance of cyclic base-stock policies for this problem.Other recent papers on the SELSP include Markowitz et al.(1995), Sox and Muckstadt (1995), Qiu and Loulou (1995),and Ha (1997a). However, in all of these papers, the onlyproblem addressed is the dynamic production=sequencingproblem; admission control is not considered. Murti (1994)considers a make-to-stock=make-to-order system similar tothe one in this paper and focuses on optimal static and dy-namic scheduling policies. However, once again, admissiondecisions are not considered.

Several recent papers have considered joint admissioncontrol and production decisions in single product settings.Li (1992) considers a &rm producing a single product andexplores the economic environmental factors that lead the&rm to hold inventory. Ha (1997b) considers stock rationingin a single-class, make-to-stock production system with sev-eral demand classes and lost sales, while Ha (1997c) consid-ers the case with backordering. Our work is di0erent fromthese papers in that we consider customer classes that re-quire di0erent products, whereas in these papers all cus-tomers require the same product, and therefore we have toaddress the resulting sequencing problem jointly with the

admission control problem. Duenyas (1995) considers thejoint problem of quoting due dates and sequencing jobs in amake-to-order queue with multiple customer classes, wherea customer’s probability of placing orders depends on thedue date quoted. Glasserman and Yao (1994) and Veatchand Wein (1992) provide general structural results for opti-mal control of queueing system.

Finally, recent work has focused on performance eval-uation of make-to-stock=make-to-order systems as well ason the problem of how a &rm decides whether to operatein a make-to-stock or make-to-order mode. Nguyen (1995)develops Nuid and di0usion approximations for the perfor-mance of a make-to-order=make-to-stock system with FIFOservice discipline. Federgruen and Katalan (1994) use theframework of a cyclic polling system with setups to addressthe decision of how many products to make to order andhow many to make to stock. Their analysis can be used bya &rm to decide which products to o0er, whether to makethem to stock or to order and in what annual quantities. Ourpaper di0ers in that we assume the decision of which prod-uct to make to stock or to order has already been made (inthe environment described above, this decision is severelyconstrained by contractual considerations for the OEM mar-ket, and the diOculty of making to stock the huge varietyof products that is demanded by the aftermarket demandsegment) and we allow for dynamically deciding which par-ticular orders to accept=reject and which class of jobs toproduce next.

The rest of this paper is organized as follows. In §2,we introduce the notation and formulation of the problem.In §3, we completely characterize the structure of the optimaladmission control and sequencing policies. These policiesare characterized by monotonic switching curves. We thenconsider how changes in problem parameters are reNected inthe switching curves and overall pro&tability. Because opti-mal policies are rather complex, in §4 we describe a simpleand easily implementable admission control and sequencingpolicy and show that, for a wide variety of examples, thispolicy performs very well. We then demonstrate the impor-tance of considering all three decisions that the &rm faces si-multaneously and show how making these decisions withoutconsidering their interdependence can result in signi&cantloss of pro&tability. In §5, we explore the e0ects of lowervariability in the demand arrival or production processes onpro&ts as well as the optimal level of OEM demand the &rmshould commit to. Section 6 concludes the paper.

2. PROBLEM FORMULATION

We consider the optimal control of a single-server queuethat produces two types of products. Demand for product oftype i (i= 1; 2), arrives according to a Poisson distributionwith rate �i. Type 1 products are made to stock, demandfor them is satis&ed from inventory, and each type 1 saleearns a revenue of R1. It is not possible to backorder demandfor type 1 products; if there is no type 1 product in stockwhen a demand occurs, the &rm satis&es the demand by

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purchasing the product from another supplier at an additionalcost (penalty) of � for each such occurrence.

Type 2 products are produced to order, and the &rm hasthe option of to accept or reject each type 2 order. (Wewill later generalize to the case where there are multipleclasses of products made to order which are only di0erenti-ated by price; for simplicity of presentation, we &rst presentthe case where there is a single product type which is made-to-order.) Each accepted order generates a revenue (alter-natively, margin) of R2. Inventory costs are incurred at therate of c1 per unit per time for each unit of type 1 productin stock, and at the rate of c2 per unit time for each unitof type 2 product in process (c2 may also serve as a proxyfor the cost to the &rm of delaying production of type 2 or-ders and thus becoming uncompetitive in that market). Weassume that production of a unit of either type 1 or type 2product takes an exponentially distributed amount of timewith mean 1

. (In §5, we allow type 1 and type 2 productionto have di0erent processing time distributions.) We furtherassume that preemptions are permitted, and that no set-upis required to switch from producing one type of product toanother.

The set of decision epochs allowed in our model corre-spond to the set of all arrival epochs and service comple-tion epochs. A policy speci&es at each decision epoch thatthe server idle or produce a unit of type 1 or type 2. Fur-thermore, in decision epochs corresponding to the arrival ofa customer for a type 2 product, a decision must be madewhether to accept the customer’s order. The objective is to&nd a policy that maximizes the average pro&t per unit timeover an in&nite horizon, where pro&t is the revenue fromtype 2 orders minus the inventory costs for type 1 and type 2products and the penalties incurred when type 1 demandscannot be met from stock. We note that because all demandfor type 1 products is satis&ed, adding the revenue associ-ated with type 1 products, R1, to the objective function hasno e0ect on the optimal control policy.

We can formulate the optimal admission and productioncontrol problem as a Markov decision process. We de&nethe state (n1; n2) as the number of units of type 1 inventory instock and the number of units of type 2 products in process.We let v(n1; n2) be the relative value function of being instate (n1; n2) and g(�1) denote the average pro&t (excludingtype 1 revenue) per transition (where a transition is an arrivalor service completion). Then, we can write

g(�1) + v(n1; n2) = 1R

{−c1n1−c2n2 + �1[v(n1−1; n2)

· I(n1¿0) + (v(0; n2)−�) · I(n1 = 0)]

+�2 max[v(n1; n2+1) + R2; v(n1; n2)]

+ max[v(n1 + 1; n2); v(n1; n2−1)

· I(n2¿0) + v(n1; n2) · I(n2 = 0)]};

(1)

where I(·) denotes the indicator function, and R =�1+�2+.

