Optimal and Asymptotically Optimal Policies for Assemble-to-Order
N- and W-Systems
Lijian Lu,1 Jing-Sheng Song,2 Hanqin Zhang3
1
Graduate School of Business, Columbia University, New York, 10027 New York
2
Fuqua School of Business, Duke University, Durham, 27708 North Carolina
3
NUS Business School, National University of Singapore, Singapore
Received 25 October 2014; revised 30 November 2015; accepted 1 December 2015
DOI 10.1002/nav.21671
Published online 5 January 2016 in Wiley Online Library (wileyonlinelibrary.com).
Abstract: We consider two specially structured assemble-to-order (ATO) systems—the N- and W-systems—under continuous
review, stochastic demand, and nonidentical component replenishment leadtimes. Using a hybrid approach that combines samplepath analysis, linear programming, and the tower property of conditional expectation, we characterize the optimal component
replenishment policy and common-component allocation rule, present comparative statics of the optimal policy parameters, and
show that some commonly used heuristic policies can lead to significant optimality loss. The optimality results require certain
symmetry in the cost parameters. In the absence of this symmetry, we show that, for systems with high demand volume, the
asymptotically optimal policy has essentially the same structure; otherwise, the optimal policies have no clear structure. For these
latter systems, we develop heuristic policies and show their effectiveness. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 62:
617–645, 2015
Keywords: inventory control; assemble-to-order system; asymptotic analysis
1.
INTRODUCTION
This article concerns optimal control policies for two
specially structured, continuous-review, assemble-to-order
(ATO) systems as shown in Fig. 1. In these systems, components are kept in stock, but final products are assembled only
after customer orders are realized. Customer orders arrive
randomly. The components are procured from outside suppliers with positive leadtimes. The assembly times of the products are negligible relative to the component replenishment
leadtimes.
The ATO approach allows manufacturers to only build
products that customers desire and hence to eliminate
finished-product inventories, achieving great flexibility at
low cost. Well-known examples include Dell Computer’s
online service and the “build-to-order” practices of BMW,
Toyota, and GM. Incidentally, the same system structure
also applies to mail-order and online retailers. In such
businesses, customers may order different, but possibly
overlapping, subsets of items that are kept in stock and
Correspondence to: Lijian Lu ([email protected])
© 2016 Wiley Periodicals, Inc.
customer satisfaction is based on the fulfillment of the entire
order.
Despite the widespread use of ATO systems, the structure
of the optimal control policy remains largely unknown. The
difficulty arises because the system topology comprises both
the assembly structure (for each product) and the distribution
structure (for each common component). While the form of
an optimal policy for an assembly system with a single final
product has been identified [28], for systems with multiple
final products this is still an open problem.
As stated in [34], control of an ATO system consists
of two decisions: inventory replenishment and inventory
allocation. Until now, only simple heuristic policies have
been implemented. Correspondingly, the academic literature almost exclusively focuses on these simple policies,
such as independent base-stock (IBS) replenishment policies
(i.e., each component follows a standard base-stock policy)
and the first-come-first-served (FCFS) component allocation
rule.
This article addresses the structure of an optimal policy
for two basic ATO systems—the N- and W-systems (see
Fig. 1)—in the following setup: demands are generated by
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Figure 1. Illustration of ATO systems.
a stationary process with independent increments. Unsatisfied demand is fully backlogged and replenishment orders
for components face a constant component specific leadtime.
Our objective is to minimize the expected long-run average
system cost, under linear holding and procurement costs for
the components and linear backlogging costs for the final
products. For conciseness and ease of presentation, the main
body of the article focuses on the N-system and Appendix
presents the extension to the W-system.
In the N-system, product-1 requires one unit of component1 only, while product-12 requires a unit of component-1
and one of component-2. In the following, we often refer
to component-1 as the common component and component2 the product-specific component. In the context of computers, for example, component-1 may be a hard disk and
component-2 may be a memory card. The hard disk may
be sold directly to a customer as a portable hard disk drive
(product-1) or may be assembled together with the memory
card into a laptop (product-12). Similarly, component-1 may
be a LCD monitor, which may either be sold directly to a customer (product-1) or be assembled with a CPU (component2) into a desktop (product-12). In the context of automobiles,
component-1 may be an engine and component-2 may be
the body of the car. The engine may either be sold separately as product-1 or be assembled with the car body into a
car (product-12). Clearly, the N-system possesses the basic
elements of an ATO system—with both assembly and distribution structures. Thus, it serves as an important building block
for general multiproduct, multicomponent ATO systems.
We now summarize our main findings. These depend on
the cost structure of the system.
1.1.
Cost Symmetry
We first study a system in which the cost parameters satisfy
a certain symmetry condition: The unit backorder cost rate
for product-1 equals the sum of the unit backorder cost rate
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for product-12 and the unit holding cost rate of component-2.
Therefore, when component-2 is available, there is no immediate cost difference for allocation of a unit of component-1
among different product-demands. This cost structure is plausible in practice if product-1 serves an external market that
enjoys a higher profit margin or faces a higher shortage cost,
while product-12 serves internal production, as an intermediate product (or a subassembly). For instance, in the computer
example mentioned above, product-1 may be a high-end
portable hard disk competing with several substitutes in the
market, while product-12 is used to assemble a laptop in a
medium price range. In the automobile example, the engine
may be used to satisfy an emergency supply contract for a
spare part that incurs a high delay penalty, while product-12
is used to assemble a car which has a smaller penalty cost.
For this system, we obtain the following full characterization
of the optimal policy:
a. Allocation Rule: For any given replenishment policy, a no-holdback (NHB) allocation rule is optimal.
A NHB rule allocates a unit of a component to an
outstanding product demand if and only if this allocation can result in the fulfillment of that demand.
Note that there are multiple NHB rules; under the
above symmetry condition, any NHB rule is optimal.
b. Replenishment Policy: Assuming this optimal allocation rule is followed, the optimal replenishment
policy is a coordinated base-stock (CBS) policy.
Under a CBS policy,
i. The component with the longer leadtime
uses a standard base-stock policy, acting on
inventory position = inventory in stock and
on order minus backlogs.
ii. The component with the shorter leadtime
adopts a state-dependent base-stock policy.
Remarkably, the base stock level depends
only on a one-dimensional state variable,
namely, the aggregate demand for one of
the two products during the most recent time
interval of a duration given by the difference
between the components’ leadtimes.
We also provide a simple procedure to determine the parameters of the CBS policy. In addition, we show that, in general,
a higher demand rate for either product increases the optimal
inventory levels for both components. This conforms with
conventional wisdom. However, counter-intuitively, when the
product specific component-2 has a shorter leadtime, a higher
product-1 demand rate lowers the optimal component-2
inventory.
The above findings are significant. The structure of the
optimal policy is remarkably simple, given the fact that a
basic representation of the problem as a Markov Decision
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
process requires a complex, multidimensional state description, including vectors of outstanding orders and the times at
which they are placed, along with inventory levels for the two
components, and backlog sizes for the two final products. Our
results show that, the combination of a simple allocation rule
(NHB) and the above stated replenishment policy (CBS) is
truly optimal. The CBS policy synchronizes the component
inventory levels. (Clearly, when the component leadtimes are
identical, the CBS policy reduces to an IBS policy.) This policy is closely related to the well-known balanced-base-stock
(BBS) policy, proved to be optimal by [28] for the singleproduct assembly system that consists of product-12 only. It
is noteworthy that a similar policy is optimal with the addition
of a second final product, that is, product-1.
