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Optimal Control of a High-Volume Assemble-to-Order SystemAuthor(s): Erica L. Plambeck and Amy R. WardReviewed work(s):Source: Mathematics of Operations Research, Vol. 31, No. 3 (Aug., 2006), pp. 453-477Published by: INFORMSStable URL: http://www.jstor.org/stable/25151739 .Accessed: 29/04/2012 12:58

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MATHEMATICS OF OPERATIONS RESEARCH lAf THTlH

Vol. 31, No. 3, August 2006, pp. 453-477 OMILU?

issn 0364-765X | eissn 1526-547110613103 10453 doi 10.1287/moor. 1060.0196

?2006 INFORMS

Optimal Control of a High-Volume Assemble-to-Order System

Erica L. Plambeck Graduate School of Business, Stanford University, Stanford, California 94305, [email protected]

Amy R. Ward Department of Industrial and Systems Engineering, Georgia Institute of Technology,

Atlanta, Georgia 30332, [email protected]

We consider an assemble-to-order system with a high volume of prospective customers arriving per unit time. Our objective is to maximize expected infinite-horizon discounted profit by choosing product prices, component production capacities, and

a dynamic policy for sequencing customer orders for assembly. We prove that a myopic discrete-review sequencing policy, which allocates scarce components among orders for different products to minimize instantaneous physical and financial

holding costs, is asymptotically optimal. Furthermore, we prove that optimal prices and production capacity nearly balance

the supply and demand for components (i.e., it is economically optimal to operate the system in heavy traffic), so system

performance is characterized by a diffusion approximation. The diffusion approximation exhibits state-space collapse: Its

dimension equals the number of components (rather than the number of components plus the number of products). These

results complement the existing assemble-to-order literature, which focuses on managing component inventory and assumes

FIFO sequencing of orders for assembly.

Key words: assemble-to-order systems; functional limit theorems; diffusion limits; Brownian motion; state-space collapse; discrete-review policy; control policy

MSC2000 subject classification: Primary: 90B05, 60F17; secondary: 90B22, 90B35, 60J70

OR/MS subject classification: Primary: queues, diffusion models, limit theorems, inventory/production, policies, review/lead

times; secondary: probability, stochastic model applications

History: Received January 16, 2003; revised December 15, 2003, January 15, 2005, and October 5, 2005.

1. Introduction. Consumers are heterogeneous, and demand a variety of products. Manufacturers have

difficulty in forecasting consumer choice, and therefore incur high costs in carrying finished-goods inventory in

the wrong amounts, for the wrong products. For modular products like computers, an efficient alternative is to

assemble to order; i.e., to hold inventories of components that can be rapidly assembled into a wide variety of

finished products in response to customer orders. As the Internet and new information technologies have enabled manufacturers to sell directly to customers

rather than through retail outlets, assemble-to-order manufacturing has become increasingly prevalent. Dell, a

leader in direct sales and assemble-to-order (ATO) manufacturing, has grown by 40% per year in recent years,

although the PC industry as a whole has grown by less than 20% per year (Economist [14]). In addition to Dell,

companies as diverse as General Electric, American Standard (Bylinsky [10]), BMW (Economist [14]), Toyota (Economist [15], Forbes [17]), Timbuk2 (Plambeck [30]), and National Bicycle (Agrawal and Cohen [2]) either have adopted or are considering adopting an ATO approach.

In an assemble-to-order system, pricing, capacity management, and dynamic execution are very challenging. To maximize profit, the first-order challenge is to choose product prices (which determine revenue and the demand for each component) and to choose the production capacity (or supply contract) for each component.

However, due to stochastic fluctuations in supply and demand, component shortages will sometimes occur. Then,

the manufacturer must dynamically ration scarce components among customer orders for various products, and/or

pay to expedite component production. In practice, most firms adopt simple static rules to sequence customer

orders for assembly, such as FIFO or proportional allocation (Agrawal and Cohen [2]) and expedite production on an ad hoc basis (Perman [29]). Even Dell has only recently begun to sequence orders for assembly dynamically, using real-time information about component availability (Perman [29]). Dynamic control is complex because Dell offers thousands of different products (computer configurations) from hundreds of different components. Therefore, theory is needed to guide business practice.

This paper and its sequel undertake a holistic analysis of ATO manufacturing. Specifically, we show how to set product prices and component production capacity, and then dynamically sequence customer orders for

assembly and manage component inventory, in order to maximize expected infinite-horizon discounted profit. In this paper, the component production capacity equals the component production rate. However, in our sequel paper (Plambeck and Ward [31]) we allow for expediting extra components and salvaging excess components, so that the effective production rate of components differs from the component production capacity.

Our analysis begins with the deterministic analog of the system, ignoring stochastic variability in supply and demand for components. We solve a static planning problem: Set product prices and component production

453

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System 454 Mathematics of Operations Research 31(3), pp. 453-477, ?2006 INFORMS

capacities to maximize profit in the deterministic system. At the optimal solution to the static planning problem, the long-run average production rate for each component is equal to the long-run average rate of demand.

Hence, with stochastic variability, customers will experience delays and components will be held in inventory. Discounted expected profit is reduced from the optimal objective value in the static planning problem because

components are purchased before they are needed, and customers will not pay until their products are assembled.

However, assuming a high demand rate, we prove that the optimal prices and component production capacities are close to the solution of the static planning problem. That is, heavy traffic is the optimal operating regime.

Then, we can employ a diffusion approximation to the dynamic control problem. First, we characterize a

simple, near-optimal rule for allocating scarce components to outstanding customer orders (dynamic sequencing). Next, we slightly perturb the static planning problem solution to increase expected infinite-horizon discounted

profit. The key assumption in our analysis is that a high volume of potential customers arrives per unit of time.

Under this assumption, because the optimal production capacity for each component approximately equals the

optimal demand rate, the functional central limit theorem dictates that the inventory position of each component

(the number in stock minus the number required to assemble outstanding orders) changes gradually. However, the system manager can quickly adjust the customer order queue lengths. Consider Dell computer as a concrete

example. Its OptiPlex assembly plant assembles more than 20,000 computers per day (an order for a computer arrives every 4.3 seconds), but the queue for a specific configuration of computer will typically be less than 100.

By prioritizing that specific configuration for assembly, Dell could eliminate the queue in minutes. However,

queues for other products would increase.

We propose to review the system and release orders for assembly at short intervals of time. If one or more

components are in shortage, products are prioritized for assembly according to price and component require ments, so that the resulting configuration of queues minimizes the instantaneous financial "holding cost" for

delaying assembly (and hence payment). Because the inventory position changes slowly, queue lengths will

almost continuously track the minimum cost configuration. This discrete-review approach to dynamic control

follows that of Harrison [21] and Maglaras [24, 25], except that the decision at each review time point pairs available components with outstanding orders instead of allocating server time amongst customer classes.

The following summarizes the main contributions of this paper,

(i) We prove that for a high-volume assemble-to-order system, optimal prices and component production

capacity are close to the solution of the static planning problem; see Theorem 5.1. In particular, as in the systems studied in Maglaras [26], Maglaras and Zeevi [28], and Plambeck [30], heavy traffic is the optimal operating

regime, meaning that diffusion approximations are relevant.

(ii) With dynamic assembly sequencing, the system exhibits state-space collapse; see Proposition 4.1. Hence, in managing component production and inventory, one need not track the customer order queue length for every

product, only the component inventory positions. This reduction in problem dimensionality is very important, because a typical assemble-to-order system is designed to support thousands of different products from tens of

different components.

(iii) We provide an asymptotically optimal policy for statically setting prices and component production

capacity, and dynamically sequencing orders for assembly; see Theorem 5.1.

The remainder of this paper is organized as follows. We first review some relevant literature. Next, in ?2, we discuss the model formulation. Section 3 sets up the static planning problem and introduces our assembly

policy. Section 4 analyzes the asymptotic behavior of the system under the "first-attempt" policy specified in ?3.

Finally, we show that nearly balanced systems are optimal, and provide the appropriate "stochastic adjustment" factor to prices and component production capacity in ?5. Appendix A contains all of our lemma proofs, and

Appendix B provides a table of notation.

1.1. Literature review. Song and Zipkin [36] provide an excellent survey of the literature on managing ATO systems. This literature is focused on the management of component inventory, taking the demand process and distribution of component lead times as given, and assuming that customer orders must be filled FIFO.

The subset of this literature that adopts a discrete-time formulation must further specify how to allocate scarce

components among orders that arrive simultaneously, i.e., in the same period. Each of the following three

papers assumes a different rule. In Agrawal and Cohen [2], scarce components are allocated "fairly" so that

in each period, the fraction of demand that is satisfied is the same for every product. In Zhang [41], orders

that arrive in the same period are prioritized according to price. Akcay and Xu [1] optimize the allocation of

components (among orders that arrive in the same period) to maximize the fraction of orders assembled within

the quoted maximum delay. All three papers assume independent base-stock control of component inventory,

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System Mathematics of Operations Research 31(3), pp. 453-477, ?2006 INFORMS 455

and optimize the base-stock levels. They find very different optimal base-stock levels, which suggests that the

optimal inventory policy is very sensitive to the rule for allocating scarce components among orders for various

products. Akcay and Xu [1] recommend that "inventory replenishment and component allocation optimizations must be made jointly" (p. 110). Clearly, these decisions are also sensitive to product prices and component lead

times. This motivates our model formulation, which incorporates product prices, component production capacity, and the dynamic sequencing of orders for assembly as decision variables. We find that the optimal allocation

of components to products (sequencing) collapses the state space, so the joint optimization recommended by

Akcay and Xu [1] becomes more tractable than inventory optimization in isolation.