In Equation (1), the cost terms ( 1R{−c1n1−c2n2}) rep-

resent the expected holding and waiting costs per decisionepoch; the terms multiplied by �1 represent transitions andpenalties associated with the arrival of a type 1 customer;the terms multiplied by �2 represent transitions and revenuesgenerated by the arrival of a type 2 order; and the termsmultiplied by represent transitions generated by a servicecompletion opportunity. (The &rm makes a new productiondecision every time an event (e.g., a type 1 demand or aproduction completion) changes the state of the system.)We note that because preemptions are allowed, idling whenn2¿0 cannot be optimal, and g(�1) is the pro&t per transi-tion when the manufacturer faces a long-run class 1 demandrate of �1. Because transitions occur with rate R, g(�1)Rrepresents the pro&t per unit time.

The second problem that the &rm faces is that of choosingthe optimal level of �1 and is formulated as

max�1

g(�1)(�1 + �2 + ) + R1�1: (2)

The &rm makes this decision only once (for example, whensigning a contract to provide products for the OEM market).Note that the choice of �1 is interrelated to the solutionof (1). It is therefore important to understand the structure ofthe optimal solution to (1) and how this structure is a0ectedby problem variables. We &rst focus on the solution to thisproblem.

3. THE STRUCTURE AND MONOTONICITY OFTHE OPTIMAL ADMISSION CONTROL ANDSEQUENCING POLICIES

In this section, we focus on the problem described by (1) andcharacterize the structure of the optimal admission controland sequencing policy that the &rm would use given that theoptimal level of �1 has already been &xed. The main resultof this section is the following theorem, which completelydescribes the structure of the optimal policy. We use theterms “increasing” and “decreasing” in their weak senses,nondecreasing and nonincreasing, respectively.

THEOREM 1. The optimal production policy is de%ned by aswitching curve h(n2) such that for n1¿h(n2); the optimalpolicy is to produce type 2 items if n2¿0 and to idle ifn2 = 0; for n16h(n2) the optimal policy is to produce type 1items. We refer to this switching curve as the productionthreshold curve. Furthermore; h(n2) is decreasing is n2.The optimal type 2 order acceptance policy is also de-

%ned by a switching curve; f(n2) such that if n1¿f(n2); theoptimal policy accepts type 2 orders but otherwise rejectsthem. We refer to this switching curve as the acceptancethreshold curve. Furthermore; f(n2) is increasing in n2.

PROOF. The proof is provided in the appendix.

The structure of the optimal policy in a typical problem isillustrated in Figure 1. The optimal policy is characterized by

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Figure 1. Structure of the optimal admission andsequencing policies.

four regions corresponding to the combinations of producingeither type 1 or type 2 items and either accepting or rejectingarriving type 2 customers. We conjecture that Theorem 1holds even when the production rate for products 1 and 2are di0erent. Furthermore, our numerical results have shownthat the admit=reject curve is linear for high traOc intensities,as suggested by the heavy traOc analysis of Markowitz andWein (1996).

Having characterized the structure of the optimal policyfor (1), we next proceed to analyze how the optimal pro&tand admission and sequencing decisions change as a func-tion of the problem parameters. Our &rst result, character-izing the monotonicity of the expected pro&ts as a functionof the problem parameters, is intuitive.

THEOREM 2. The optimal average pro%t per unit timeg(�1)R of (2:1) is increasing in ; �2; R2; and decreasingin �; c1; c2; �1.

PROOF. See the appendix.

We note that the formulation in (1) does not considerrevenues for type 1 products but considers overall pro&tsfrom acceptance of type 2 products and sequencing deci-sions once the choice of �1 has been made. Therefore, theexpected pro&t in (1) is decreasing in �1 because the higherthe value of �1, the fewer type 2 orders the &rm can ac-cept. Having characterized the monotonicity of the averagepro&t per unit time in (1), we next proceed to character-ize the monotonicity of the switching curves with respect tothe input parameters. Consider two instances of the admis-sion control and sequencing problem described by (1). Todi0erentiate the second instance from the &rst one, we usethe prime (′) symbol for costs and switching curves in thesecond case.

THEOREM 3. (a) Suppose that �1 = �′1; �2 =�′2; c1 = c′1;c2 = c′2; �= �′; = ′; and R′

2¿R2. Then; f′(n2)6f(n2)for all n2; and h′(n2)6h(n2) for n2¿1; and h′(0)¿h(0).

(b) Suppose that �1 = �′1; �2 = �′2; c1 = c′1; c2 = c′2;= ′; R′

2 =R2; and �′¿�. Then; f′(n2)¿f(n2) andh′(n2)¿h(n2).

PROOF. See the appendix.

In words, the second part of Theorem 3 states as � getslarger, it may become optimal to switch from accepting type2 orders to rejecting orders in any given state and to switchfrom producing type 2 orders to type 1 orders in states withn2¿0, and &nally to switch from idling to producing type 1orders in states with n1 = 0. This result can be explainedintuitively. As stocking out of type 1 units now costs more,the policy switches towards giving higher priority to type 1orders, keeping more type 1 units in stock and acceptingfewer type 2 orders.

The &rst part of Theorem 3 states that as R2 gets larger,it may become optimal to switch from rejecting type 2 or-ders to accepting them in any given state, to switch fromproducing type 1 orders to producing type 2 orders in stateswith n2¿0, and to switch from idling to producing type 1orders in states with n1 = 0. As type 2 orders become morepro&table when R2 is increased, the policy tends to switchto accepting more of them and to give higher priority totheir production rather than the production of type 1 orders.However, when there are no type 2 orders to work on, thepolicy works towards building a larger &nished goods in-ventory of type 1 units so that it can have the opportunityto accept and work on type 2 units when orders for themarrive. Figure 2 displays how the switching curves shift asa function of a change in R2.

Although the switching curves f and h have nice mono-tonicity properties with respect to � and R2, they are notnecessarily monotonic with respect to the other parameters.We show this by way of counterexample. Consider the fol-lowing problem which we will refer to as “base case.” Thebase case has the following values: R2 = 20, �= 35, c1 = 0:5,c2 = 2, �1 = 1:5, �2 = 1, and = 2. We will analyze theoptimal decisions in states S1 = (2; 2) and S2 = (1; 6) under

Figure 2. Change in policy as a result of an increasein R2.