With the form of optimal policy identified, we are able to
investigate the effectiveness of the commonly used IBS policies. Our numerical experiments indicate that the optimality
loss incurred using the best IBS policy can be significant.
The IBS policy performs well only when the two component
leadtimes are similar. (Recall that IBS is optimal when the
leadtimes are identical).
At the same time, we make a methodological contribution.
Our optimality proof introduces a novel, hybrid approach for
identifying an optimal policy. The standard approach for characterizing the structure of an optimal policy is to formulate the
problem as a dynamic program or Markov Decision process.
However, as mentioned before, the complexity of the system
state and the nonexponential feature of the leadtime for each
component precludes this approach. Our approach consists of
two steps. The first step establishes the optimality of a NHB
allocation policy under an arbitrary replenishment policy. To
this end, we derive a sample-path lower bound of the system
cost and construct a NHB allocation rule to achieve the lower
bound via a linear programming model. Thereafter, a NHB
allocation policy can be used without loss of optimality. The
second step is to find the optimal replenishment policy to minimize the expected cost lower bound obtained from the first
step, and show that a CBS policy achieves this expected lower
bound. We accomplish this by invoking the tower property of
conditional expectations—that is, the informed expectation
on average equals the uninformed expectation; see Eq. (25)—
for the expected cost that arises in any finite time interval.
In particular, the tower property enables us to sequentially
find the optimal replenishment policy for the component
with the longer leadtime and then that for the component
with the shorter leadtime by leveraging the realized product
demand information during the time difference between the
two component leadtimes.
1.2.
Cost Asymmetry
Next, we consider systems with non-symmetric costs. We
obtain the following results:
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a. Asymptotically Optimal Policy: Under the highvolume demand regime (demand rates for both
product-1 and product-12 are significantly high), we
use heavy-traffic limiting arguments to identify the
asymptotically optimal policy by allocating components at discrete review points separated by an
infinitesimal interval. Under this asymptotic regime,
we show that the optimal replenishment policy is a
CBS policy, and the optimal allocation policy is the
periodic-review NHB policy with a priority based
backorder clearing (PBC) rule. Thus, interestingly,
the form of the asymptotically optimal policy is
essentially the same as that of the optimal policy in
the symmetric-cost case.
b. Bounds: When demand is in the small to medium
range, there is no clear structure for the optimal policy
and we construct two systems with symmetric costs
whose long-run average costs form a lower bound
and an upper bound on the long-run average cost of
the original system. This allows us to exploit the optimal policies of these bounding systems based on our
results for the symmetric-cost case to devise heuristic
policies for the original system. In addition, we use
the cost lower bound as a benchmark to evaluate the
performance of the heuristics.
c. Heuristics: We develop various heuristic policies
based on our results for the symmetric-cost case (as
applied to the bounding systems) and the asymptotic analysis for high-volume demand systems with
asymmetric costs. Numerical results show that, in
general, when cost parameters are asymmetric, the
most effective heuristic policy consists of a CBS
replenishment policy combined with a rationing allocation policy. Under a rationing allocation policy,
one of the two final products is treated with a higher
priority than the other. Any unit of the common component is used to fill an outstanding demand with
high priority if the allocation results in an immediate fulfilment. A unit of the common component
is allocated to fill an outstanding demand with low
priority only when the common component inventory level exceeds a threshold value characterized
by a piecewise linear function of the component2 inventory. (For the special case of ample supply
of component-2, the system reduces to a single-item
inventory system with two demand classes. For such
systems, it has been shown that a simple base-stock
policy is optimal for replenishment and a rationing
policy is optimal for inventory allocation among the
two products; see [34]).
Finally, we show that our techniques to derive cost bounds
can be extended to W-systems. Moreover, under specific cost
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and leadtime conditions, we specify a policy (a CBS replenishment policy combined with a NHB allocation rule) whose
cost equals the lower bound, and thus, is optimal.
The rest of the article is organized as follows. Section 2
reviews the relevant literature. Section 3.1 introduces the
basic notation and model formulation, Section 3.2 outlines
our basic methodology for the optimality proof. Section
4 introduces our sample path methods and characterizes
the optimal common-component allocation rule. Section 5
focuses on the form of the optimal replenishment policy
and its comparative statics under symmetric cost. Section
6 analyzes the system with asymmetric costs. In Section 7,
we develop bounds and heuristics based on the results for
systems with symmetric and asymmetric costs. We also investigate the effectiveness of the IBS policy for systems with
symmetric cost. Finally, Section 8 summarizes the article.
Appendix contains the extensions to the W-system and the
online companion contains all proofs.
2.
LITERATURE REVIEW
As mentioned above, the majority of studies in the ATO
literature focus on specific classes of policies. In particular, for continuous-review models, most papers assume the
FCFS allocation rule, and develop tools to evaluate and
optimize the IBS replenishment policies. See, for example,
[17–19, 30–33, 37, 42]. For periodic-review models, because
the demand during each period is batched and filled at the
end of the period, different allocations rules have been considered. These include a fixed priority rule [41], the FCFS
rule [12], and a fair-share rule [3]. [4] study a product-based
allocation rule that makes optimal or near-optimal allocation decisions within each period. Similarly, [14] study optimal allocation decisions within each period using the idea
of “multi-matching,” in the sense that multiple components
must be matched with multiple products. However, all these
studies apply FCFS to demand between periods. In other
words, these alternative allocation rules, if implemented in
a continuous-review environment, reduce to FCFS.
A second allocation policy considered in the literature is
the NHB allocation rule, which was first described and analyzed by [35], followed by [8, 20]. (It is worth mentioning
that both FCFS and NHB are commonly seen in practice;
[15] describe an example of NHB at Dell and [40] describe
an example of FCFS at Amazon.com.).
The NHB rule describes a class of allocation rules, which
allocates a component to a demand only if such an allocation will result in the fulfillment of the demand. When there
are multiple types of backorders waiting for the same common component, however, how to allocate that component
among different backorders (i.e., the backorder clearing rule)
still needs to be specified. [35] assume the FCFS backorder
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clearing rule, and, therefore, they call the allocation policy
a modified FIFO rule. [16] show that, for the N-system and
W-system (and their generations) under the IBS policy and
symmetric cost, the NHB rule is sample-path optimal among
all possible allocation rules, regardless of the types of backorder clearing rule. However, when the costs are asymmetric,
PBC outperforms FCFS. [8] provide extensive discussions
on effectiveness of various backorder clearing rules, including PBC and priority with reservation (PR). For systems with
asymmetric costs, we employ NHB, PBC, and PR allocation
rules to construct heuristic policies. Different from [8, 20],
however, we use the CBS policy for replenishment, instead
of the IBS policy. We also show the impact of alternative component allocation rules, together with a CBS replenishment
policy, on the system cost.