In an ATO system with only one component in shortage, the assembly-sequencing decision is similar to the

decision of which class to serve in a multiclass, single-server queue. Much of the research in this area combines

sequencing with dynamic lead-time quotation. Two representative publications are Wein [39] and Duenyas [13], and each of these papers provides an extensive review of related literature. When customers must be served

within a class-specific maximum delay but production cannot be expedited (our follow-up paper Plambeck and

Ward [31] has delay constraints and expediting), then some customers must be turned away. Maglaras and Van

Mieghem [27] and Plambeck et al. [32] consider dynamic sequencing and admission control in this setting. In practice, many firms accept all customer orders, but set a lower bound on the fill rate (fraction of customer

orders assembled within the quoted maximum delay). Researchers have provided methods for optimizing the

base-stock level for each component in order to minimize long-run average inventory holding costs, subject to

the lower bound on the fill rate. For example, when component lead times are i.i.d. random variables (the relevant

model formulation when transportation comprises most of component lead time), Song [35] and Lu et al. [23] show how to calculate fill rates, and Cheng et al. [12] go on to optimize base-stock levels. When component

production capacity is limited by supply contracts and/or physical capacity constraints, lead time increases in the

number of components ordered. To capture this, as in our formulation, Song et al. [37], Glasserman and Wang

[19], and Wang [38] explicitly model the component production facilities as single-server queues. Glasserman and Wang [19] undertake an asymptotic analysis as the fill rate approaches one, and establish a linear relationship between the target maximum delay and the component base-stock levels.

Our follow-up paper (Plambeck and Ward [31]) assumes a fill rate of exactly one. To make this feasible, we

allow the system manager to expedite components instantaneously at a high per-unit cost. In this setting, we

provide an asymptotically optimal policy for statically setting prices and component production capacity, and

dynamically sequencing orders for assembly and expediting components. Our control problem becomes even more challenging when the system manager can dynamically sell excess components at a low "salvage" price. We heuristically derive and numerically solve an appropriate approximating diffusion control problem.

2. Model formulation. Consider a system in which J different components are assembled into K different finished products, as shown in Figure 1. Product k = 1,... ,K requires a positive, integer amount of type 7

= 1,...,/ components equal to

akj. Every component j is required for the assembly of at least one product, so that for any k, akj

> 0 for at least one /. Assembly is instantaneous, given the necessary components.

Stochastic demand

X V V V

- - - AT products

f Assembly )

/ \ / \ / \ / \ J components

A A A A

y

Stochastic production

Figure 1. The assemble-to-order system.

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System 456 Mathematics of Operations Research 31(3), pp. 453^77, ?2006 INFORMS

At time t = 0, the system manager chooses product prices px,... ,pK and component production capacity

yx,..., jj. Then, given the vector of product prices p,x customers order product k according to a renewal

process Ok with rate \k =

kk(p). Specifically, Ok(t) denotes the cumulative number of orders for product k that

arrive before time t. A customer pays pk when his order for product k is filled. Components of type j arrive

at the assembly facility according to a renewal process C; with rate y- (the component j production capacity).

Specifically, Cj(t) denotes the cumulative number of components that arrive before time t. Each component has

an associated unit production cost, c; > 0, paid upon the delivery of the component. Pricing and component

production capacity decisions are irreversible, and remain fixed throughout the time horizon. Observe that our

formulation allows for customer migration between products; i.e., customer demand for any one product may

increase as the prices of other products increase.

Next, the system manager must dynamically determine when and in what sequence to assemble outstanding

product orders. Depending on relative prices, holding costs, outstanding orders, and component inventory levels, it may be advantageous to prioritize assembly of one product over another. Without loss of generality, we can

assume that customer orders for each product are filled FIFO. Therefore, specification of the cumulative number

of type k orders assembled in [0, t], which we denote by Ak(t), uniquely determines the order queue lengths at

time t,

Qk(t) = Ok{t)-Ak{t), k=\,...,K, (1)

and the component inventory levels at time t,

Ij(t) =

Cj(t)-j:akjAk(t)>0, j=l,...,J. (2) k=\

Component inventory incurs a linear physical holding cost. Let hj denote the physical holding cost per component of type j per unit of time.

The objective is to maximize the expected value of the infinite-horizon discounted profit2

a oo J p oo

n ^ E / Pke~8t dAk(t) -

? / e-8t(cjdCj(t) + hjlj(t) dt),

which, using the Reimann-Stieltjes integration by parts theorem and the definition of the queue length and

inventory processes in (1) and (2), can also be written in the following form, which is more convenient for

analysis

.oo/* J \ -oo

/ K J \

n=/ ^e-8t(Y,PkOk(t)-j:cjCJ(t))dt- e-8t(j:8pkQk(t) +

j:hjIj(t))dt. (3) J? \k=i j=\ / J? \k=\ j=\ /

Because each customer pays when his order is filled, delay in order assembly decreases discounted revenue. The

system manager could avoid this loss of revenue by holding large component inventories. However, he would

then purchase components long before using them, causing him both to tie up a large amount of capital in

component inventory, and to incur excessive physical inventory holding costs.

An admissible policy u = (pu, yu, Au) specifies the product prices, component production capacity, and the

sequencing rule for assembly. We require that pu and yu are nonnegative. Also, the process Au is integer-valued,

nondecreasing, nonanticipating, and has Au(?t) = 0 for all t > 0. For clarity of presentation, we do not subscript

the system processes associated with a particular policy whenever the policy under consideration is clear from

the context. The reader is to understand the implicit dependence of the system processes on the policy. In general, our stated goal of maximizing (3) by choosing product prices, component production capacity, and

the sequencing rule for assembly appears intractable. However, under the high-volume conditions introduced in

?4.1, our goal is achievable.

Our analysis requires a few standard assumptions on the demand function A. First, $p) is continuously

differentiable, and the Jacobian matrix [dXk/dpm]k m=1 K is invertible everywhere. The inverse function the

orem then guarantees that the inverse demand function p($ is unique, continuous, and differentiable. Second,

customer demand for any one product is strictly decreasing in the price of that product, but is nondecreasing

1 We do not explicitly use notation, such as v, to indicate v is a vector; the reader should be able to differentiate between scalars and vectors

from the context.

2 All integrals appearing in this paper should be interpreted as Riemann-Stieltjes integrals in the usual sense.

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System Mathematics of Operations Research 31(3), pp. 453^177, ? 2006 INFORMS 457

in the price of different products, so d\k(p)/dpk < 0 while d\k(p)/dpm > 0, m^k. Third, ?*=1 d\k/dpm < 0

for each k = 1,..., K, which means that demand for each product decreases when all products' prices increase

by the same amount. Finally, we assume that the revenue rate r(A) =

J2k=i ^kPkW *s strictly concave, as in

Gallego and van Ryzin [18]. We conclude by stating a lemma useful for later analysis. We delay its proof, which mimics that of Lemma 2

in Bernstein and Federgruen [8], until the appendix, where the reader can find the proofs of all the lemmas in

this paper.

Lemma 2.1. The assumptions

_^M<0, ^M>0, m#* and ?^<0 for each k=l,... ,K (4) tyk dPm m=l dPm

imply

% < 0 and

^- < 0, m^k.

3. The static planning problem and the proposed assembly policy. Subsection 3.1 derives prices and com

ponent production capacities that maximize the profit rate, assuming that demand and production flow at their

respective mean rates. This yields a first-order approximation to the optimal prices and production capacities in the system with stochastic variability. In ?3.2, we propose a discrete-review assembly policy that minimizes instantaneous financial holding costs at each review point by distributing components to product orders.

3.1. An initial policy for setting prices and component production capacities. Suppose that in initially set

ting product prices p and component production capacity y, the system manager ignores the discrete and stochas tic nature of customer orders and component production, and simply assumes that demand and production flow at their long-run average rates. Then, to maximize the profit rate, he chooses prices and component production

capacities according to the solution of the following static planning problem: K J

* - p^a*>0EPkh(p)

~ ? 7jcj> (5)

subject to the constraint that all customer orders must be filled

K

Y,akjh(p)<7j> ; = 1,...,/. (6) *=i

In preparation for later analysis, we state the following lemma, which establishes properties of the solution to

(5H6). Lemma 3.1. The unique solution to the static planning problem (p*, y*) has

j

pl > ? cjakj > 0 for every k = 1, . . . , K, and y* > 0. (7)

j=\

Furthermore, suppose we relax the constraint (6) to yield the following perturbed problem:

if ($)= max nE/>A(p)-?y,c,, (8)

subject to K

EVl(P)^; + ?j. j=l,...,J. (9)

If Oj < y*, j = 1,. . . , /, the perturbed problem (8)-(9) has a unique optimal solution (p*(0), y*(6)) such that

j

P\0)=p\ y\B) =

y*-0, and W(0) -Tr = Y<cjej

The optimal objective value in (5) upper bounds the expected profit rate, which implies 8~l7f upper bounds

expected infinite-horizon discounted profit in the stochastic system. Due to stochastic variability, customers will

experience delay, and components will sit in inventory, meaning the upper bound is not, in general, achieved. However, ?5 establishes that under high-volume conditions, the optimal prices and component production capac ities are close to (p*, y*), and infinite-horizon discounted expected profit is close to 8_177.