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the base case and other cases. Under the base case, the op-timal production decision is to produce type 1 items in S1

and type 2 items in S2. Changing only �1 to 0.75 from 1.5but keeping all the other parameters the same, will in theoptimal policy result in producing type 2 items in S1 andtype 1 items in S2. Similarly, changing only to 5 from 2will result in once again the optimal policy producing type2 items in S1 and type 1 items in S2. These examples clearlydemonstrate that the switching curves are not necessarilymonotonic with respect to all parameters.

A simple extension of our model also has a nice structure.Consider the setting of a single make-to-stock class butan arbitrary number of make-to-order classes. Assume thattype 1 orders are the make-to-stock orders and type 2through −N orders are make-to-order. If we further assumethat all products have the same mean production rate andthat all the make-to-order products have the same waitingcost c2, and are numbered such that R2¿R3¿ · · ·¿ · · ·Rn,are again able to characterize the structure of the policy.First of all, we note that since waiting costs for all make-to-order products are the same, once we accept a make-to-orderproduct, we do not need to keep di0erentiating it from othermake-to-order products. Thus, we can represent the stateonce again as (n1; n2) where n1 is the number of units oftype 1 in stock, and n2 denotes the sum of type 2 through Norders accepted but not yet processed. Then, using the sameapproach used for proving Theorem 1, it is easy to show thatthere exists a production threshold curve h(n2) such that ifn1¡h(n2) then the optimal policy is to produce a type 1 unitand otherwise to serve any of the make-to-order units wait-ing. Similarly, for each make-to-order class there exists anacceptance switching curve fi(n2) such that if n1¿fi(n2)then a type i order will be accepted, and it will be rejectedotherwise. Furthermore, h(n2) and fi(n2) are monotonicwith respect to n2 as in Theorem 1. Finally, it is easy to showthat fi(n2)¡fj(n2) for j¿i. Figure 3 demonstrates thestructure of the optimal policy in this case. We note that allthe production rates and waiting costs being equal is a crucialassumption in this case because if those strong assumptionsdid not hold, one would need to di0erentiate betweentypes thereby increasing the dimensionality of the statespace.

We close this section by noting that the &rm can &nd theoptimal level of �1 by solving (1) by discretizing the inter-val between [0; ) and performing a one-dimensional search.(Because convexity cannot be guaranteed, (1) would need tobe solved for all the discrete values in the interval.) We notethat this procedure is very di0erent from procedures usedby most &rms, including the one that motivated this study.The above-described way of setting �1 levels takes into ac-count the opportunities for satisfying aftermarket demandas well as optimal production and acceptance policies. Forexample, a typical &rm will have separate and relatively un-coordinated OEM sales, aftermarket sales, and operations.In our numerical study in the next section, we focus on theimportance of coordinating all three functions by showinghow even a &rm which is uncoordinated with respect to one

Figure 3. Structure of the optimal admission andsequencing policies—multiple make-to-order types.

of these functions (say, OEM sales) operates signi&cantlysuboptimally compared to a &rm that coordinates all threefunctions.

4. NUMERICAL STUDY

The optimal policies described in the previous sectionare rather complex. In particular, the production and ac-ceptance policies are de&ned by switching curves, whichmake their actual implementation problematic. However,they also point to the importance of considering thesedecisions simultaneously, i.e., “coordinating” these deci-sions. In this section, we &rst explore the performance ofa simpler, more implementable policy for production andacceptance decisions. However, this policy still considersthese decisions simultaneously. We then give an exampleof a relatively sophisticated, but somewhat uncoordinated,policy for controlling the three decisions, which ultimatelyfails.

We &rst focus on the sequencing and acceptance deci-sions after �1 is set. Consider the following simple policythat is described by two integer parameters I and W . I de-notes an “inventory limit”; type 1 production is allowed onlyfor n1¡I . W is a “work limit.” Type 2 orders are acceptedwhen the “work” in the system, de&ned as I−n1 +n2, is lessthan W . When either the inventory or the queue is empty(i.e., when n1 = 0 or n2 = 0), type 1 items are produced(unless n1 = I , in which case the system is idled); in allother states, type 2 items are produced. For any I and W ,this gives rise to a Markov chain with an associated valuethat is easy to calculate. It remains to &nd appropriate valuesfor I and W . For this, we perform a simple two-dimensional

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Table 1. Comparison of optimal and heuristic admission and sequencing policies.

No. R1 R2 � c1 c2 �1 �2 Opt. Heur. I W % dif.

1 10 20 25 1 2 1 1 2 20.11 20.10 7 9 0.32 5 20 25 1 2 1 1 2 15.11 15.10 7 9 0.13 10 5 25 1 2 1 1 2 11.58 11.52 6 6 0.54 10 50 25 1 2 1 1 2 49.74 48.78 8 16 1.95 10 20 50 1 2 1 1 2 18.53 18.23 9 8 1.66 10 20 100 1 2 1 1 2 17.33 16.94 10 8 2.37 10 20 25 0.5 2 1 1 2 22.15 22.15 10 11 0.08 10 20 25 2 2 1 1 2 17.66 17.65 4 7 0.19 10 20 25 1 0.5 1 1 2 22.46 22.14 6 13 1.4

10 10 20 25 1 5 1 1 2 17.75 17.63 7 7 0.711 10 20 25 1 2 0.5 1 2 19.32 19.31 3 9 0.112 10 20 25 1 2 1.75 1 2 17.01 16.94 13 12 0.413 10 20 25 1 2 1 0.5 2 9.44 9.37 4 5 0.714 10 20 25 1 2 1 1.75 2 23.89 23.82 10 10 0.315 5 10 25 1 2 1 1 2 6.58 6.52 6 6 0.816 50 10 25 1 2 1 1 2 51.58 51.52 6 6 0.117 10 10 5 1 2 1 1 2 15.3 14 3 5 6.818 10 10 25 1 2 1 1 2 11.57 11.52 6 6 0.419 5 25 500 2 2 1 3 10 64.92 63.45 3 15 2.320 5 25 32 0.5 4 4 3 6 57.37 57.06 45 40 0.0521 50 25 10 10 5 5 4 10 320.3 319.3 2 10 0.322 10 20 40 1 2 1.5 1 2 16.66 16.62 15 12 0.2

search. Although this is computationally ineOcient (espe-cially because we cannot guarantee concavity), it is certainlya very simple policy to describe and implement in practice.