Only very few papers discuss the form of the optimal control policy for ATO systems. [5] consider the special case
of a single end-product and multiple demand classes. They
assume the supply process for each component is a singleserver queue with exponentially distributed processing times.
They show that under Markovian assumptions on demand
and production, the optimal replenishment policy is a statedependent base-stock policy, and the optimal allocation rule
is a multilevel rationing policy that depends on the inventory
levels of all other components. [9] extend the above results to
nested systems and [21] extend them to general M-systems,
assuming at most one job in the supply system at any time. We
consider a very different component supply process, which
is an infinite-server queue with constant processing times.
For the W-system with identical deterministic component
leadtimes (a special case of the supply system that we consider), [8] show that when a symmetric cost condition or a
“balanced capacity” condition is satisfied, the optimal replenishment policy is a simple base-stock (i.e., IBS) policy, and
the optimal allocation policy is a specific NHB policy, which
is state-dependent. For the same system, that is, when the
leadtimes are identical, [26] show that an IBS plus an allocation rule with a “target” backlog level (which is similar to
the above rationing policy) becomes asymptotically optimal
when the leadtimes go to infinity.
In most ATO systems, leadtimes are not identical. Very
little is known about the structure of an optimal policy under
nonidentical leadtimes, both for N- and W-systems. [20]
show, for a W-system with symmetric cost, that under an IBS
replenishment policy, the common component is optimally
allocated with a NHB allocation rule. Using a different proof
technique, we generalize the result for arbitrary replenishment policies. This extension is important because, as mentioned, the optimal replenishment policy is not, in general,
IBS.
[25] independently derive a lower bound for the optimal
cost in a W-system, by solving a stochastic program. We
derive a lower bound for the W-system as well, based on the
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
approach described above. Our lower bound is much easier to compute. In addition, they present conditions (called
verification lemma) that the optimal policy should satisfy to
achieve the lower bound under symmetric cost and leadtime
conditions whereby L2 = L3 > L1 , see Fig. 1. However,
these conditions involve a set of complex equations and
require solving a stochastic program, which is computationally challenging. Different from their approach, we explicitly
characterize the optimal policies (a CBS replenishment policy together with a NHB allocation rule) with an intuitive
and easy-to-implement ordering policy. This characterization
also enables us to conduct various comparative statics analyses, which would be challenging with Reiman and Wang’s
verification lemma. Moreover, we show that our policy is
asymptotically optimal in the high-demand region. This, in
turn, helps us construct effective heuristic policies.
The concept of a CBS policy is closely related to the BBS
policy, shown by [28] to be the optimal replenishment policy for a single product assembly system. [6] provide an
alternative proof using a lower bound argument.
Various authors have studied a single-item system with
multiple demand classes. Assuming periodic review and leadtime equals the review cycle, and assuming all previous
backorders are cleared at each review point, [36] shows that a
base-stock policy is optimal for replenishment and a rationing
policy is optimal for stock allocation. In continuous review
systems, assuming a make-to-stock production facility and
exponentially distributed production time, [10] shows that
a base-stock policy is optimal for the production decision
and a rationing policy is optimal for stock allocation for lost
sales systems. For backlogging systems with two demand
classes, [11] shows that the optimal production policy is still
base–stock-type and that the optimal stock allocation policy follows a monotone switching curve. [38] extend [11]
to backlogging systems with multiple demand classes and
further show that the optimal allocation is a rationing policy. Assuming a constant leadtime and economies of scale in
replenishment, [1, 7] propose a (Q, r) policy for replenishment and PR for stock allocation. In their concluding remarks,
[7] also suggest using the same policy for managing the common component inventory in the ATO environment. We also
study a heuristic that employs the same allocation policy.
The asymptotic studies by [22–24] on high-volume
demand ATO systems consider different settings from ours:
Each component is produced by a production facility with
finite capacity and the production times are random. The
decision variables include the component production capacities, assembled product production sequence, and product
prices. Once a component’s production capacity is chosen,
the production facility produces the component at full capacity, so there is no inventory decision. Nevertheless, we adopt a
similar approach as these authors (i.e., restricting allocation
decisions to discrete points) for our system with constant
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leadtimes and asymmetric costs. We derive asymptotically
optimal replenishment and allocation policies under the highvolume demand regime. Recently, [39] consider a system
similar to that in [23] and derive asymptotically optimal
continuous-review component allocation policies.
3.
MODEL AND PRELIMINARIES
In this section, we first present the model setting. We
then provide an outline of our approach to establishing the
optimality results.
3.1.
Model Setting
For convenience, we introduce the notation for a general continuous-review ATO system so that both N- and
W-systems can be specified as special cases. The system
has multiple components and multiple products. Each product uses several components. Components are indexed by
i ∈ I = {1, 2, · · · , m}, and products are indexed by K, a
subset of I, if it requires one unit of each component in K
and none in I \ K. Let K be the set of all products. We use
subscripts i to index the quantities related to component i,
and superscripts K to index the quantities related to product
K. For each component i, let Ki be the set of products that
require component i. For any t ≥ 0, i ∈ I and K ∈ K,
denote
DK (t) = cumulative demand of product K in (0, t],
Di (t) =
DK (t) = cumulative demand for
K∈Ki
component i in (0, t],
For any process {X (t), t ≥ 0}, denote
X (t − a, t] = X (t) − X (t − a), 0 ≤ a < t,
X (t − a, t] = X (0, t], 0 < t < a,
X (t) = X (0), t < 0.
We assume that the demand process (DK (t), K ∈ K), t ≥ 0
is stationary and has independent increments. This implies
that the component-i demand process {Di (t), t ≥ 0} is also
stationary and has independent increments. We assume continuous demand. A similar analysis can be carried through
to the discrete demand case with the derivative operation
replaced by the difference operation.
For each component i, let L i be the replenishment leadtime for component-i, a positive constant. Denote L =
min {Li : i ∈ I} and i = Li − L. Let
Di (t) = Di (t − Li , t] = leadtime demand of component i
(ending at time t),
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Di (t) = Di (t − i , t],
Di = a generic random variable having the same
distribution as Di (L) with pdf ψi (·),
cdf i (·)and ccdf i (·),
Di = a generic random variable having the same
distribution as Di (i ).
For each product K, let D K (t) = DK (t − L, t],
B K (t, π ) = B K (0) + DK (t) − AK (t),
I Pi (t, π ) = I Pi (0) + Oi (t) − Di (t),
where I Pi (0) is the initial inventory position. Without loss of
generality, we assume I Pi (0) = Ii (0) and B K (0) = 0. The
net inventory of component i can be calculated from the state
variables as follows:
Ii (t, π ) −
B K (t, π ) = Ni (t, π ), i ∈ I.
(2)
K∈Ki
D K = a generic random variable having the same
distribution as DK (L)
ψ(·) = the joint pdf of(D K , K ∈ K).
We now describe the state of the system and the control
policies. Let Ii (0) and B K (0) be the initial on-hand inventory for component-i and initial backorders for product-K,
respectively. For any t ≥ 0, define
Oi (t) = cumulative replenishment orders in[0, t],
with Oi (0) = 0,
AK (t) = total product K demands satisfied in [0, t],
with AK (0) = 0,
Ft = DK (s), 0 ≤ s ≤ t, K ∈ K .