3.2. The proposed assembly policy. The assembly policy we propose is a discrete-review policy that

releases orders for assembly at review time points, and does nothing at all other times.3 We let /, 21, 31,... be

the discrete-review time points, where the review-period length

VIA-M

depends upon the average time between order arrivals, and || is the Euclidean norm. Hence, the review-period

length will become short in high volume.

Initially, the number of orders assembled by time 0, A+(0), is zero. At each subsequent review time point, we

allocate available inventory to product orders so as to minimize instantaneous holding costs, given the shortage of each component. The shortage4 process tracks the difference between the number of components required to

assemble all outstanding orders and the number of components in inventory, defined as

Sj(t) =

Y,?vOk(t)-Cj(t), j = i,...,j, (io)

k=\

= Z?uQk(t)-ij(t), j=h...,J. (li)

k=\

Let the functions (which we derive in the paragraphs following) Q*(S, O, A) and /*(?, O, A) denote the arrange ment of queue lengths and inventory levels that minimize instantaneous holding costs when the shortage process is S, the cumulative number of order arrivals is O, and A is the number of orders that have been assembled so

far. Then, the assembly process A+ is defined recursively as follows:

A.(il) = O(il) -

Q*(S(il), O(il), A.((i -

1)/)), (12)

recalling that A+(0) = 0. No order assembly occurs between review time points. Our discrete-review policy

contrasts with those in the work of Harrison [21] and Ata and Kumar [4] because a time allocation is not relevant

to our situation.

When the solution to the linear program

nun ??>;& +?>,./; (13) ^' -

k=\ 7=1

subject to

Ij =

T,*kjQk-Sj>0, j = l,...,J, (14) k=\

0-Q>A (15)

is unique, Q* and /* are exactly the solution to (13)?(15). Here, constraint (14) requires that inventory levels be

nonnegative. Constraint (15) prevents already-assembled orders from being disassembled.

Otherwise, when multiple solutions to (13)?(15) exist, we examine constraint (15). If (15) binds for any one

of the solutions, we arbitrarily choose a solution to use to define Q* and /*. Otherwise, if (15) is slack for every

solution, we carefully select a solution (q*, i*) to the linear program in (13)?(14), ignoring constraint (15) as

follows.

Assumption 3.1. The function q* is chosen as a Lipschitz continuous function ofthe shortage process S. In

particular, for any Sl e diJ and S2 e$iJ,

max \qUSl) -

qUS2)\ < k max \S) -

S2\, (16) k=l,...,K]

* * ' ./=1,...,/'

J J '

where k is a finite constant.

3 For analytic convenience, as is standard in the assemble-to-order literature, we assume that assembly is instantaneous. If assembly time is

a constant rk for each product k, then our analysis goes through with e~8Tkpk substituted for pk. 4 The shortage is the negative of the inventory position, the term commonly used in the operations management literature. We use the term

"shortage" to focus attention on how scarce components are allocated to outstanding orders.

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System Mathematics of Operations Research 31(3), pp. 453-177, ? 2006 INFORMS 459

In the case that the solution to (13)?(14) is unique for every S, Theorem 10.5 in Schrijver [34] (which is based

on Hoffman's lemma) establishes (16). To see when uniqueness for every S occurs, observe that the objective function (13) can be written as

m-^hjSj,

where

k=\ \ j=\ /

subject to K

ZakjQk>Sj, j=\,...,J. k=\

The linear program defining / has a unique solution whenever the vector of "cost coefficients"

(ri + ?Viy..--.rt + EW) (17)

is not parallel to (<z*/)*=i,...,je>

tnQ vector of products requiring component j, for every j =

1,.. . , /.

If the vector in (17) is parallel to (akj)k=u_K for some j, then the solution to (12)?(13) is not unique for some S. Nevertheless, one can identify a Lipschitz solution q*(S) by perturbing the price vector p* very slightly, so that the vector in (17) is not parallel to

(akj)k=\,...,K f?r anv ./= 1,...,/, yet the (unique) optimal solution to the perturbed linear program is an optimal solution to the unperturbed linear program. Once equipped with the q*(S), one should proceed with the original (unperturbed) p* for the remainder of the analysis. Existence of a Lipschitz q*(S) is proven rigorously in Proposition 2 of Bassamboo et al. [5]. For additional explanation of

the issue regarding the construction of a Lipschitz continuous function, see Berens et al. [7]. The linear program in (13)?(15) does not account for the effect of assembly in the current period on future

holding costs due to constraint (15). Nevertheless, the myopia of the proposed policy does not cause problems in the high-volume limit. Intuitively, when the system experiences a high volume of demand and component

production, the functional central limit theorem dictates that the shortage process in (10) changes gradually compared to the rate at which orders arrive and component production occurs. Therefore, review-period lengths can be set so that even though many orders and components arrive in each review period, the shortage position only experiences small changes in each review period. Hence, one expects constraint (15) to often be inactive in high volume, which means queue lengths and inventory levels generally track (q*(S(t)), i*(S(t))). In other

words, they are almost deterministic functions of the shortage process; see Proposition 4.1.

4. System behavior under the static planning problem solution and the proposed assembly policy. To

analyze the performance of a policy that sets prices and component production capacities using the solution to the static planning problem in ?3.1, and assembles orders according to the proposed assembly policy in

?3.2, we undertake an asymptotic analysis as the customer arrival rate increases. We use the colloquial term

"high volume" to refer to a system with a high order arrival rate as defined in ?4.1. Subsection 4.2 shows that our proposed assembly policy forces queue lengths and inventory levels to track deterministic functions of the

shortage process with very high probability. Finally, ?4.3 establishes the asymptotic behavior of the shortage, queue length, and inventory processes, and of expected infinite-horizon discounted profit.

4.1. High-volume conditions. The one modelling assumption necessary for our asymptotic analysis is that the system experiences a high volume of demand. Specifically, consider the sequence n = 1, 2,... . Let order arrival rates tend to infinity in a manner that preserves the structure of the demand functions, as follows:

K(p) = n\k(p), k=\,...,K. (18)

Henceforth, when we wish to refer to any process or other quantity associated with the assemble-to-order system having order arrival rate function A", we superscript the appropriate symbol by n. An admissible policy refers to an entire sequence, u =

(pnu, ynu, AJJ), that specifies an admissible policy for each n. We let AJJ^ ==

A?(/?JJ). In

high volume, the review-period length is / 1 \2/3

in~\W\) '


The following technical specifications are needed. Let (H, W, P) be a probability space. For each positive

integer /, let Dl be the space of all functions <o: [0, oo) -> 9ft1' that are right continuous with left limits. Consider

Dl to be endowed with the usual Skorohod-^ topology (see, for example, Ethier and Kurtz [16] or Billingsley

[9]), and let Ml denote the Borel ^--algebra on Dl associated with this topology. All stochastic processes in this

paper are measurable functions from (Ci, SF, P) into (Dl, Ml) for the appropriate dimension /. For a sequence of stochastic processes {?n}, each of dimension /, the notation gn => ? means the probability measures induced

by ?n on (Dl, Ml) converge weakly to the probability measure induced by ? on (Dl, Ml). We represent the renewal processes Ox,..., 0K and Cx,... ,Cj in terms of the K + J independent sequences

of mean nonnegative random variables {xk(i), i = 1, 2,...}, k = 1,..., K and {yfi),

i = 1,2,...}, j =

1,.. . , J having

Var(*t(l)) = oS,t and Var(y/1))

= <7^

as follows. Let

m m

9?t(?0 =

?**('). * = 1,...,AT and <y/m) =

?y/i), j = \,...,J, 1=1 1=1

and define

Onuk(t) = max{m

> 0: 9?t(m) <

AJy} and CnuJ(t)

ee max{m > 0: %(m)

< nynujt}.

We assume that E|jc^(l)|2+e < oo, k = 1,..., K and E|^(l)|2+e

< oo, j = 1,..., / for e = 3. Normal, truncated

normal, uniform, and Erlang random variables satisfy this condition, as do Weibull and gamma random variables

for a restricted set of parameters. We use the assumption of finite 2 + s moments in the proof of Lemma 4.1

to obtain a probabilistic bound on the number of orders and components arriving within a review period. The

work of Ata and Kumar [4] implies that the minimal assumption necessary for our analysis is s > 0, but we set

s = 3 to avoid defining the review-period length in terms of s, which simplifies and shortens our presentation. It is convenient for our analysis to also define the scaled and centered processes

?:*? =

V^n-'O^M -

WK>

and

C"uJ(t) =

Vi(n-lC"uJt)-y]t).

4.2. Reduction in problem dimensionality. If the constraint (15) that orders cannot be disassembled did

not exist, our assembly policy's myopia would not be worrisome. Then, queue lengths and inventory levels

would minimize instantaneous holding costs at all review points iln, i = 0,1, 2,..., meaning they would equal

(q*(Sn(iln)),i*(Sn(iln))). In high volume, our next proposition shows that with very high probability, this is

exactly the case. In particular, our assembly policy achieves a reduction in problem dimensionality because queue

lengths and inventory levels are, with very high probability, deterministic functions of the shortage process.

Define/n=Ll//nJ =

LI?A*|2/3J.

Proposition 4.1. Under any policy u with pnu ?

p* and y% -> y* as n -> oo, there exists a constant /3 such

that under the proposed assembly policy

P(Qn(iln) =

q*(Sn(iln)) for all i = 1, 2,. . ., F) > 1 - /3n"1/6.