Table 1 compares the pro&ts obtained by the optimal pol-icy and by the simple heuristic policy described for 22 exam-ples. In the table, the last &ve columns, respectively, showoptimal pro&ts, pro&ts when using the heuristic policy, theparameters for the heuristic, and the percent di0erence be-tween optimal and heuristic pro&ts. Example 1 representsthe base case, and in Examples 2 through 14 we systemati-cally increased or decreased one of the problem parametersto test the performance of the described policy under a vari-ety of conditions including high versus low type 1 or type 2demand, and high or low holding and penalty costs. Wealso added eight additional examples to cover other cases.The results in Table 1 clearly show that this simple, imple-mentable policy performs very well. The average di0erencebetween the performance of the optimal and heuristic poli-cies was only 1.1%. Although the computation of I and Wis somewhat time-consuming, in practice this would need tobe done only once; and the policy is very easy to describeto both manufacturing and sales departments. It is interest-ing that although the heuristic’s policy of giving priority totype 2 items (except when n1 = 0 or n2 = 0) seems unintu-itive at &rst, we note that when penalties for not meetingtype 1 demand are high, the policy accepts fewer type 2 or-ders, and thus makes it less likely that type 2 items will haveto be worked on when type 1 inventory is low. ConsiderExamples 1, 5, and 6, in which all the variables remain thesame except for �, which takes the values 25; 50; and 100.When �= 25, the heuristic sets I = 7 and W = 9. When �is increased to 50, I is increased to 9, while W is decreasedto 8. Thus, the heuristic is rejecting more type 2 orders andis actually producing more inventory when it is not working

on type 2 orders. Similarly, when � is increased to 100, I isincreased to 10, while W remains at 8.

We note that the above policy still considers sequenc-ing and acceptance decisions simultaneously in setting theparameters for the heuristic. As we noted in the previoussection, in many companies these decisions, as well as thedecision of how much OEM demand to contract for, areuncoordinated. In fact, many companies, including the onewith which we are most familiar, have separate divisionsfor OEM and aftermarket sales but have common manufac-turing facilities where both types of products are produced.Furthermore, the divisions have an incentive to maximizedivision pro&ts. In many other &rms, OEM contracting isperformed &rst and the aftermarket sales are subsequentlyconsidered. We show through a simple example how suchuncoordination may be very costly. Consider a &rm’s OEMsales department, which decides how much OEM demand toaccept (i.e., the optimal level of �1) without taking into con-sideration the existence of aftermarket demand. Assumingthis division was doing everything else optimally, it wouldsolve (1) and (2) by setting �2 = 0, and &nd the optimallevel of �1. Furthermore, assume that the aftermarket salesand scheduling departments are then noti&ed of the choiceof �1 and make their acceptance and sequencing decisionsoptimally and in coordination by solving (2:1) (with thevalue of �1 given to them) and implementing the optimalpolicy for sequencing and aftermarket order acceptance. Inthis example, all decisions except the initial decision of thechoice of �1 by the OEM sales department are coordinatedand optimal.

We numerically tested the performance degradation ofthis behavior on several of the examples from Table 1. In Ex-ample 20, the OEM sales department would actually choosea �1 value of 4.60 if it behaved in the above-described

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manner, whereas the optimal value of �1 is 2. Even thoughall decisions are made optimally thereafter, the uncoordi-nated behavior of this one department would result in a 38%decrease in &rm pro&ts! We note that in Example 20, after-market sales are nearly &ve times as pro&table as a OEMsales. However, even in Example 18, where aftermarket andOEM margins are equal, the decrease in pro&ts would benearly 6%. Not unexpectedly, we have observed that as after-market margins become signi&cantly higher than OEM mar-gins (exactly the scenario observed in industry), the pro&tdiscrepancy increases. Finally, we note that in these cases,overestimating the optimal value of �1 usually results in amore signi&cant pro&t degradation than underestimating it.

5. EXTENSION TO ERLANG ARRIVAL ANDSERVICE DISTRIBUTIONS

In the previous sections, we assumed that all distributionswere exponential. However, in some cases, signing a largecontract with a &rm might also result in a less variable de-mand pattern. It is easy to extend our formulation to handleErlang distributions for interarrival and service times. Thisallows us to handle service and arrival distributions whichhave signi&cantly less variability than the exponential. Forexample, if we assume that the arrival process of type 1demand is Erlang-z, with all other distributions still expo-nential, we can de&ne v(n1; n2; k) to be the relative valuefunction of having n1 type 1 units in stock, n2 type 2 orders,and k phases until the next type 1 arrival, where each phaseis of duration 1=(z�1). We can then write the following re-cursive optimality equation:

g(�1) + v(n1; n2; k)

= 1R

{−c1n1 − c2n2 + z�1[v(n1; n2; k − 1) · I(k¿1)

+ v(n1 − 1; n2; z) · I(k=1; n1¿0)

+ (v(0; n2; z) − �) · I(k=1; n1=0)]

+ �2 max[v(n1; n2 + 1; k) + R2; v(n1; n2; k)]

+ max[v(n1 + 1; n2; k); v(n1; n2 − 1; k) · I(n2¿0)

+ v(n1; n2; k) · I(n2=0)]}; (3)

where g(�1) denotes the average reward per transition asbefore, and R = z�1 + �2 + . In a similar manner, we caneasily extend the formulation to allow Erlang processingtimes for type 1 or type 2 production. This also allows us toformulate situations where type 1 and type 2 items do nothave the same processing time distribution.