1 (Admissible
Policy). A policy π =
DEFINITION
(Oi (t), AK (t)), t ≥ 0 is called admissible if it satisfies, for
any t > 0,
(i) Oi (t) and AK (t)are jointly determined by the
system’s
available information until time t, Ft ,
K
(ii)
A
(t) ≤ Ii (0) + Oi (t − Li ),
K∈Ki
K
(iii) A (t) ≤ B K (0) + DK (t).
The set of all admissible policies is denoted as A. Any
process that satisfies the property in Definition 1 is called
Ft -adaptive. We call {(Oi (t), i ∈ I), t ≥ 0} a replenishment
policy or process, and {(AK (t), K ∈ K), t ≥ 0} an allocation
rule or process.
For any π = {(Oi (t), AK (t)), t ≥ 0} ∈ A, let
Ii (t, π ) = on − hand inventory of component i at time t,
B K (t, π ) = backorders for product K at time t,
I Pi (t, π ) = inventory position of component i,
Let hi be the unit holding cost rate of component-i and bK be
the unit backorder cost rate of product-K, the total inventory
cost at time t is
C(t, π ) =
hi · Ii (t, π ) +
bK · B K (t, π ), (3)
i
Our objective is to find an admissible π ∈ A to minimize
the long-run average expected total inventory cost
T
1
lim sup E
C(t, π )dt .
(4)
T →∞ T
0
Let C ∗ be the optimal system cost. Our optimization problem
can be stated as follows:
T
1
(P1)
C ∗ = min lim sup E
C(t, π )dt .
π ∈A
T →∞ T
0
A policy π ∗ that achieves the above minimum is an optimal
policy.
The focus of this article is to study two special but important building blocks of the ATO system. In particular, in
Section 3-5, we study the N-system, namely, I = {1, 2} and
K = {{1} , {1, 2}}. In Appendix, we show that our analysis is applicable to the W-system, that is, I = {1, 2, 3} and
K = {{1, 2} , {1, 3}}.
For any real number x, we denote x + = max {x, 0} and
−
x = max {−x, 0}.
3.2.
The system dynamics are:
Ii (t, π ) = Ii (0) + Oi (t − Li ) −
K∈Ki
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AK (t),
Sketch of the Main Approach
Because our approach to the optimality proof is nonstandard and takes several steps, to facilitate understanding of
the development in the rest of the article, we now provide a
sketch of the main idea by using the N-system.
Let C(t) denote a sample-path lower bound, that is,
after ordering at time t,
Ni (t, π ) = I Pi (t − Li , π) − Di (t) = net inventory of
component i at time t.
(1)
K
minC(t, π ) ≥ C(t)
(P2)
π ∈A
Note that
and
T
0
C(t, π )dt ≥
T
1
C ∗ ≥ lim sup E
T
T →∞
0
for all t ≥ 0.
C(t)dt for all π ∈ A, T > 0
0
T
C(t)dt ≡ C.
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
Thus, a sample-path lower bound generates a lower bound
on the expected cost, which, in turn, yields a lower bound
on the optimal cost. If the expected lower bound E[C(t)] is
attainable by a policy in A (this attainability and (3) imply
that C(t) is nonnegative with probability one), in view of
E
T
C(t)dt =
0
STEP 3: Assuming cost symmetry and taking the expected
value of the above sample-path lower bound, we obtain
a lower bound on the expected system cost:
E[C(t, π )] ≥ E[ϕ(N1 (t, π ), N2 (t, π ))]
= E[ϕ(I P1 (t − L1 , π ) − D1 (t),
T
I P2 (t − L2 , π ) − D2 (t))],
E[C(t)]dt
0
(by Fubini’s theorem, see p. 518 of [13]), then this policy
must be optimal.
Recall that N i is the component net inventory defined in
(1). To characterize the optimal policy, our approach is to first
develop a conditional sample-path lower bound and identify conditions under which a structured allocation policy
achieves this conditional lower bound for all t. Given the
current inventory status (N1 (t, ·) and N2 (t, ·)), this step essentially finds an optimal allocation rule. Next, we restrict our
attention to this optimal allocation policy and proceed to characterize the structure of a replenishment policy that achieves
the expected lower bound
E[C(t)] = E{E[C(t)|Ni (t, ·), i = 1, 2]}
for all t. More specifically, we take the following steps:
STEP 1: For any given policy π ∈ A, we analyze how
it affects the cost C(·, π) on each sample path. This
involves state variable decomposition and reexpressing
the cost function and system dynamics; see Section 4.1.
STEP 2: For any given t, applying the results in Step 1 and
conditioning on the net component inventory Ni (t, ·) =
ui , i = 1, 2, we obtain a linear program which yields a
conditioned sample-path lower bound
ϕ(u1 , u2 ) = min{C(t, π) : π ∈ Â(t, u1 , u2 )},
where
Â(t, u1 , u2 ) = {π : Ni (t, π) = ui , i = 1, 2} .
Thus, for any given t,
C(t, π ) ≥ ϕ(N1 (t, π), N2 (t, π)),
for all π ∈ A
We then show that under cost symmetry (Condition 1
defined later), the NHB allocation rules can achieve this
(conditioned) lower bound for all t, and are, thus, optimal. However, without the cost symmetry, we are not
able to identify any allocation policy of simple structure
to attain the lower bound for all t. This is accomplished
in Section 4.2.
623
t ≥ 0, (5)
where the last equality follows from (1). Using conditional expectations and the tower property of the conditional expectations, we identify the structure of the
replenishment policy [in terms of I P1 (t, ·) and I P2 (t, ·)]
that minimizes (5) over all π ∈ A. This completes
the characterization of the optimal policy π ∗ that minimizes the expected cost (4), under cost symmetry; see
Section 5.1.
STEP 4: Without cost symmetry, we construct two bounding systems with symmetric costs and then use the
optimal policies for these systems to suggest several
heuristic policies for the original problem.
4.
N-SYSTEM: SAMPLE PATH RESULTS
4.1.
Sample-Path State Decomposition
For any π ∈ A, without loss of generality, we assume
that replenished inventory units will be used according to
the sequence of their arrivals, that is, according to the firstavailable-first-use rule. Also, the demands for each product will be satisfied on a FCFS basis. This assumption is
supported by an observation made by [20]. These authors
show that any allocation rule can be modified in the above
described fashion without changing the number of backorders
of each product or the on-hand inventory of each component. With this assumption, the product-specific component-2
is assigned (or committed) to the product-12 demands on
the FCFS basis. Similar to [20], we decompose the on-hand
inventory and backorders at time t as follows:
• Y2 (t, π ): on-hand inventory of product-specific
component-2 not yet assigned to a demand of product12;
• B212 (t, π ): backorders of product-12 due to missing
component-2, including the backorders missing both
components 1 and 2;
• B112 (t, π ): backorders of product-12 due to missing only the common component-1, either because
component-1 is missing or because it is committed to
product-1 according to π .
Note that
I2 (t, π ) = Y2 (t, π ) + B112 (t, π ),
(6)
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B 12 (t, π) = B112 (t, π) + B212 (t, π),
(7)
Y2 (t, π) × B212 (t, π) = 0.