The proof of Proposition 4.1 requires the following lemma, which bounds the number of orders and compo

nents arriving during each review period.

Lemma 4.1 (Ata and Kumar [4]). Under any policy u with pnu ->

p* and ynu ->

y* as n -> oo, for each

k = 1,. .. , K and j

= 1,... ,J, and for any finite constant a > 0, there exists a constant /3 > 0 such that

P( max max \Onk((i + l)ln) -

Onk(iln) -

n\k(pnu)ln\ < anl/3)

> 1 -j8n"1/6 (19) V=0,1,...,/"

? 1 k=l,...,K /

and

p( max max |C?((i + 1)/B)

- C"(iln)

- nynu Xn\ <

anx/3) > 1 -j8n"1/6. (20)

\/=0,l,...,/"-l ;'=!,...,J J J >J /

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System Mathematics of Operations Research 31(3), pp. 453-^77, ? 2006 INFORMS 461

Proof of Proposition 4.1. Fix n, let

. i_a;_i_ a =

.=^.%2|A12/3l+Kmax;=1,...(y(l+EL^),

and consider a sample path co e ft on which

max max \Onk((i+ \)ln) -

Onk(iln) -

Xk(p")nlnI < an1'3 (21) i=0,l,...,/"-l k=\,...,K* kKX / ' kX ' kyru/ '

and

max max |C?((i + l)/n) -

CUil") -

yu ,/iZn| < anl/3. (22)

Because for every i = 0,1,..., In

? 1

XX*<?WO) ^ XX*?(S"('H. 0"(?7"), A?((i -

1)/")), k=\ k=\

if

own -

qtmin) > >w - i)n (23) for each k = 1,..., K and every / = 1,. . ., In, so that constraint (15) is not violated at ql(Sn(iln)), then

Qn(iln) =

Q*(Sn(iln), On(iln), An((i -

1)/M)) =

q*(Sn(iln)).

We use mathematical induction to show (23) holds for every / = 0, 1,.. ., In ? 1. For / = 1, observing that

g*(0) = 0, use (21), (22), and the Lipschitz continuity of q* to find

Onk{l") -

q'k(S"(n) >

hiPDnl" ~ an1" -k max

|s;(/")| 7=1,-..,./ J

> \k(pnu)nln - anl/3 - Kanl/3 max

( 1 + ?># )

= ""1(^F-?(1+^?J(1+I',?)))

> 0 = Ank{Q) (for large enough ?),

for k = 1,..., K. Hence, ?>*(/") =

q*(S"(ln)). Next, assuming 2*(?7") =

q*(Sn(iln)) so that A"(i7") =

On{iln) -

q*(S"{iln)), the following similar argument

o"k((i+i)n-qi(s"(a+i)n) = oj((i+1)/?)

- o;(i/-) + own+ isn{un))

- q;(s"(ir))

- q*k(s"((i+1)/"))

= o;((i +1)/") -

own + 4{sn{iin)) -

q'k(sn((i + i)i")) + A'k(un)

^B,^(^-K1+v^X1+liatf)))+Ail(,'n > Ank(iln) (for large enough n)

establishes Q*((i+ l)ln) =

q*(Sn((i+ l)ln)). Lemma 4.1 guarantees there exists a constant /3 such that the prob ability of having a sample path that satisfies (21) and (22) exceeds 1 ?

/3n~1/6, which completes the proof.

4.3. Asymptotic behavior of the system. Proposition 4.1 is the key to showing the proposed assembly policy achieves a reduction in problem dimensionality in high volume. In particular, our first theorem, which establishes the asymptotic behavior of the system, shows the following.

(i) Queue lengths and inventory levels viewed under diffusion scaling are approximately deterministic func tions of the shortage process in high volume.

(ii) Expected infinite-horizon discounted profit in the nth system is within y/n of the upper bound from the static planning problem, 8~xnif.


In preparation for the statement of Theorem 4.2 and later results, define T to be the / x / matrix whose

(/, y)th entry is

r,? = E WkjK<k + YylMi=j}. (24)

*=1

The matrix T is the covariance matrix for the Brownian motion appearing in Proposition 4.2 as the heavy-traffic limit for the shortage process. The following uniform integrability result, which employs arguments similar to

those necessary to establish (155) in the proof of Theorem 5.3 in Bell and Williams [6], is also useful.

Lemma 4.2. Under any policy u with pnu ?

p* and y" -*

y* as n -> oo, for each k = 1,..., K and j

=

j l??e~8t sup \Onuk(s)\dt, n>0\ and j f?? e~8t sup \Cnu Js)\dt, n>0\ \h 0<s<t

' J [J0 0<s<t

' J

are uniformly integrable families. Furthermore, as n -? oo, for each k = 1,. . . , K and j =1,. . . ,J,

E j"e~8t6nk(t)dt-+0

and E f??e~8tCnu .(f)dt -> 0.

Jo Jo

Recall that H"pit r An) is the infinite-horizon discounted profit (defined in (3)) for the system having arrival

rate n\*k, operating under the policy (/?*, y*, A").

Proposition 4.2. Under the policy (p*,y*, A"), for B a Brownian motion with 0 drift and covariance

matrix T, as n ? oo,

(a) (Sn,Q\In)/V^ => (B,q*(B),i*(B)), (b)

[nn ? 8~ln7f~\ r r?? / K J \ "I

(P''r'^-J -

-E[/o e-s,(ZptM(B(t)) +

T,hji*(B(t)))dt .

Proof. Proof of (a): By the functional central limit theorem (see, for example, Theorem 5.11 in Chen and

Yao [11]),

Ol = ^K<kB?k

and CJ => fi^ltf,

where Bk, k = 1,..., K and Bf, j = 1,..., J are K + J independent standard Brownian motions. Therefore,

by the continuous-mapping theorem, any linear combination of the components of n~l/2Sn weakly converges as

follows:

J Sn

J / K \

j=\ vn j=\ \k=\ /

= il*j(i:?Vyf^tf-y[^jBf) j=\ \k=\ /

for each (xx,..., Xj) e diJ. Furthermore, for any / = 1,..., J and j = 1,..., J,

Cov(E akiJA^X ~ Jji<iB?, E akjyJK<kB0k

- fitfCtf) \k=l k=\ '

K

= E wvKvo.k + y^lM1=J}

k=\

Theorem 7.7 in Billingsley [9] (the Cramer-Wold device) now establishes

cn ~ => B, (25)

<s/n

where B is a /-dimensional Brownian motion with mean 0 and covariance matrix T defined in (24).


Because assembly only occurs at discrete-review time points, for any k = 1,. . ., K,

s^y^Ql(t)-ql(^)

< max n-x/2Ql(iln)-ql(^P-\ +n~l/2 max \Onk((i+\)ln)-Onk(iln)-n\lln\ ~/=o,i,...,i//?

* k\ Jn ) /=o,i,...,i//wl

* k\ / k \

JSn([t/ln\ln)\ JSn(t)\ 1/6 A* + sup qU VL/_J

J )-qk\ ?)=- +? 7

rrr o<r<i k\ Vn / V yfn ) |A*| ->0

in probability, as n -> oo, by Proposition 4.1, the observation that for any S e di, q*(n~l/2S) =

n~l/2q*(S), Lemma 4.1, and the fact that iln, / = 0,1, 2,..., (ln)~l becomes dense in (0,1). Because by the continuous

mapping theorem q*(n~l/2S) => q*(B), we conclude

-?= => q*V)9 (26) yjn

as n ? oo.

Recall from (11) that scaled inventory levels are represented as a linear combination of scaled queue lengths and the scaled shortage process. Hence, the continuous-mapping theorem establishes

4= =* T.^q\B)-B

= i\B) (27) vn k=\

as n -> oo. Finally, the stated joint convergence follows because q* and i* are deterministic functions.

Proof of (b): The representation for infinite-horizon discounted profit in (3) shows

n; -s-'nw _

,.

.? 0;(,) ?a;,

_? c;M-?t;,n t

Vn Jo \~ yfn ]T[ J

Vn J

4> \k=l Vn j=} V" J

From our assembly policy, because akj

assumes integer values, for any t > 0,

e*w * ^^Mli}")}+?"kit)

_ KliJ'")

<E^(^J'")+^w-^([^J/") ^ E(XX suP |Ojf(j)-nA^|)+ sup |C;(5)-ny;*|

7=1 \k=l 0<5<r / 0<5<r

+ 2 sup \Ol{s)-nKs\+n^-^-. (29)

Hence, Lemma 4.2 guarantees {/0?? e~8tn~l,2Qnk(t) dt) is less than the sum of uniformly integrable families, and so forms a uniformly integrable family for each k = 1,..., K. Also, because


the bound in (29) and Lemma 4.2 establish that {/0?? e~8tn~l/2IJ(t)dt] forms a uniformly integrable family for

each j = 1,..., J. Therefore, part (a), the continuous-mapping theorem, and the now-justified interchange of limit and expectation show, as n -> oo,

E fMe-8t^^dt]Ê\fê'8tql(B(t))dt\

k=l,...,K, (30)

E J e-t'^jJ-dtl'+El] e-8ti](B(t))dt\, j=l,...,J. (31)

Lemma 4.2, and the convergences in (30) and (31), establish the convergence of the expected value of (28), i.e., that

rlT?. . ,?, -8-lnrrl r r?? / K J

\ "I

Ep^->-E / e-* Eri8?W) + EW0) )dt L V" J L/o V=1 y=1 / J

Observe that the proof of Proposition 4.2 goes through with only a small technical modification when prices and component arrival rates are adjusted slightly in the nth system, but the assembly policy A" is unchanged. Define the capacity imbalance of the system under a given policy u:

k=l

Because Sn (t) K

-^ =

EÂ(t) -

cj(t) + V^teij, (32)

provided

y<,->0,-, (33)

as n ~> oo, the same arguments used to establish the weak convergence in (25) show

cn

^=*Be, ?Jn

as n -> oo, where Be is a Brownian motion with drift 8 and covariance matrix T defined in (24). When pnu -> p* and y% -> y* as n -> oo, Proposition 4.1 and Lemma 4.1 hold. Therefore, the arguments establishing the weak

convergence of the scaled queue length and inventory processes in (26) and (27) remain valid. Furthermore, the

arguments establishing the convergence of expected values in (30) and (31) remain valid, and we can conclude

Eff We<*S%^ + I>;^#) A -+ *W. (34) yo V?J v? ,t? v? / J

where V?{6) represents the expected discounted "holding cost" caused by stochastic imbalance in supply and

demand for components

X(0) s e[ ("V'JEKHW')) + EhjiWt))) dt\ (35)

Define

rrn __ v/^m ? 7m ? ^ /_

= fse-t'tZpl&Jt)-icjCljit))

dt

+s-iMi:pnu,Mpnu) -i:^ -

*). V=i ;=i / .