We are interested in how improvements in arrival or pro-duction process variability a0ect the &rm’s pro&tability andthe size of contracts the &rm would be willing to sign withan OEM. Intuition suggests that as the arrival process oftype 1 demand becomes less variable, perhaps through bet-ter customer-supplier communication or contractual agree-ment, the &rm’s pro&ts would increase, and the &rm wouldalso be willing to dedicate more of its capacity to that cate-gory of customer. Similarly, the large cumulative production

Figure 4. Optimal pro&ts as a function of �1 withchanges in arrival variability.

volumes resulting from a long-term relationship with theOEM customer might lead to a more eOcient, less variableproduction process for the product demanded by the OEM.It is also common practice for some &rms, such as Toy-ota, to assist their suppliers in variability reduction. Intuitionagain suggests that such improvements in type 1 productionprocess would have the same results as the reduction in thevariability of the type 1 demand arrival process. Numericalexamples have veri&ed this intuition. We give two examplesof such behavior.

As a base case, consider the situation where all inter-arrival and processing times have exponential distribu-tions and R1 = 8, R2 = 15; �= 25, c1 = 1, c2 = 2, �2 = 1,and = 2. Figure 4 shows the impact of decreases in thevariability of type 1 demand arrivals. In this &gure, weplotted the pro&t of the &rm as a function of its choice of �1

when the type 1 demand arrival process was exponential,Erlang-2, and Erlang-5. As the arrival process becomes lessvariable, the pro&t level at any �1 increases, and as we ex-pected the optimal level of �1 also increases. In particular,the optimal level of �1 is 0.85 when the arrival process isexponential and it increases to 0.93 for Erlang-2 and to 0.99for Erlang-5 distributions. Similarly, Figure 5 demonstratesthe e0ect of reducing type 1 processing time variability.We plotted the optimal pro&ts as a function of �1 for ex-ponential, Erlang-2 and Erlang-5 type 1 processing timedistributions (the processing time for type 2 was &xed tobe exponential with the same mean for all examples. Onceagain, we note that as the production process becomes lessvariable, both pro&ts and the optimal value of �1 increase.In particular, the optimal �1 changes from 0.85 to 1.00 forErlang-2 and 1.02 for Erlang-5 distributions.

These results clearly show the bene&cial e0ects to thesupplier of decreasing arrival or processing time variability.In fact, as lower variability leads to higher pro&ts at alllevels of �1, a supplier might be willing to o0er prices tothe OEM in exchange for guarantees in demand variabilityor improvements in the design of products which lead todemonstrated decreases in production variability.

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Figure 5. Optimal pro&ts as a function of �1 withchanges in process variability.

6. CONCLUSIONS AND FURTHER RESEARCH

In this paper, we addressed the joint admission control andsequencing decisions faced by a &rm producing products formultiple segments. We were able to characterize the struc-ture and sensitivity of optimal policies in the context of amulticlass M=M=1 queue. Because the structure of the opti-mal policy is rather complex, we explored the performanceof a simpler policy and also presented examples showing theimportance of simultaneously considering these decisions.

Many open problems remain to be explored. First, ourformulation easily extends to general processing times withdi0erent rates for di0erent products and we conjecture thatour structural results continue to hold. Second, in many casesin practice, switching from one type of product to anotherrequires a set-up; exploring how optimal admission controland sequencing decisions are a0ected by the presence ofsetups is a very interesting question. For example, is a &rmmore likely to accept orders for a product for which itsmachines are already set up, if set-ups are very long? Third,we have so far explored these questions only in the contextof a single-server queue. It would be interesting to considermore complicated multiserver or network settings.

ACKNOWLEDGMENTS

The authors would like to thank Larry Wein, the asso-ciate editor, and two anonymous referees for comments thatgreatly improved the paper. This research has been sup-ported by grants from the National Science Foundation andthe Tauber Manufacturing Institute.

APPENDIX----PROOFS OF THEOREMS

PROOF OF THEOREM 1. Glasserman and Yao (1994) developgeneral results for optimal control of queueing systems. Inour model, type 2 order arrivals are supermodular in bothproduction events; this prevents us from using their results

directly. The use of their results would require that theevent of type 2 arrivals be supermodular in type 1 produc-tion events and submodular in type 2 production events.Nonetheless, we followed the same path in that we provethe optimal policy has a special structure by showing thatsub- and super-modularity conditions that induce the struc-ture survive the single-step optimality recursion and are in-herited by the optimal value function. Much of our analysisfollows the framework of Ha (1997a–1997c).

We &rst de&ne some additional notation that substantiallyshortens the analysis that follows. We use the symbols ↑ and↓ as abbreviations for “nondecreasing” and “nonincreasing,”respectively. For any real valued function t on the state space(denoted S), de&ne the following:

T1t(n1; n2) = t(n1 − 1; n2) · I(n1¿0) + (t(0; n2) − �) · I(n1=0);

T2t(n1; n2) = max[t(n1; n2 + 1) + R2; t(n1; n2)];

T3t(n1; n2) = max[t(n1; n2 + 1); t(n1; n2)];

T4t(n1; n2) = max[t(n1 + 1; n2); t(n1; n2 − 1)

· I(n2¿0) + t(n1; n2) · I(n2=0)];

Tt(n1; n2) = 1R [−c1n1 − c2n2 + �1T1t(n1; n2)

+�2T2t(n1; n2) + T4t(n1; n2)];

D1t(n1; n2) = t(n1 + 1; n2) − t(n1; n2);

D2t(n1; n2) = t(n1; n2 + 1) − t(n1; n2);

D3t(n1; n2) = t(n1 + 1; n2 + 1) − t(n1; n2):

D1 represents the additional value (under function t) ofan additional unit of type 1 inventory, D2 is the value ofan additional type 2 order in queue, and D3 is the valueof an additional inventory unit relative to the value gainedby &lling an existing type 2 order (when the current state is(n1; n2 + 1)).

Let V be the set of functions on S such that if t ∈V thenfor every R2¿0:(i) D1t(n1; n2) ↓ n1; ↑ n2, and 6�,(ii) D2t(n1; n2) ↑ n1; ↓ n2, and 60,

(iii) D3t(n1; n2) ↓ n1 and ↓ n2.The above conditions are sub- and super-modularity

conditions on t. For example, consider the conditionthat D2t(n1; n2) ↓ n2. This means that when a type 2order is accepted, the state space transitions from thecurrent state, say (n1; n2), in direction (0,1) to state(n1; n2) + (0; 1) = (n1; n2 + 1); i.e. that D2t(n1; n2) ↓ n2 im-plies that as the queue length increases the marginal valueof each new order gets smaller (more negative). We showbelow that the optimal value function v is in V . It followsthat for any state in which it is optimal to reject arrivingorders (i.e., D2v(n1; n2)¡−R2) it remains optimal to rejectorders at any state with the same type 1 inventory but largertype 2 queue.