(8)
+
−
It is easy to verify that v1 + v2 = (u−
1 − u2 ) at optimality.
Stating the optimal solution of the linear program (P3) in the
original problem data, we obtain:
Thus, (3) can be reexpressed as
C(t, π ) = h1 I1 (t, π) + h2 Y2 (t, π) + b1 B 1 (t, π)
(1)
+ b12 B212 (t, π) + (h2 + b12 )B112 (t, π),
(9)
and (2) becomes
i. If b1 < b12 + h2 , then
I1 (t, π ) − B (t, π ) −
1
PROPOSITION 1: If there exists an admissible policy π̂
such that the conditional sample-path lower bound given by
the linear program (P3) can be achieved, it has the following
sample-path properties.
B112 (t, π)
−
B212 (t, π)
= N1 (t, π ),
(10)
Y2 (t, π ) − B212 (t, π ) = N2 (t, π).
(11)
Combining (6)–(8) and (10)–(11) yields
+
Y2 (t, π ) = [N2 (t, π)] ,
B212 (t, π)
+
B 1 (t, π̂ ) = ([N1 (t, π̂ )]− − [N2 (t, π̂ )]− ) , (15)
B112 (t, π̂ ) = 0,
(16)
C(t, π̂ ) = h1 N1 (t, π̂ ) + h2 N2 (t, π̂ )
−
= [N2 (t, π)] , (12)
I1 (t, π ) = B 1 (t, π) + B112 (t, π) + N1 (t, π) + [N2 (t, π )]− .
(13)
Furthermore, from the nonnegativity of I1 (t, π) and (13), we
obtain
+
B 1 (t, π ) + B112 (t, π) ≥ ([N1 (t, π)]− − [N2 (t, π)]− ) .
(14)
+ (h1 + h2 + b12 )[N2 (t, π̂ )]−
+ (h1 + b1 )([N1 (t, π̂ )]−
− [N2 (t, π̂ )]− )+ .
ii. If b1 = b12 + h2 , then
B 1 (t, π̂ ) + B112 (t, π̂ )
+
= ([N1 (t, π̂ )]− − [N2 (t, π̂ )]− ) ,
(17)
Thus, a sample-path lower bound can be generated by
(P2 )
C(t) = min C(t, π) subject to (12)– (14).
π ∈A
4.2. Sample-Path Lower Bound: A
Linear-Programming Approach
C(t, π̂ ) = h1 N1 (t, π̂ ) + h2 N2 (t, π̂ )
+ (h1 + h2 + b12 )([N1 (t, π̂ )]−
∨ [N2 (t, π̂ )]− ).
(18)
iii. If b1 > b12 + h2 , then C(t, π̂ ) equals (18) and
For any fixed t, let
B 1 (t, π̂ ) =0,
(v1 , v2 , v3 , v4 , v5 )
= (B
1
+
(t, π ), B112 (t, π), I1 (t, π), Y2 (t, π), B212 (t, π )).
Conditioning on (N1 (t, π), N2 (t, π)) = (u1 , u2 ), the system
equations with these two variables fixed can be expressed
as a polyhedron and Problem (P2 ) can be expressed as the
following linear program:
ϕ(u1 , u2 ) =
(P3)
=
⎧
⎪
min
⎪
⎪
⎪
⎪
⎪
⎨s.t.
⎪
⎪
⎪
⎪
⎪
⎪
⎩
min
{C(t, π)}
π ∈Â(t,u1 ,u2 )
b1 v1 + (h2 + b12 )v2 + h1 v3 + h2 v4 + b12 v5
v3 − v1 − v2 − v5 = u1 ,
v4 = u+
2,
−
v 5 = u2 ,
vi ≥ 0, i = 1, · · · , 5.
Naval Research Logistics DOI 10.1002/nav
B112 (t, π̂ ) = ([N1 (t, π̂ )]− − [N2 (t, π̂ )]− ) .
iv. A necessary condition for an admissible π̂ ∈ A to
be sample-path optimal is (17).
Proposition 1 (iv) indicates that, at sample-path optimality,
the lower bound in (14) is tight. Thus, from (12) and (17), if
a sample-path optimal admissible π̂ exists, it will yield the
lowest possible total number of backorders under π̂ for every
sample path:
Bmin (t, π̂ ) = B 1 (t, π̂ ) + B112 (t, π̂ ) + B212 (t, π̂ )
= [N1 (t, π̂ )]− ∨ [N2 (t, π̂ )]− .
(19)
Next, we identify the type of policy that leads to (19).
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
4.3.
NO-Holdback Policies
The following definition is a generalization of the NHB
allocation policy introduced by [20].
2: We say that an admissible policy π =
DEFINITION
(Oi (t), AK (t)), t ≥ 0 ∈ A is NHB if the following
conditions are satisfied for all t ≥ 0,
B K (t, π ) × min {Ii (t, π), i ∈ K} = 0,
K = 1, 12. (20)
When
K π is NHB, we also call the allocation rule
A (t), t ≥ 0 an NHB allocation rule.
Thus, under an NHB policy, a demand is backordered if
and only if there is no on-hand inventory of at least one of its
components. Obviously, for any given replenishment policy
{Oi (t), t ≥ 0}, (20) prescribes an allocation rule at t (even
though it leaves some degree of freedom as for how exactly
the backorders should be satisfied), which justifies the last
statement of the definition. It is straightforward but tedious
to prove the following result:
PROPOSITION 2: For any Ft -adaptive replenishment
policy {(O1 (t), O2 (t)), t ≥ 0}, there exists a Ft -adaptive
allocation process {(A1 (t), A12 (t)), t ≥ 0} such that
{(Oi (t), AK (t)), t ≥ 0} is NHB.
REMARK 1: The above definition of the NHB policy can
be generalized to the general ATO system defined in Section
3. Proposition 2 also holds.
Let N be the set of all NHB policies. We have
LEMMA 1: For any π ∈ N ,
I1 (t, π ) × (B 1 (t, π) + B112 (t, π)) = 0,
policy π̂ . This result relies on the following cost symmetry
condition:
CONDITION 1: The cost parameters in the N-system
satisfy b1 = b12 + h2 = c.
This condition indicates that, backlogging a product-1
demand incurs the same cost as backlogging a product-12
demand while holding one unit of component-2 in inventory.
Therefore, when component-2 is available, there is no immediate cost difference for allocation of a unit of component-1
among different product-demands. We can either use this
unit to satisfy a product-1 demand (and hence backlog a
unit of product-12, incurring cost b12 + h2 ), or use it to satisfy a product-12 demand (and hence backlog a product-1,
incurring cost b1 ). This cost structure is reasonable in certain
situations in practice as discussed in the Introduction.
4.3.2.
Asymmetric Costs
The next question is, is the sample-path cost lower bound
given by Proposition 1 achievable when Condition 1 does not
hold? Consider, for example, b1 < b12 + h2 . This condition
implies that, to reduce cost, whenever component-2 inventory
is available, we should give product-12 higher priority when
allocating component-1. This is exactly what the solution (16)
in Proposition 1 (i) indicates: under a sample-path optimal
policy π, we should never backlog product-12 only because
of a shortage of component-1. As a result, Proposition 1 (i)
implies that I1 (t, π ) ≥ Y2 (t, π ) for any t. (Otherwise, if there
exists t 0 such that I1 (t0 , π ) < Y2 (t0 , π ), on the event when at
least I1 (t0 , π ) + 1 units of demand of product-12 have arrived
and nothing else has changed, there would be at least one unit
of backlog of product-12 due to stockout of component-1).