When pnu and ynu solve the perturbed static planning problem so that pnu =

p*(6nu) and ynu =

y*(d"), by Lemma 3.1,

k=\ j=\ j=\

and so Lemma 4.2 and the convergence in (34) shows

E[wu}-+8-xY,cjeJ-w(d), 7=1

as n -? oo. Observe from the definition of I1JJ that E[njJ] and its limiting value are negative because 8~lnTT is

an upper bound on the expected infinite-horizon discounted profit. We have now established the following corollary to Proposition 4.2.

Corollary 4.1. Let Be be a Brownian motion with drift 6 and covariance matrix T defined in (24). Under a policy u =

(pnu,ynu, A?) having

Vh~0nuj-+dj, j=\,...,J

as n ? oo, and /?JJ =

p*(0D and y? =

y*(0^) so that pnu and y" solve the perturbed static planning problem in

(8)-(9), as n-+oo,

(a) (Sn,Q\In)/Vn~ => (Be,q*(Be),i*(Be)),

(b) E[njj]-+8-*e*j-iCj0j-x(0).

5. An asymptotically optimal policy. Adjusting product prices and component production capacities poten

tially improves the performance of the policy (/?*, y*, A") in high volume. In particular, Corollary 4.1 suggests that any asymptotically optimal policy u should have

lim E[ii^] = sup JS"1 ^CjOj

- X(6)\. (36)

eenn j=i J

Unfortunately, guaranteeing the existence of a unique maximizer of the right-hand side of (36) is difficult, because for a nontrivial portion of the parameter space, the function

*=1 7=1

is not convex in S. (We show a large region of the parameter space in which convexity does not hold in ?5.2 of Plambeck and Ward [31].) However, the following lemma provides the basis for the definition of a policy that is asymptotically optimal.

Lemma 5.1. The set of maximizers

@* = {6

e diJ: 8-lJ2cjdj -

X(0) > 5"11>^

- X(ff) for all 6'emJ\

I 7=1 7=1 >

is nonempty and

?* C [-Cil8X(0)(l +nutt{c/}/A), 8X(0)/a]

x x [-cJl8X (0)(l

+ max{c/}/A), 8X(0)/b\,

where

4=?r.4("Mi:^)/(|;%)). Recall from Lemma 3.1 that p\

- Y!j=\ akjCj

> 0 for all k, so A > 0. The main result of this paper both establishes that optimal pricing and capacity decisions create heavy-traffic

conditions, and provides an asymptotically optimal policy. Definition 5.1. A policy is said to be asymptotically optimal if it is admissible and

liminf E[fi;] > limsupE[f[^] (37) n-?oo n?xx>

for any other admissible policy u.


Our definition of asymptotic optimality captures deviations from the upper bound on infinite-horizon expected discounted profit in the stochastic system of order yfn. To see that this scaling is the appropriate one, first

realize that the functional central limit theorem applied to the shortage process in (32) shows shortage process fluctuations are inherently of order yfn. Hence, order queues and component inventory levels also may be of

order yfn. Therefore, under any admissible policy, infinite-horizon discounted profit will not be closer than order

yfn to its upper bound.

Theorem 5.1. For 0* e 0*, the policy * = (p\ y*

- n~l/26*, AnJ having

j

lim E[ii;] = y;c;^-?(?*) n^-oo f?f J J

y=i

is asymptotically optimal. Furthermore, under any asymptotically optimal policy a (in particular, under any

policy that is optimal in the nth system for every n),

Pl ~+ P*> 7na -> r*> and dist[y/7tdna, 0*] -> 0, (38)

where

IrE^W.)-):,;, 7 = 1,..., J. k=\

The notation

dist[p, S] = inf \p

? s\ seS

denotes the Euclidean distance between a point p and a set S.

We devote most of the remainder of this section to proving Theorem 5.1. After its proof, we discuss how to

compute 76(d) for a given 6 e #t. To begin, we establish the following intermediary result that shows prices and

component production capacities must converge to the static planning problem solution.

Proposition 5.1. Under any admissible policy a that is asymptotically optimal,

pl -* p* and yna -> y\

as n-+ oo.

Proof. An asymptotically optimal policy u has

liminfE[n;]> limEtfV^.)] n->-oo n-?oo VK ' *'

L^o \k=\ j=\ / J

where the equality follows from Proposition 4.2(b). Dividing both sides of the above inequality by yfn implies

[ftn "i rn* "i ?? = liminf E ^^^ -8-{7f >0. (39)

yfn] n^??

|_ n J

The representation for infinite-horizon discounted profit in (3) implies

n Jo \k=i vn j=\ \ln /

+5-1(EK,At(^)-EcX7-^) \*=i j=\ '

- rse-s'(f:p:,k^^-)dt. (40) Jo \k=1 n }


Define

K

f(S,p)= min YsPkQk Q k=\

K

subject to: ? akjQk >

[Sj]+, j=l,...,J, k=i

Qk>0, k=l,...,K.

The representation for the shortage process in (11) and the fact that queues are nonnegative imply

for all t > 0. Therefore, because the functional strong law of large numbers (FSLLN; see, for example, Theorem

5.10 in Chen and Yao [11]) implies

Qn Qn

-^->0, k =

\,...,K and -^

-> 0, j =

\,...,J, a.s., Vn Vn

uniformly on compact sets, as n ?> oo, by (40),

By the FSLLN, for any t > 0,

sup . max n~lSnuj(s)

-

(e^M

" r^V

-* 0, a.s.,

and so, also a.s.,

liminff^-^-S-'iA

minr1 (E<,(A(K)

- ft) -][>,<;) q \k=\ 7=1 /

subject to: _ _ < liminf? ?6 !7T. (41) n^oo K r K -i +

E ?*,<?*> E?*A(pi:)-y?", 7=1,....-/, Jt=l U=l J

qk>0, k = l,...,K,

By comparing the linear program in (41) with the static planning problem (5)-(6), we conclude that (41) is nonpositive. Furthermore, because (/?*, y*) is the unique optimal solution of the static planning problem, (41) equals zero if and only if

Pu -> P* and yl ~> 7* as n -> oo. (42)

Taking expected values of both sides of (41) and observing that (39) lower bounds

liminfEr^-^^-g-1^ n^oo \_ n

by 0, we conclude any asymptotically optimal policy must satisfy (42). Proof of Theorem 5.1. We first obtain an upper bound on limiting expected infinite-horizon discounted

profit, centered and scaled, under any admissible policy u. We then show that the policy * achieves this upper

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System 468 Mathematics of Operations Research 31(3), pp. 453-477, ? 2006 INFORMS

bound, which establishes its asymptotic optimality. Finally, we establish that any asymptotically optimal policy also satisfies (38).

Proposition 5.1 implies that we can assume the policy u has pnu ->

p* and ynu ?

y* as n ?> oo. Observe from

the representation for infinite-horizon discounted profit in (3) that

n: = f sWfxA%(o-EoQ,wV

+vî(ea,(/>?>;u -E^K ,-*?) \k=l 7=1 /

-oo / K Qn J In (t)\ ~ e-s'(ZSplk^dt

+ Zhj-^J-)dt.