The inequality conditions imply that (1) the value of eachunit of type 1 inventory is less than the penalty cost, and (2)once admitted to the queue each order has a negative value.These conditions are necessary to show that the requiredstructure is not violated at the boundaries of the state space.

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The following lemma shows that these relations are pre-served under the operator T .

LEMMA 1. If t ∈V then T1t; T2t; T3t; T4t; Tt ∈V .

PROOF. For brevity, we provide only the proof that D1 ap-plied to the various functions is decreasing in n1. Proofs ofthe other conditions are similar, so including them here isnot additionally illustrative.

We &rst show that D1T1t(n1; n2) ↓ n1. Considering theborder of the state space &rst—for any n2, D1T1t(0; n2) =T1t(1; n2)−T1t(0; n2) = �¿D1T1t(1; n2). The inequality fol-lows from (i). For n1¿1, D1T1t(n1; n2) =D1t(n1 − 1; n2)which is ↓ n1 because t ∈V .

We next require that D1T2t(n1; n2) ↓ n1. For every(n1; n2),

D1T2t(n1; n2) = max[t(n1 + 1; n2 + 1) + R2; t(n1 + 1; n2)]

−max[t(n1; n2 + 1) + R2; t(n1; n2)]; (4)

D1T2t(n1 + 1; n2)

= max[t(n1 + 2; n2 + 1) + R2; t(n1 + 2; n2)]

− max[t(n1 + 1; n2 + 1) + R2; t(n1 + 1; n2)]; (5)

and we require that (4) minus (5) is ¿0. For (4), there arefour possible outcomes from the two max functions. Oneof these is t(n1 + 1; n2) − t(n1; n2 + 1) − R2, but this vio-lates D2t ↑ n1 which follows from t ∈V and is therefore notfeasible. The other three possible outcomes are: D3t(n1; n2)+ R2, D1t(n1; n2 + 1), and D1t(n1; n2).

Suppose that D3t(n1; n2)+R2 is the actual outcome of (4).This implies that t(n1 + 1; n2 + 1) +R2 came out of the &rstmax (equivalently stated and used below, D2t(n1 + 1; n2)+ R2¿0). This also implies that these same terms comeout of the second max of (5) because it is the same asthe &rst max of (4). This, together with D2t ↑ n1, impliesthat the only feasible outcome of (5) is D1t(n1 + 1; n2 +1). Subtracting (5) from (4) thus gives D3t(n1; n2) + R2

−D1t(n1 +1; n2 +1), which also equals (seen by expandingthe de&nitions and cancelling terms) D3t(n1; n2)−D3t(n1 +1; n2) + (D2t(n1 + 1; n2) + R2), and this is ¿0 by D3 ↓ n1.Similar arguments show that (4) minus (5) is ¿0 for theother outcomes of (4); for brevity, here and below we donot give a complete case-by-case analysis.

Identical arguments, but with R2 replaced by zero, showthat D1T3t(n1; n2) ↓ n1.

To show that D1T4t(n1; n2) ↓ n1, we again start at theboundary of the state space (n2 = 0). For every n1,

D1T4t(n1; 0) = max[t(n1 + 2; 0); t(n1 + 1; 0)]

−max[t(n1 + 1; 0); t(n1; 0)];

which is also D1T3t(n1; 0) and is thus ↓ n1.Away from the boundary (n2¿1),

D1T4t(n1; n2) = max[t(n1 + 2; n2); t(n1 + 1; n2 − 1)]

−max[t(n1 + 1; n2); t(n1; n2 − 1)]; (6)

and

D1T4t(n1 + 1; n2) = max[t(n1 + 3; n2); t(n1 + 2; n2 − 1)]

−max[t(n1 + 2; n2); t(n1 + 1; n2 − 1)]:

(7)

Feasible outcomes of (6) are D1t(n1 +1; n2), D1t(n1; n2−1),and −1 ·D2t(n1 + 1; n2 − 1). The other potential outcome,t(n1 + 2; n2) − t(n1; n2 − 1), violates D3t ↓ n1. As above, itcan be shown case-by-case that (6) minus (7) is always¿0,so D1T4t(n1; n2) ↓ n1.

It remains to see that D1Tt(n1; n2) ↓ n1. This followsimmediately from the de&nition of T—inventory andholding costs are linear, and Tt is otherwise constructedby addition and multiplication of positive constants withconstituent functions (D1T1t; D1T2t, and D1T4t) that arethemselves ↓ n1.

Now, consider a value iteration algorithm to solve for theoptimal policy for (1) in which v0(n1; n2) = 0 for every state(n1; n2), and

vk+1(n1; n2) =Tvk(n1; n2): (8)

Here, vk(n1; n2) can also be viewed as the optimal valuefunction when the problem is terminated after k transitions.In order to prove Theorem 1, we &rst show that (1) has awell-de&ned solution, and the vk(n1; n2) values in (8) con-verge to v(n1; n2) in (1).

LEMMA 2. There exists an integer J; a constant g(�1) and afunction v such that vkJ+r(n1; n2)−(kJ+r)g(�1)→ v(n1; n2)for all r = 0; : : : ; J − 1 as k→∞.

PROOF. We &rst note that, without loss of optimality, we canadd the constraint to the original problem that we cannot ac-cept a customer when R2¡c2n2=R, and we cannot produceanother unit of type 1 when �¡c1n1=R. (For example, ifc1n1=R¿�, this indicates that the expected amount of hold-ing cost incurred by holding the n1 units of type 1 productsin stock until the next transition is greater than the penaltycost that would be incurred because of a type 1 product notbeing available if the next event were the arrival of demandfor a type 1 product.) Therefore, we convert the originalproblem to one with &nite state space. We also have &niteaction spaces and the model is unichain. The result followsfrom Puterman, 1994.