Plugging (15) and (16) into (13), it follows from (11) that
+
([N1 (t, π )]− − [N2 (t, π )]− ) + N1 (t, π ) + [N2 (t, π )]−
+
I1 (t, π ) = (N1 (t, π ) + [N2 (t, π)]− ) ,
+
B 1 (t, π ) + B112 (t, π) = ([N1 (t, π)]− − [N2 (t, π)]− ) .
Lemma 1 shows that the NHB policies satisfy the necessary
condition (17), and hence, they lead to the lowest total number of backorders (19) on every sample path. These results
are extensions of those in [16]. While these authors derive
the total number of backorders under any given IBS replenishment policy and NHB allocation rule, our results apply to
any Ft -adaptive policy that is NHB. In addition, we obtain
an exact expression for the on-hand inventory of the common
component as well.
4.3.1.
625
Symmetric Costs
Lemma 1 also shows that the sample-path lower bound
in Proposition 1 (ii) is achievable by an NHB admissible
≥ [N2 (t, π )]+ .
This, in turn, implies either (a) [N1 (t, π )]− ≥ [N2 (t, π )]−
and [N1 (t, π )]+ ≥ [N2 (t, π )]+ , or (b) N1 (t, π ) ≥
[N2 (t, π )]+ − [N2 (t, π )]− for any t.
However, recall from (1) that Ni (t, π ) (before allocation)
is controllable only through I Pi (t − Li , π ) by replenishment
orders before time t − Li . If D1 (t − L1 , t] turns out to be
big, there is no way we can adjust N 1 to guarantee (a) or (b).
Therefore there does not exist a Ft -adaptive allocation policy such that (15) and (16) hold. Consequently, we conclude
that, in this case, there does not exists a sample-path myopic
allocation policy such that the sample-path cost lower bound
is achievable.
Similarly, we can argue that no matter which kind of
replenishment policy and allocation rule are used, we cannot always keep B 1 (t, π ) = 0. The reason is the same as the
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above: we cannot control D 1 (t − L1 , t]. Hence, the solution
in Proposition 1 (iii) is not implementable.
To summarize, we have:
THEOREM 1 (Lower Bound’s Attainability) Assume that
the component net inventory is given for any time t.
i. Under Condition 1, there exists an NHB policy such
that the conditional sample-path lower bound given
by the linear program [P3; see Proposition 1 (ii)]
can be achieved.
ii. When Condition 1 does not hold, no policy can
achieve the conditional sample-path lower bound
given by the linear program [P3; see Proposition
1 (i) and (iii)].
5.
SYMMETRIC COSTS: EXACT OPTIMAL
POLICY
In this section we assume Condition 1 holds. We first characterize the form of the optimal policy and then discuss the
properties of the optimal policy.
5.1.
Form of Optimal Policy
Lemma 1 shows that the conditional sample path lower
bound of Proposition 1 (ii) is achievable by an NHB policy π̂ . Because the NHB allocation rules depend on the
system state at time t only (after receiving the replenishment shipment), the next step is to see whether we can
identify a type of replenishment policy that can minimize
E[ϕ(I P1 (t −L1 , π )−D1 (t), I P2 (t −L2 , π)−D2 (t))] among
all π̂ ∈ N (see (5)).
Note that the best an admissible replenishment policy can
do to influence the net inventory of component i at t is through
an order decision at t −Li so to increase the inventory position
I Pi (t − Li , π ), using the latest possible information Ft−Li .
The problem then becomes how to synchronize the replenishment decisions of both components. For ease of presentation,
we divide our analysis into two cases:
(SL)
Product-specific component with longer leadtime:
L2 = L + ≥ L1 = L;
(CL) Common component with longer leadtime: L1 =
L + > L2 = L.
A special case of (SL) is
(IL)
Identical leadtimes: = 0, L1 = L2 = L.
First we look at SL case, L2 ≥ L1 . Let
G(y1 , y2 ) = h1 E(y1 − D1 ) + h2 E(y2 − D2 )
+ (c + h1 )E((D1 − y1 )+ ∨ (D2 − y2 )+ ),
(21)
Naval Research Logistics DOI 10.1002/nav
which is the expected inventory (i.e., holding and backorder)
cost at time t, given the inventory position of component-i at
t − Li is yi , i = 1, 2. Note that, because the Di have densities,
G is C 2 , twice continuously differentiable.
LEMMA 2: G(y1 , y2 ) is L convex. That is, (i) it is jointly
convex and submodular, and (ii) its Hessian matrix is a
diagonally dominant M-matrix.
Note that after the order decision for component-2 is
made at t − L2 , by the time we make an order decision on
component-1 at t − L1 , some random demand has occurred
in the time interval (t − L2 , t − L1 ] = (t − L − , t − L],
which should be taken into account. Observe that
I P2 (t − L − , π ) − D2 (t)
= I P2 (t − L − , π ) − D2 (t − L) − D2 (t − L, t].
Because Ft−L− ⊆ Ft−L , by the tower property of conditional expectations and Proposition 1 (ii),
E[C(t, π )|Ft−L− ]
= E {E[C(t, π )|Ft−L ]|Ft−L− }
≥ E{E[ϕ(I P1 (t − L, π ) − D1 (t), I P2 (t − L − , π )
− D2 (t))|Ft−L ]|Ft−L− }
= E[G(I P1 (t − L, π ), I P2 (t − L − , π )
− D2 (t − L))|Ft−L− ]
= E{E[G(I P1 (t − L, π ), I P2 (t − L − , π )
− D2 (t − L))|Ft−L ]|Ft−L− }.
(2)
(22)
Our objective is to find π ∈ A to minimize (22). Noting
that I P1 (t − L, π ), I P2 (t − L − , π ) and D2 (t − L) are
measurable to Ft−L , we can first find y1 to minimize
G(y1 , I P2 (t − L − , π ) − D2 (t − L)).
(23)
Denote this minimizer by s̃1 (I P2 (t − L − , π ) − D2 (t −
L)). Because I P2 (t − L − , π ) is measurable to Ft−L−
and D2 (t − L) is independent of Ft−L− , we can find y2
to minimize
E[G(s̃1 (y2 − D2 (t − L)), y2 − D2 (t − L))].
(24)
As each step of (22)–(24), the construction reaches the lower
bound. A more rigorous proof is given in the Appendix.
This analysis establishes the optimality of a CBS replenishment policy of the form (s̃1 (·), s̃2 ), where s̃2 is a constant
and s̃1 (·) is a real function. The policy works as follows: the
product-specific component-2 follows a standard base-stock
policy with base-stock level s̃2 . The common component-1
follows a state-dependent base-stock policy with base-stock
level s̃1 (·) that depends on the demand realization in the last Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
time periods. That is, for any time t > 0, keep the component2 inventory position at s̃2 and the component-1 inventory
position at s̃1 (s̃2 − D2 (t − , t]).