(43)

Because the modified static planning problem upper bounds the maximum revenue of a system having capacity imbalance 0",

E a*(p:k* -

E<y>?, - * < * W)

- *=E*AV (44)

*=1 ;=1 y=l

where the equality follows from Lemma 3.1. Combining (43) and (44), and applying Lemma 4.2, shows

/ J r r?? / K On (t) J In (t)\ 1\ limsupE[n^]<limsup r'VÊ^y-E / ^'(?E<i^7^ + E^J!T^ ) <" ) (45)

n-*oo n-?oo \ j=x \_J0 \ k=\ Vn 7=1 Vn / J/

Consider any subsequence n, having yfnfil1 -> 6 e di, as n,

- oo. On this subsequence, as in Equation (34),

limfS-'VÊ^Cy-E/ ^'(SE^^ + E^^)* W'E^-W- (46)

w'~*??\ j=\ Lô \ *=i vn; 7=1 V1/ / J/ 7=i

Next, consider any subsequence nt having yfnfi^j ? ? oo for at least one j e {1,..., J] and on which

lim sup y/rTilO"1J+ < oo for all m ^ j.

n,-ôo

Then, as nt -> oo,

S-Ê^C:, -E / WsErf*^

+ E*^)

* - -oo. (47) 7=1 L'O \ jfc=l V"' 7=1 v"? / J

Finally, consider any subsequence nt having, as nt -> oo, yfnfinu\j

-> oo for all y e T+ C {1,..., J}, yfnfinu[j

->

-oo for all j e T_ C {1,..., J} -

T+, and yfnfil'j ->

dj, a constant for all other ;. Let e be such that pnu k >

J2j T (cj + )akj f?T eac^ k = l,... ,K. (Lemma 3.1 guarantees the existence of such an e.) Then, observe that

J V r?? / K Oni (t) J Ini.(t)\ 1

7=1 LJ0 \ k=\ Wni 7=1 VWi / J

< rl<&?. &,-!'**-*? T,(cj +

e)akJE\^P]dt (48)

7=1 J? k=\jeT+ L V"; J

< tr'jhjlcfa-rte-* Z(cJ +

e)B\??}.]dt (49)

= -s-'v?7 E K'j+5"1 V^ E *A":> + 5_1 V^ E ^"A jeT+ ;6T_ ;*(T+UT_)

- f??8e~s' E(^+e)E[Efly^t(o-c;:;(ol^ (50) "/? ,6T+ U=l J

- -oo, (51)


as nt -> oo. The bound in (48) follows from our choice of e, and the fact that inventory levels lj1

are positive for every nt. To see (49), observe from the shortage process representation in (11) that

- E *?#:*(')=-s::j(t)

- c,?

< -s;;/*)

for all t > 0, every nt, and any policy u. Substituting the expression for nJl/2S^tj(t)

in (32) yields the equality

(50). Finally, Lemma 4.2 and our assumption on the limiting behavior of VîK1 shows the limiting behavior in

(51). We conclude from (43), (46), (47), and the sequence of inequalities in (48)-(51) that

limsupE[nj;] < sup [8-lf2cjej ~

^(e)) n-*oo eed\J \ j=l /

= 8-lJ2cje*-W(d*), (52)

7=1

for some 0* e ?*, which is nonempty by Lemma 5.1. By Corollary 4.1, the policy achieves the upper bound

in (52). To conclude the proof, because by Proposition 5.1 any admissible policy a that is asymptotically optimal has

Pa ~^ P* anc* Ja ~^ T*? it is enough to establish di\st[Vnd^, ?*] -? 0 as n -> oo, which we do by contradiction. If

the policy a contains a subsequence on which V^l\â\ ~* ??? ̂ rom (47) anc^ (48)-(51), liminfn_>00E[II^] -> ?oo,

and so the policy a cannot be asymptotically optimal. Otherwise, on any subsequence n{ having Vn~fia ~^W & 6*, from (43), (44), and (46),

lim E[n;]<s-!5:c^-*(?), n:-+oo ' J J 7=1

which implies j

limsupE[Il"] < r1 Yc:Q* -

W(6*) = lim E[fi;], n-oo

j=l J n-ôo

and so again the policy a cannot be asymptotically optimal. If policy a is optimal in the nth system for every n, then policy a is admissible and E[II?]

> E[IIJJ] for any

other admissible policy u, for every n. The asymptotically optimal policy is admissible, so

liminf E[n^] > liminf E[ff;] > limsupEpTJ] n-*oo n-+oo n^QO

for any admissible policy u, which establishes that the optimal policy a is asymptotically optimal. The key to implementing our proposed policy is to find a 6* e @*, which requires computing E[^f(f9)]. This

expected value can be computed by numerically solving a partial differential equation, as the following lemma

(whose proof is given in the appendix) establishes.

Lemma 5.2. Let Ld be the generator of the diffusion Be so that

1 J J d2 J d 2 .=lm=1 dXjdxm ?i dxj

Suppose the function f is twice continuously differentiable on diJ, has bounded f, f, and f", and solves

Lefe -Sfe + Sj: p'kq; + ? hfi = 0. (53)

k=\ 7=1

Then,

fe(0) = nO). (54)

6. Conclusions and future research. For a high-volume assemble-to-order system, optimal pricing and

capacity investment decisions result in utilization of component production capacity near 100%. In this "heavy traffic" regime, the inventory position for each component evolves according to a Brownian motion. Optimal dynamic allocation of scarce components to outstanding orders (equivalently, optimal sequencing of orders for

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System 470 Mathematics of Operations Research 31(3), pp. 453-477, ? 2006 INFORMS

assembly) forces order queues and component inventory levels into a configuration that minimizes instantaneous

physical and financial holding cost. In other words, the system exhibits state-space collapse. In reality, customers are impatient, and want to know their waiting time before placing an order. In response,

most ATO manufacturers guarantee that orders will be filled within some specified maximum delay. Our analysis

suggests that delays are short compared to order arrival rates under the policy =(/?*, y* ?

n~l/26*, A"). To see

this, observe that when order arrival rates A1?..., A^ are large, a sample-path version of Little's law, analogous

to Reiman's "snapshot principle" [33], shows that for all t > 0,

d On(t)

Dnk(t)^^, (55) Ak

where Dk(t) is the delay in the nth system associated with filling a product k order that arrives at time t, and ^D denotes approximately equal in distribution. Because Corollary 4.1 shows that under the policy * queue

lengths are of order Vn, the approximation (55) implies that delays are of order n~1/2 when order arrival rates

Aj,..., A^ are of order n. Hence, it will be economical to quote a maximum delay of order n~l/2.

Our follow-up paper (Plambeck and Ward [31]) considers an ATO system having delay constraints. To ensure

that the system manager can fill all product orders quickly enough, we allow him to expedite components at

a high per-unit cost. We provide an asymptotically optimal policy for setting prices, component production

capacity, sequencing orders for assembly, and expediting components. Additionally, we allow the system manager to sell excess components for a low "salvage" value when inventories grow large, then heuristically derive

and numerically solve an approximating diffusion control problem that motivates a policy for expediting and

salvaging components. The state-space collapse demonstrated in this paper greatly simplifies the inventory policy: One need not track the number of outstanding orders for every product, only the component inventory positions.

We have assumed (both in this paper and Plambeck and Ward [31]) that when the system manager invests

in component production capacity, he knows the mean arrival rate for customer orders as a function of prices. In practice, capacity investment occurs under considerable uncertainty regarding future demand rates. Further

research is needed to address dynamic pricing in response to shocks in the order arrival rate. Although closed

form solutions are rare in dynamic pricing models (the only papers we know of that obtain closed-form dynamic

pricing strategies are Xu and Hopp [40] and Gallego and van Ryzin [18]), an asymptotic analysis in a high volume regime may provide some insight. We have also taken a monopolist's perspective. Research is needed

on capacity, pricing, and inventory management in an assemble-to-order system with oligopolistic competition. An essential assumption for our analysis is that the assembly operation is not capacitated. This is a reasonable

assumption in consumer electronics and PC manufacturing. However, automobile assembly is capacitated and yet automobile manufacturers are straining to assemble to order (Economist [15], Forbes [17]). The potential increase

in expected profit has been estimated at $80 billion per year for the auto industry as a whole (Economist [14],

Agrawal et al. [3]). Further research is needed to determine how to optimally control systems with capacitated

assembly.

Appendix A. Lemma proofs. Proof of Lemma 2.1. Because k(p) is continuously differentiable and its Jacobian matrix / =

[dkk/

dpm]k m=1 K is invertible everywhere, the inverse function theorem guarantees that the inverse demand function

p(k) is unique, continuous, and differentiable, and has Jacobian matrix

'-'-[frl Define the K x K matrices:

r d\x Idx^ dx^/dk^ dkx fdkx 1

dp2/ dpx dp3/ dpx dpKf dpx

dA2 ldk2 dk2 ldk2 _dk2_ ldk2

dpj dp2 dp J dp2 dpK/ dp2

T=z\ : :

Idk^/dk^ _dkjLldk]L ^dk^/dk^ Q L dP\ I 3Pn dP\ I 8Pn dPl I dPN J

A = diag(|M \dPk/k=l,...,K

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System Mathematics of Operations Research 31(3), pp. 453-477, ? 2006 INFORMS 471

and observe that

J = A(I-T).

T is a substochastic matrix because (4) guarantees that all of its elements are nonnegative and its row sums are less than one. Therefore, oo

(/-r)-' = E^

i=0

and (I ?

T)~l has all nonnegative elements. Furthermore,

J-l=(I-T)-lA~\

where A"1 is a diagonal matrix with strictly negative elements. Specifically, A-1 =zdmg((dXk/dpk)~x)k=x K where dkk/dpk < 0 by assumption. We can therefore conclude that all elements of J~l are negative, and all

diagonal elements of J~l are strictly negative, which completes the proof. Proof of Lemma 3.1. Because the demand function A has a unique inverse demand function p, it is equiv

alent to formulate the perturbed static planning problem as a maximization over order arrival and component production capacities as follows:

^)=A>T,X>0r(A)-^^ (56)

7=1

subject to

E?v*k<yj + 0j, y = i,...,/. (57) *=i

Our assumption that r(\) is strictly concave guarantees that (56)-(57) is a convex program, and so has a unique optimal solution for all 6 e diJ. The Lagrangian associated with (56)-(57), when equivalently viewed as a

minimization problem, is

SP(A, y, u) = -r(A) + E W + E uj (E *yAt

- 7j

- d), 7 = 1 7 = 1 \k=\ /

which yields the Karush-Kuhn-Tucker (KKT) conditions

d% dr(\) '

/ x ^-

= -^

+ E^7W;=0' m=l,...,K, (58)

? =Cj-Uj

= 0, (59)

K

E?*A-r,-0,<o> j = \,...,J, (60) k=\

uj (E

"kjK -yj-e^=o,

7 = 1,..., J, (61)

u > 0. (62)

Because Uj =

Cj>0 by (59), and our assumption that component production costs are positive, from (58),

0 < l^amjCj

= ?? = 2. A4?