To complete the proof of Theorem 1, we &rst note that (byLemma 1) it also follows that v∈V . Conditions (i)–(iii)are suOcient for the structural requirements on the optimalpolicy to hold as follows. If it is optimal to accept a customerin state (n1; n2) (i.e. v(n1; n2 + 1) + R2¿v(n1; n2)), thencondition (ii) implies that v(n1+1; n2+1)+R2¿v(n1+1; n2),and it is optimal to accept customers in state (n1 + 1; n2).Similarly, if it is optimal to reject a customer in state (n1; n2),then is also optimal to reject a customer in state (n1; n2 + 1)by the fact that D2v(n1; n2) = v(n1; n2 +1)−v(n1; n2) is ↓ n2.

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The existence and monotonicity of the production thresholdcurve is similarly implied by conditions (i) and (iii).

PROOF OF THEOREM 2. The results for R2; c1; c2, and � di-rectly follow from the fact that using the policy that is opti-mal before the change in these parameters after the changeas well will result in higher pro&ts. We give the proof for�2; the proofs for and �1 are similar.

Consider two systems, labeled System 1 and System 2,that are identical except for di0ering type 2 order arrivalrates, �[1]

2 and �[2]2 for systems 1 and 2 where �[1]

2 ¡�[2]2 .

Suppose that System 2 is operated in the following manner:1. Whenever a type 2 order arrives, it is immediately re-

jected with probability (�[2]2 − �[1]

2 )=�[2]2 :

2. Acceptance=rejection of the orders not rejected instep 1; as well as the sequencing of orders is doneusing the optimal policy for System 1.

Under this policy, System 2 now exactly replicates the op-timal operation of System 1 in all respects and will thusrealize the same pro&tability as System 1. Because thepro&tability for System 1 is feasible to System 2, the opti-mal pro&tability for System 2 must be greater than or equalto that of System 1.

PROOF OF THEOREM 3. We give the proof for the policy sen-sitivity with respect to R2. The proof for the sensitivitywith respect to � is similar. We again consider two sys-tems, denoted System A and System B, that are identical ex-cept for di0ering type 2 pro&ts that are respectively R2 andR′

2; R2¡R′2. We apply Equation (8) to both systems simulta-

neously; the algorithms are initialized with v0(·) = v′0(·) = 0.We &rst prove the following lemma.

LEMMA 3. For every k = 0; 1; : : : and for every (n1; n2):(iv) D1v′k(n1; n2) − D1vk(n1; n2)¿0:(v) D2v′k(n1; n2) − D2vk(n1; n2) + R′

2 − R2¿0:(vi) D3v′k(n1; n2) − D3vk(n1; n2)60:

PROOF. Using proof by induction, we note that the lemmaholds trivially for k = 0 and assume that (iv)–(vi) hold atk. Note that from the previous proofs, vk ; v′k ∈V , so (i)through (iii) hold for vk and v′k . Here we show that (v) holds;veri&cation of (iv) and (vi) is similar and is omitted forbrevity. We now have two di0erent type 2 pro&ts, R2 andR′

2; we &rst rede&ne the operator T2 such that R′2 replaces R2

within T2 when it has “prime” arguments.For (v) at k + 1 we need that

D2v′k+1(n1; n2) − D2vk+1(n1; n2) + R′2 − R2¿0;

or equivalently that

D2Tv′k(n1; n2) − D2Tvk(n1; n2) + R′2 − R2¿0: (9)

Inventory and holding costs are identical for the two sys-tems and can henceforth be ignored. It follows that suOcient

conditions for (6:9) to hold are that it holds when T isreplaced by each of T1; T2; and T4. This is trivial for T1, sowe proceed to T2; substituting T2 for T in (9) and expandingyields the requirement that

max[v′k(n1; n2 + 2) + R′2; v

′k(n1; n2 + 1)]

−max[v′k(n1; n2 + 1) + R′2; v

′k(n1; n2)]

−max[vk(n1; n2 + 2) + R2; vk(n1; n2 + 1)]

+ max[vk(n1; n2 + 1) + R2; vk(n1; n2)] + R′2 − R2 (10)

is ¿0. De&ne functions w and w′ on {0; 1}× S as

w(u; n1; n2) ={vk(n1; n2 + 1) + R2 if u= 1;vk(n1; n2) if u= 0;

and de&ne w′ in the same manner with primes (′) attachedto each vk and R2. Thus, T2vk(·) = maxu∈{0;1} w(u; ·). Letu(n1 ; n2) = argmaxu w(u; n1; n2) and u′(n1 ; n2) = argmaxu w

′(u;n1; n2). It follows from (ii) that u′(n1 ; n2+1)6u′(n1 ; n2) andu(n1 ; n2+1)6u(n1 ; n2); it follows from (v) that u′(·)¿u(·). Thus,the vector (u′(n1 ; n2+1); u

′(n1 ; n2); u(n1 ; n2+1); u(n1 ; n2)) has six

possible values: (1,1,1,1), (0,0,0,0), (1,1,0,1), (1,1,0,0),(0,1,0,0), (0,1,0,1). (10) is considered for each of these:

(1,1,1,1): (10) equals D2v′k(n1; n2 +1)−D2vk(n1; n2 +1)+R′

2 − R2, which is ¿0 by (v).(0,0,0,0): (10) equals D2v′k(n1; n2)−D2vk(n1; n2)+R′

2−R2,which is ¿0 by (v).

(1,1,0,1): (10) equals v′k(n1; n2 + 2) − v′k(n1; n2 + 1) + R′2;

which is ¿0 because u′(n1 ; n2+1) = 1.(1,1,0,0): (10) equals v′k(n1; n2 + 2) − v′k(n1; n1 + 1) −

vk(n1; n2 + 1) + vk(n1; n2) + R′2 − R2¿0. The in-

equality holds because u′(n1 ; n2+1) = 1 implies thatv′k(n1; n2 +2)+R′

2¿v′k(n1; n2 +1) and u(n1 ; n2) = 0implies that vk(n1; n2)¿vk(n1; n2 + 1) + R2.

(0,1,0,0): (10) equals−vk(n1; n2+1)+vk(n1; n2)−R2 whichis ¿0 because u(n1 ; n2) = 0.

(0,1,0,1): (10) equals 0.