∗
, and s1 (y) be determined by
More specifically, let s1∗ , s12
c
c
∗
=
, Pr(D 1 + D 12 ≤ s12
)=
,
Pr(D ≤
c + h1
c + h1
(25)
h1
1
12
1
s1 (y) = min x : Pr(D + D ≥ x, D ≥ x − y) =
.
c + h1
(26)
1
s1∗ )
As for any given y, Pr(D 1 + D 12 ≥ x, D 1 ≥ x − y) is
decreasing in x, s1 (y) is well-defined. Let
⎧
∗
⎪
⎨y + s1 , y ≤ 0,
∗
s̃1 (y) = s1 (y),
(27)
0 < y ≤ s12
,
⎪
⎩ ∗
∗
s12 ,
y > s12 ,
G̃(y) = E[G(s̃1 (y − D2 ), y − D2 )],
s̃2 = argminy G̃(y).
(28)
(29)
∗
]. We have
Clearly, y ≤ s̃1 (y) ≤ y + s1∗ for any y ∈ (0, s12
LEMMA 3: Assume SL. (i) For any fixed y, s̃1 (y) given
by (27) minimizes G(·, y). Moreover, s̃1 (y) is increasing and
continuous, and 0 ≤ d s̃1 (y)/dy ≤ 1. (ii) G̃(y) defined by
(28) is convex. Thus, its minimal point s̃2 is unique.
LEMMA 4: Assume SL. Suppose that the initial system
satisfies I P2 (0) = s̃2 and I P1 (0) = s̃1 (s̃2 ). Then, under the
admissible policy π̃ determined by the CBS replenishment
policy (s̃1 (·), s̃2 ) and the NHB allocation rule, the system is
fully coordinated in the following sense : for any t ≥ 0,
which is determined by
Pr(D 1 + D 12 ≥ s̃1 (s̃2 ), D 1 ≥ s̃1 (s̃2 ) − s̃2 ) =
Pr(D 1 ≤ s̃1 (s̃2 ) − s̃2 , D 12 ≥ s̃2 ) =
G̃(y) = minG(y1 , y).
h2
.
c + h1
h2
,
c + h1
s2 (y) = min x : Pr(D1 (L1 ) ≤ y − x,
h2
D 12 ≥ x) =
.
(32)
c + h1
Pr(D1 (L1 ) ≤ s2∗ ) =
Let
s̆2 (y) =
y − s2∗ , y ≤ s2∗ ,
y > s2∗ ,
s2 (y),
G1 (y1 , y2 ) = EG(y1 − D 1 , y2 ),
Ğ(y) = E[G (y − D2 , s̆2 (y − D2 ))],
1
y1
s̆1 = argminy Ğ(y).
Hence, we have
(s̃1 (s̃2 − D2 (t − , t]), s̃2 ) = (s̃1 (s̃2 ), s̃2 ) = argminG(y1 , y2 ),
y1 ,y2
(31)
Obviously, a CBS policy requires more information than
an IBS policy. It requires recording the demand for product12 in the last interval (t − , t]. Because component-1 serves
product-1 and product-12, at any time t, the inventory of
component-1 will be updated in such a way that its available net inventory will be larger than the available amount of
component-2 at time t + L1 , that is, s̃1 (s̃2 − D2 (t − , t]) ≥
s̃2 − D2 (t − , t] (see Lemma 3). When the leadtimes are
identical, a CBS policy reduces to an IBS policy. In particular, when the net inventory position of component-2,
∗
s̃2 −D2 (t −, t], is higher than s12
, we order component-1 up
∗
to s12 , which serves both products. When component-2 will
definitely be in shortage (s̃2 − D2 (t − , t] ≤ 0), we raise the
inventory position of component-1 higher than component2’s, with amount s1∗ , to serve the demand for product-1. When
component-2 is not definitely in shortage but there is not a
sufficiently large inventory of it, we order component-1 in a
way that balances between these two extremes.
For the CL case, following a similar line of analysis as
in Theorem 2, we can show that a similar type of policy is
optimal. Define
Based on the above two lemmas, we can prove
REMARK 2: In the case of IL, (28) degenerates into
h1
, (30)
c + h1
Denote s̄1 = s̃1 (s̃2 ), s̄2 = s̃2 as the solution of (30)–(31), the
IBS policy with base-stock levels (s̄1 , s̄2 ) and an NHB allocation rule are optimal among all admissible policies A. The
minimum long-run average inventory cost is C ∗ = G(s̄1 , s̄2 ).
I P1 (t, π̃ ) = s̃1 (s̃2 − D2 (t − , t]), I P2 (t, π̃) = s̃2 .
THEOREM 2 (Optimal Policy SL). Assume Condition
1. When the common component has a shorter leadtime
(L1 < L2 ), the optimal admissible policy constitutes a CBS
replenishment policy with parameters (s̃1 (·), s̃2 ) and an NHB
allocation rule. The minimum long-run average inventory
cost is C ∗ = G̃(s̃2 ).
627
(33)
(34)
(35)
(36)
It is not difficult to see that s̆2 (y) ≤ y − s2∗ for any y. Different from Theorem 2, in order to make the minimizer (see
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(33), s̆2 (·)) an implementable policy, we need to filter out
the demand information for product-1 in the last interval
(t − , t], i.e., D 1 (see (34)). To make this possible, we
need to assume independent product demands. Parallel to the
SL case, we have the following results.
LEMMA 5: Assume CL. (i) G1 (y1 , y2 ) is L convex. (ii)
For any fixed y, s̆2 (y) given by (33) minimizes G1 (y, ·).
In addition, s̆2 (y) is continuous, increasing, and 0 ≤
d s̆2 (y)/dy ≤ 1. (iii) Ğ(y) defined by (35) is convex. Thus,
its minimal point s̆1 is unique.
LEMMA 6: Assume CL. Suppose that the initial system
satisfies I P1 (0) = s̆1 and I P2 (0) = s̆2 (s̆1 ). Under the
admissible policy π̆ given by the CBS replenishment policy
(s̆1 , s̆2 (·)) and the NHB allocation rule, the system is fully
coordinated in the following sense: for any t ≥ 0,
I ˘P 1 (t, π̆ ) = s̆1 , I ˘P 2 (t, π̆) = s̆2 (s̆1 − D2 (t − , t]).
THEOREM 3 (Optimal Policy CL) Assume Condition
1. When the common component has a longer
1leadtime (L
1>
L
),
and
the
product
demand
processes
D
(t),
t
≥
0
and
2
12
D (t), t ≥ 0 are independent, the optimal admissible policy constitutes an NHB allocation rule and a CBS replenishment policy with parameters (s̆1 , s̆2 (·)) (namely, for any time
t > 0, keep the component-1 inventory position at s̆1 and the
component-2 inventory position at s̆2 (s̆1 −D2 (t −, t])). The
minimum long-run average inventory cost is C ∗ = Ğ(s̆1 ).
It is worth mentioning that the policy requires only recording the demand for product-12 in the last interval (t − , t].