? +/>m(A),

7=1 dAm k=i ?*m

and so pm(X) > Ej=i <*mjcj because, by Lemma 2.1, dpk(k)/d\m

< 0. By (60), when 6 = 0,

K

k=\

because p(k) > 0 implies X(p) > 0. Next, for 0 > 0, observe that conditions (58)-(62) are satisfied when

A*(0) = A* and y*(0)

= y* -

0j9 j = 1,..., /. Hence, the KKT theorem ensures that (A*(0), y*(6)

= y*-0)

is a global maximizing point for (56)-(57), which implies that (p*(6) =

p(\*), y*(6)) is a global maximizing point for the original perturbed problem in (8)-(9), and

7f(e)-7f =

Y,cjej. u

7=1

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System 472 Mathematics of Operations Research 31(3), pp. 453^77, ? 2006 INFORMS

Proof of Lemma 4.1. We have assumed that E|jc*(l)|2+e < oo, k=\,...,K and E|^(l)|2+e < oo, j =

1,.. ., / for s = 3. Therefore, from Lemma 9 and bound (113) on the residual interarrival time at the start of a review period in Ata and Kumar [4], for any finite constant a > 0 there exists a constant C(a) such that for

nln > 2/a,

p( max max \Onk((i + \)ln) -

Onk(iln) -

kk(pnu)nln\ > anln)

< In / fff ._

V=0,l,...,/"-l k=l,...,K* *VV ' ' k\ / k\ru' l

J (nlny+e/2

< C(a)\k*\2/5n~l/6.

Setting jg =

max(C(a)fl-1/6|A*|2/5, (2/a)1/2) establishes (19). Similar arguments yield (20). Proof of Lemma 4.2. For brevity, we suppress the subscript indicating the policy u. We first show uniform

integrability, and then argue the convergence to zero. Observe that for any k = 1,..., K and t > 0,

Onk(t)-nklt -Mk(Onk(t) + \) , <%k{<yk(t) + \)-n\?kt 1

-7=-=-7=-+-T-7=' t63^ V? Vn Vn Vn

where Mk(m) =

^k(m) ? nt is a square-integrable martingale relative to the filtration a(xk(\), xk(2),...,

xk(m)), and kk =

kk(pnu) -+ k*k as n -> oo. Furthermore, bounding the residual interarrival time at t with the full

interarrival time encompassing t yields

9?t(Qg(Q + l)-nAt(p?)r ^xk(Onk{t) + \)

Vn ~

Vn

_Mk(Q?k(t)+l)-Mk(0?k(t)) | 1

Vn Vn

\Mk(Onk(s) + l)\ 1 < 2 sup

' *v k]J-- + ?. (64)

0<s<t Vn Vn

Now use (64) to bound (63):

Onk{t)-nkk{p")t \Mk(Q?k(s) + \)\ 2 ? "^ SUP l? ?~ I??

V? o<5<? Vn \Jn

and observe that for n > 1,

( sup W-?W)> V < U sup l^"(f) +

l)l+2V \0<5<r V*2 / V 0<s<t Vn )

<18sup|M-(?^) +

1)|2+8. (65) 0<S<f W

Because5

E( sup \Mk(Onk(s) + 1)|2) < 4E\Mk(Onk(t) + 1)|2 (Z/ maximum inequality)

\0<5<r /

< 4E(jc*(1) -

l)2E(Onk(t) + 1) (Wald's 2nd moment identity) = 4E(;cit(l)

- l)2E[^ek(Onk(t) + 1)] (Wald's lst moment identity)

= 4E(**(1) -

l)2E[kk(pn)nt + (%k(Onk(t) + 1) -

nkk(pn)t)] <

4E(jc*(1) -

l)2(kk(pn)nt + Exk(l)2) (Lorden's inequality),

the bound in (65) and the assumption that pn -> p* as n -> oo implies

/ \Onk(s)-nkk(pn)s\2\ E sup -^? fe </C^ + ^2

\0<J<f| Vn I /

5 Lorden's inequality is found in Lorden [22] or pages 99-100 in Gut [20].

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System Mathematics of Operations Research 31(3), pp. 453^*77, ?2006 INFORMS 473

for large enough n, where kx and k2 are finite constants. From Jensen's inequality,

\Jo o<s<t \jn )

<r(supWS)-"^)28e->>dt, Jo \o<s<t Oy/n )

and so, taking expectations and using Fubini to justify interchanging the expectation and integral, we find

E(f\-syupm?^Adt)2 \Jo o<s<t \/n )

<rSe-?>E(supMis)-"^)2dt Jo \o<s<t Oy/n )

<8~2(Kxt + K2).

Therefore, for any k = 1,..., K, the family

\re-By*v\di{s)\dt\ I/O 0<s<t J

is uniformly integrable. For any k = 1,..., K and B a standard Brownian motion, the functional central limit theorem and the

continuous-mapping theorem show

rPk8e-><m)-nKtdt =? rPk8e-s'J^lkB(t)dt, Jo y/n Jo v

as n ?> oo. Because we have just established uniform integrability,

E\rPt8e-"2i^Z^L dt] ̂ E\rpk8e-s'jh^lkB(t)dtl Jo y/n J \_Jo v '

j

as n -> oo. Observe that because e~8t\B(t)\ > 0 for all t > 0 and B(t) has a normal distribution with mean 0

and variance t, /. OO

" /. 00 -00 hlf

E / e~8t\B(t)\dt

= e~8tE\B(t)\dt

= e~8\-dt<oo. Jo J Jo Jo V 77

Therefore, Fubini justifies the interchange necessary to conclude

Ef~ Pk8e-8'J\*k<rlkB(t)dt = 0.

Identical arguments establish both that

\ Te-1" sop \CJ(s)\dt\ [JO 0<s<t J

is uniformly integrable for each j = 1,. . ., J, and that /. oo

E Cj8e~8t sup CUs)dt-+0 J0 0<s<t

for each j =

1,...,/. D

Proof of Lemma 5.1. We will show that

ldediJ: 8-lf^Cjdj-X(0)>-X(0)\ c

x[-C71S?(0)(l +

max{c,}/A), S?(0)/a1. (66)

Hence, in maximizing 8~l X^=1 CjOj

- $f(0), one may restrict attention to the compact set

X [-cJl8X(0)(l

+ max{c,}/A), 8#(0)/a1


By Fubini's theorem,

X(0) = [a?e-8'E\,?ptql(B9(t))

+ ^hji'j(B9(t))\dt J? U=l ;=1 J

= jfV?'jf^(2*)-'*|nrêxp(-(x^^

because Be(t) is a multivariate normal random variable with mean 6t and covariance matrix Ft. Assumption 3.1

guarantees that J^k=l PÔO + Xî nji*(x)

is Lipschitz continuous in x, and it follows that 8"1 ?j=1 cjOj-\-X(ff)

is a continuous function of 0. Therefore, 8"1 ]TJ=1 cjOj-\-X(ff) achieves its maximum in the compact set

*[-cJl8X(0)(l +max{c;}/A), ??(0)/a].

It remains to prove (66). Define

J+(d) = {j:Oj>0},

ft(?)= E flw(^ + A(fl)). 7ey+(0)

It follows from Lemma 1 that A(0) > A > 0 and pk(6)

< p\ for all 6 e diJ. Furthermore, define

K

f(fi;0) = min E ftWft

*: s.t.

T,av9j>Bj for jeJ+(0)

qk=0 if akj = 0 for all j J+(0).

Observe that for any 6, B 9ty and f > 0,

f(B;9)<EK?;(5)+EVi(fi) /t=l 7=1

f(tB\ 6) = tf(B; 6),

and f (B; 6) is a convex function of Z?. Another useful property is that

?c,0,-f(0;0)<max ? c^-EftWfc

S.t. E ?;> ? *, ;6/+(S) ;ey+(fl)

EWjZOj for jeJ+(9), k=\

qk = 0 if ̂ , = 0 for all j e J+(6)

= -A(fl)D^]+.

7=1

Plambeck and Ward: Optimal Control of a High-Volume Assemble-to-Order System Mathematics of Operations Research 31(3), pp. 453^77, ?2006 INFORMS 475

Employing these properties of f(B', 0) with Fubini's theorem and Jensen's inequality, we find that for any 0 e 3lJ,

J J roo r k j -i

8-l'?cj0j-X(0) =

8-lY,cJ0j- ^E EpI^W) + EWW) dt 7=1 7=1

? U=l 7=1 -I

J /. oo

<S-'EcA-/0 e-s'E[f(Be(t);e)]dt

J /. oo

< ?"' E ciei ~

/ e-s'f(E[Be(t)]; 9) dt 7=1

J?

J /. OO

= 8-lJ2cj9j- e-s'f(t0;9)dt

7=1 'o

J /. oo

= S-lJ2cjej- e-8ttf(6;6)dt

7=1 J?