It remains to show that (9) holds with T4 replacing T ,and doing so includes the complication that the state spaceboundary must be checked. At the boundary (n2 = 0), sub-stituting T4 for T in (9) yields the requirement that

06D2T4v′k(n1; n2)|n2=0 − D2T4vk(n1; n2)|n2=0 + R′2 − R2

= max[v′k(n1 + 1; 1); v′k(n1; 0)]

−max[v′k(n1 + 1; 0); v′k(n1; 0)]

−max[vk(n1 + 1; 1); vk(n1; 0)]

−max[vk(n1 + 1; 0); vk(n1; 0)] + R′2 − R2: (11)

To keep notation to a minimum, rede&ne w and w′ by

w(u; n1; n2) =

vk(n1 + 1; n2) if u= 1;vk(n1; n2 − 1) if u= 0; n2¿1;vk(n1; n2) if u= 0; n2 = 0;

with w′ again de&ned in the same manner with primes at-tached to each vk ; also, the new w′ and w are applied to the

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de&nitions of u and u′. Now, T4vk(·) = maxu∈{0;1} w(u; ·).It follows from (ii) that u(n1 ;0)¿u(n1 ;1) and u′(n1 ;0)¿u′(n1 ;1).It follows from (iv) that u′(n1 ;0)¿u(n1 ;0) and from (vi) thatu′(n1 ;1)6u(n1 ;1). The vector (u′(n1 ;1); u

′(n1 ;0); u(n1 ;1); u(n1 ;0)) has

&ve feasible values, and we consider the right-hand side of(11) for each:

(1,1,1,1) gives D2v′k(n1 + 1; 0)−D2vk(n1 + 1; 0) +R′2 −R2

which is ¿0 by (v).(0,0,0,0) gives R′

2 − R2¿0.(0,1,0,1) gives v′k(n1; 0) − v′k(n1 + 1; 0) − vk(n1; 0) +

vk(n1 + 1; 0) + R′2 − R2. This is equal to

D2v′k(n1 + 1; 0) − D2vk(n1 + 1; 0) + R′2 − R2 −

D3v′k(n1; 0) + D3vk(n1; 0); which is ¿0 by (v)and (vi).

(0,1,0,0) gives v′k(n1; 0) − v′k(n1 + 1; 0) − vk(n1; 0) +vk(n1; 0) +R′

2 −R2; which (because u(n1 ;0) = 0) is¿v′k(n1; 0) − v′k(n1 + 1; 0) − vk(n1; 0) + vk(n1 +1; 0) + R′

2 − R2; and this is ¿0 as immediatelyabove.

(0,1,1,1) gives v′k(n1; 0) − v′k(n1 + 1; 0) − vk(n1 + 1; 1) +vk(n1 +1; 0)+R′

2−R2; which (because u′(n1 ;0) = 1)is ¿D2v′k(n1 + 1; 0)−D2vk(n1 + 1; 0) +R′

2 −R2;and this is ¿0 as for (1,1,1,1).

Away from the boundary, the analog of (11) is

06max[v′k(n1 + 1; n2 + 1); v′k(n1; n2)]

−max[v′k(n1 + 1; n2); v′k(n1; n2 − 1)]

−max[vk(n1 + 1; n2 + 1); vk(n1; n2)]

−max[vk(n1 + 1; n2); vk(n1; n2 − 1)] + R′2 − R2: (12)

It follows from (vi) that u′(·)6u(·) and from (iii) thatu(n1 ; n2+1)6u(n1 ; n2) and u′(n1 ; n2+1)6u′(n1 ; n2). As before, weconsider the right-hand side of (12) for each of the six feasi-ble combinations of the vector (u′(n1+1; n2); u

′(n1 ; n2); u(n1+1; n2);

u(n1 ; n2)):

(1,1,1,1) gives D2v′k(n1+1; n2)−D2vk(n1+1; n2)+R′2−R2,

which is ¿0 by (v).(0,0,0,0) gives D2v′k(n1; n2−1)−D2vk(n1; n2−1)+R′

2−R2,which is ¿0 by (v).

(0,0,0,1) gives v′k(n1; n2) − v′k(n1; n2 − 1) − vk(n1; n2) −vk(n1+1; n2)+R′

2−R2. This is¿D2v′k(n1; n2−1)−D2vk(n1; n2 − 1) + R′

2 − R2 (because u(n1 ; n2) = 1);which is ¿0 by (v).

(0,0,1,1) gives v′k(n1; n2)− v′k(n1; n2 − 1)− vk(n1 + 1; n2 +1) + vk(n1 + 1; n2) + R′

2 − R2; which equals−D3v′k(n1; n2) + D3v′k(n1; n2 − 1) + D2v′k(n1 +1; n2)−D2vk(n1 + 1; n2) + R′

2 − R2; which is ¿0by (iii) and (v).

(0,1,1,1) gives v′k(n1; n2)− v′k(n1 + 1; n2)− vk(n1 + 1; n2 +1) − vk(n1 + 1; n2) + R′

2 − R2: This is ¿v′k(n1 +1; n2 + 1) − v′k(n1 + 1; n2) − vk(n1 + 1; n2 + 1) −vk(n1 + 1; n2) + R′

2 − R2 (because u′(n1 ; n2+1) = 0),which is itself ¿0 by (v).

(0,1,0,1) gives v′k(n1; n2) − v′k(n1 + 1; n2) − vk(n1; n2)+ vk(n1 + 1; n2) + R′

2 − R2; which equals−D3v′k(n1; n2)+D3vk(n1; n2)+D2v′k(n1 +1; n2)−D2vk(n1 + 1; n2) +R′

2 −R2 and is ¿0 by (vi) and(v).

To complete the proof of the theorem, because v′k(n1; n2)−vk(n1; n2) converges to v′(n1; n2) − v(n1; n2), conditions(iv)–(vi) will hold for the value functions of the averagecost problem for both systems. Now note that condition (v)implies that if acceptance of type 2 orders at state (n1; n2)in System A is optimal, it must also be optimal in Sys-tem B. Similarly, condition (vi) implies that if productionof a type 2 product is optimal in state (n1; n2) with n2¿0in System A, then it must also be optimal in the same statein System B. Finally, note that the special case of n2 = 0 ofCondition (iv) implies that if production of a type 1 unit ispreferred over idling in state (n1; 0) for System A, then itwill also be preferred in the same state for System B.

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