7=1

Therefore, to achieve 8~l J2j=i cflj

~ X(0) >

-X(0), 0 must satisfy

j

?[0,]+ < S?(0)/A(fl) < 5? (0)/A,

7=1

which implies

8j<8W(0)/A for 7 = 1,...,/.

Also observe that

s-ii:cjej-w(0)<s-ii:cjej, 7=1 7=1

so to achieve 8~l ?j=1 cy-0y

- $f (0) > -$?(0), 0 must also satisfy

CjOj^-Z^-XiO)

> -5^f(0)(l+max{c;}/AV

which completes the proof of (66). Proof of Lemma 5.2. By Ito's formula, for any t > 0,

e-8<f(Be(t))-f(Bd(0))

= fe-8sLef(Bd(s))ds- f8e-'^(Be(s))ds

+ J:fe-bsdf{B/S)) dBj(s). Jo Jo J^Jo dxj

J

Because the stochastic integrals

r'^V^W) .,

Jo aXj J

are martingales,

E e-s'f(Be(t)) -

f(Be(0)) -

f'e-s?Lef(Be(s))ds+f'8e-Ssf(Be(s))ds] = 0.

Jo Jo


Letting r -^ oo, by the bounded convergence theorem,

E jf e-s'(Lef(B0(t))

- 8f(Be(t)))

<fr]

= -/(B?(0)),

and so

E p-5'^/(B9(0)-8/(B9(0)

+ Erf%,(^W) + E'1ji;^(0))^]

= 3?(0)-/(S,(O)). (67)

Assumption (53) applied to (67) when Bd(0) = 0 establishes (54).

Appendix B. Table of notation. In the following, k = 1,..., K and j = 1,..., /.

System parameters

pk Price of product k.

Xk(p) Price-dependent product k order arrival rate.

7y Component j production capacity.

Cj Component cost.

hj Holding cost per unit time for component j.

akj Number of type j components required to assemble product k.

8 Discount rate.

Notation associated with the proposed policy

(/?*, y*) Solution to the static planning problem in (5)-(6).

(p*(d), y*(6)) Solution to the perturbed static planning problem in (8)-(9) 0* =

argmax(5"1 ?j=1 CjSj -

7t(6)) Stochastic imbalance cost, where $f(0) is defined in (35). A* Assembly policy defined in (12).

(q*, /*) Profit-maximizing queue lengths and inventory levels, defined as the

solution to the linear program in (13)?(14). / =

(1 /1A|*)2/3 Review period length.

Acknowledgments. The authors thank Li Chen, Mike Harrison, Anton Kleywegt, Sunil Kumar, Joshua Reed, Alex Shapiro, and Ruth Williams for helpful discussions relating to the issues presented in this paper. They especially acknowledge the anonymous referees for their explicit instructions on how to justify the existence of a Lipschitz continuous function Q*. This research was supported in part by the National Science Foundation

under Grant DMI-0239840.

References

[1] Ak^ay, Y., S. Xu. 2004. Joint inventory replenishment and component allocation optimization in an assemble-to-order system.

Management Sci. 48 99-116.

[2] Agrawal, N., M. A. Cohen. 2001. Optimal material control in an assembly system with component commonality. Naval Res. Logist. 48 409^129.

[3] Agrawal, M, T. V. Kumaresh, G. A. Mercer. 2001. The false promise of mass customization. McKinsey Quart. 3 62-71.

[4] Ata, B., S. Kumar. 2005. Heavy traffic analysis of open processing networks with complete resource pooling: Asymptotic optimality of discrete review policies. Ann. Appl. Probab. 15 331-391.

[5] Bassamboo, A., J. M. Harrison, A. Zeevi. 2004. Design and control of a large call center: Asymptotic analysis of an LP-based method.

Oper. Res. 54 419^135.

[6] Bell, S. L., R. J. Williams. 2001. Dynamic scheduling of a system with two parallel servers in heavy traffic with complete resource

pooling: Asymptotic optimality of a continuous review threshold policy. Ann. Appl. Probab. 11 608-649.

[7] Berens, H., M. Finzel, W. Li, Y. Xu. 1997. Hoffman's error bounds and uniform Lipschitz continuity of best lp-approximations.

J. Math. Anal. Appl. 213 183-201.

[8] Bernstein, F., A. Federgruen. 2004. Comparative statics, strategic complements and substitutes in oligopolies. J. Math. Econom. 40

713-746.

[9] Billingsley, P. 1999. Convergence of Probability Measures, 2nd edition. John Wiley and Sons, Inc., New York.

[10] Bylinsky, G. 2000. Heroes of U.S. manufacturing. Fortune 141(March 20) 192-205.

[11] Chen, H., D. D. Yao. 2001. Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization. Springer-Verlag, New

York.

[12] Cheng, R, M. Ettl, G. Lin, D. D. Yao. 2002. Inventory-service optimization in configure-to-order systems. Manufacturing Service

Oper. Management 4 114-132.


[13] Duenyas, I. 1995. Single facility due date setting with multiple customer classes. Management Sci. 41 608-619.

[14] Economist. 2001. A long march: Mass customization. 360(July 14) 63.

[15] Economist. 2004. Ripe for revolution. 372(September 4) 12.

[16] Ethier, S. N., T. G. Kurtz. 1986. Markov Processes: Characterization and Convergence. John Wiley & Sons, Inc., New York.

[17] Forbes. 2004. Just in time manufacturing. 174(July 5) 22.

[18] Gallego, G., G. van Ryzin. 1994. Optimal dynamic pricing of inventories with stochastic demand over finite horizons. Management Sci. 40 999-1020.

[19] Glasserman, P., Y Wang. 1998. Leadtime-inventory tradeoffs in assemble-to-order systems. Oper. Res. 46 858-871.

[20] Gut, A. 1988. Stopped Random Walks: Limit Theorems and Applications. Applied Probability, Vol. 5. Springer-Verlag, New York.

[21] Harrison, J. M. 1996. The bigstep approach to flow management in stochastic processing networks. F. P. Kelly, S. Zachary, I. Ziedins,

eds. RSS Lecture Note Series. Stochastic Networks: Theory and Applications. Oxford University Press, Oxford, UK, 57-90.

[22] Lorden, G. 1970. On excess over the boundary. Ann. Math. Statist. 41 520-527.

[23] Lu, Y, J. S. Song, D. D. Yao. 2003. Order fill rate, leadtime variability, and advance demand information in an assemble-to-order

system. Oper. Res. 51 292-308.

[24] Maglaras, C. 1999. Dynamic scheduling in multiclass queueing networks: Stability under discrete review policies. Queueing Systems 31 171-206.

[25] Maglaras, C. 2000. Discrete-review policies for scheduling stochastic networks: Trajectory tracking and fluid-scale asymptotic opti

mality. Ann. Appl. Probab. 10 897-929.

[26] Maglaras, C. 2003. Revenue management for a multi-class single-server queue. Working paper, Columbia Business School, New York.

[27] Maglaras, C, J. A. Van Mieghem. 2005. Queueing systems with leadtime constraints: A fluid-model approach for admission and

sequencing control. Eur. J. Oper. Res. 1 179-207.

[28] Maglaras, C, A. Zeevi. 2003. Pricing and capacity sizing for systems with shared resources: Approximate solutions and scaling relations. Management Sci. 49 1018-1038.

[29] Perman, S. 2001. Automate or die. Business 2.0. (July) 60-67.

[30] Plambeck, E. 2004. Optimal leadtime differentiation via diffusion approximations. Oper. Res. 52 213-228.

[31] Plambeck, E., A. R. Ward. 2005. Optimal control of a high-volume assemble-to-order system with guaranteed maximum delays and

expediting. Working paper, Stanford Graduate School of Business, Stanford, CA.

[32] Plambeck, E., S. Kumar, J. M. Harrison. 2001. Asymptotic optimality of a single server queueing system with constraints on throughput times. Queueing Systems 39 23-54.

[33] Reiman, M. I. 1982. The heavy traffic diffusion approximation for sojourn times in Jackson networks. R. L. Disney, T. J. Ott, eds.

Applied Probability-Computer Science, The Interface. Birkhauser, Boston, MA, 126-128.

[34] Schrijver, A. 1986. Theory of Linear and Integer Programming. Wiley, Chichester, New York, 411-421.

[35] Song, J. S. 1998. On the order fill rate in a multi-item base-stock inventory system. Oper. Res. 46 831-845.

[36] Song, J. S., P. Zipkin. 2003. Supply chain operations: Assemble-to-order and configure-to-order systems. S. Groves, T. DeKok, eds.

Handbooks in Operations Research and Management Science, Chapter XI, Supply Chain Management, Vol. XXX, 561-593.

[37] Song, J. S., S. Xu, B. Liu. 1999. Order fulfillment performance measures in an assemble-to-order system with stochastic leadtimes.

Oper. Res. 47 131-149.

[38] Wang, Y 2000. Near-optimal base-stock policies in assemble-to-order systems under service level requirements. Working paper, Massachusetts Institute of Technology, Cambridge, MA.

[39] Wein, L. 1991. Due-date setting and priority sequencing in a multiclass M/G/l queue. Management Sci. 37 834-850.

[40] Xu, X., W. J. Hopp. 2004. Dynamic pricing and inventory control with demand substitution: The value of pricing flexibility. Working

paper, Northwestern University, Evanston, IL.

[41] Zhang, A. X. 1997. Demand fulfillment rates in an assemble-to-order system with multiple products and dependent demands. Produc

tion Oper. Management 6 309-324.

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