Because component-2 only serves product-12, at any time t,
the inventory of component-2 will be updated in such a way
that its available net inventory will be less than the remaining
inventory of component-1 after product-12’s depletion during
(t − , t]. That is, s̆2 (s̆1 − D2 (t − , t]) ≤ s̆1 − D2 (t − , t]
(see Lemma 5). In particular, this CBS policy implies that
I P2 (t) and D1 (t − , t] are independent for any t.
REMARK 3: The optimal CBS policies in Theorems 2, 2
reduce to the optimal BBS policy for the corresponding
assembly system when D1 (t) ≡ 0. The N-system is degenerated into a standard single-item inventory model consisting
of component-1 if D12 (t) ≡ 0, and the optimal CBS policy
reduces to the standard base-stock policy.
5.2.
Properties of the Optimal Replenishment Policy
We next investigate how the optimal policy parameters
change in response to changes in demand andleadtimes. We
assume
demand processes D1 (t), t ≥ 0
12that the product
and D (t), t ≥ 0 are independent. In particular, we show
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how the demand parameters and leadtime impact the optimal policies, that is, (a) (s̄1 , s̄2 ) under IL (Remark 2); (b)
(s̆1 , s̆2 (·)) under CL (Theorem 2); and (c) (s̃1 (·), s̃2 ) under
SL (Theorem 2).
DEFINITION 3: A stochastic process Ŵ(t), t ≥ 0 is
said to be stochastically larger than another stochastic process
{W(t), t ≥ 0} if for any fixed t and x, Pr(W(t) > x) ≤
Pr(Ŵ(t) > x).
Regarding the effect of demand variability, we have
PROPOSITION 3: Assume Condition 1. (i) When the
demand process for product-12 becomes stochastically
larger, the optimal base-stock levels under IL and the
CBS-levels under both SL and CL increase. That is,
s̄1 , s̄2 , s̆1 , s̃2 , s̆2 (y) and s̃1 (y) increase for any fixed y; (ii)
When the demand process for product-1 becomes stochastically larger, component-1’s base-stock levels under IL and
CL, and CBS level under SL, that is, s̄1 , s̆1 and s̃1 (y) for
any fixed y, respectively, increase, while the CBS level for
component-2 under CL, s̆2 (y), decreases for any fixed y.
The above proposition shows that we should keep more
inventory of both components when the demand process for
product-12 is stochastically larger, which is intuitive. Similarly, a stochastically larger demand process for product-1
pushes up the optimal inventory position of component-1.
However, less intuitively, when component-1 has a longer
leadtime (CL), a stochastically larger demand process for
product-1 actually pulls down the optimal inventory position
of component-2. This can be explained as follows. Recall
that one unit of component-2 needs to match with one unit
of component-1 to fulfill one unit of product-12. But a larger
demand process product-1 will consume more of component1, leaving less component-1 for product-12, so it is optimal to
lower component-2’s inventory to save the inventory holding
cost without changing the backorder cost.
The next result reveals the impact of the leadtimes and the
leadtime difference.
PROPOSITION 4: Assume Condition 1. (i) When increases, s̆1 and s̃2 increase, and there is no effect on s̃1 (y)
and s̆2 (y) for any fixed y; (ii) When L increases, s̄1 and s̃1 (y)
increase for any fixed y.
According to the above proposition, a bigger leadtime difference will always push up the optimal inventory positions
for both components. This is because a larger implies
a larger leadtime demand for the component with a longer
leadtime and bigger demand realization in the time interval of length . The latter fact affects the component that
has a shorter leadtime indirectly. However, does not affect
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
6.
629
ASYMMETRIC COSTS: ASYMPTOTICALLY
OPTIMAL POLICY
In this section, we consider systems with asymmetric costs.
That is, Condition 1 may not hold. Recall that in this case
there is no simple exact optimal policy, so our goal is to
characterize an asymptotically optimal policy as the demand
volume grows large. Following the notation and procedures
in [2, 22, 23], we consider a sequence of the N-systems,
indexed by n, in which all problem parameters remain the
same as before, but the product demand rates are linear in
n. All processes in the nth system are superscripted by n. In
particular, the product-K demand rate in the nth system is
EDK,n (1) = nλK , K ∈ {1, 12} .
Figure 2. Effect of L on s̆2 (y).
DEFINITION 4: An admissible policy π̂ is asymptotically
optimal if for any T > 0 and π ∈ An
the CBS levels s̃1 (·) and s̆2 (·). Thus, to serve the additional
demand resulting from the increased , it is optimal to
increase the inventory of the component that has a longer
leadtime.
When L increases, the optimal common-component inventory increases under IL and SL, but the optimal inventory
level for the product-specific component can increase or
decrease. The reason is as follows: a larger L increases
leadtime demand for both products. Conversely, it seems
reasonable to raise the inventory of component-2 to serve
the increased demand of product-12. We call this the first
effect. Conversely, it may be wise to lower the inventory of
component-2 because, due to increased demand for product1, fewer of component-1 may be left for product-12. We call
this the second effect. Whether L pulls down or pushes up
the optimal inventory of component-2 depends on which of
the two effects dominates, as shown in Fig. 2. When inventory of component-1 is low, the second effect dominates as a
result, a larger L reduces the inventory of component-2. On
the contrary, when the inventory of component-1 is high, the
first effect dominates, and a larger L pushes up the inventory
of component-2.
The results in this section are summarized in Table 1, in
which each row reports the impact of changing the corresponding variable unilaterally. We note that a stochastically
larger demand implies a larger mean demand. It may imply a
higher demand variability in some cases but not necessary for
all case. It would be interesting to investigate the effects of
demand variability with mean demands fixed. These effects,
however, are much more complex to characterize, because
they involve the relative magnitudes of the demand, cost,
and leadtime parameters. This is even true for a single-item
inventory system, in which the relative magnitude of cost
parameters matters; see, for example, [29]. Therefore, we
leave such a study to future research.
1
lim sup √ E
n
n→∞
0
T
⎛
2
⎝
hi · Iin (t, π̂ ) +
i=1
1
≤ lim sup √ E
n
n→∞
0
T
⎞
b ·B
K
K,n
(t, π̂ )⎠ dt
K∈{1,12}
⎛
2
⎝
hi · Iin (t, π ) +
i=1
⎞
b ·B
K
K,n
(t, π )⎠ dt.
K∈{1,12}
From the analysis in Section 5, we know that the component allocation and replenishment actions become more
and more frequent as the demand rates increase. To detect
these actions, we first equally partition time interval [0, ∞)
into subintervals such that in each subinterval there is only
one component allocation decision. Thus, similar to Proposition 1, we can again use the linear-programming approach
to develop a lower bound on the system cost and identify
an allocation rule to achieve this lower bound. Then, following the procedure described in Section 3.2, by leveraging the
tower property of conditional expectations we can find the
optimal component replenishment
policy.
n
For any π =
(Oi (t), AK,n (t)), t ≥ 0
∈ An,
K,n
we first modify the allocation process A (t), t ≥ 0
into a discrete-review policy AK,n
such that
(t), t ≥ 0
n
π = (Oin (t), AK,n
(t)),
t
≥
0
∈
A
.
The
modified
pol
icy allocates components for products at review points
l n , 2l n , 3l n , · · · with
⎛
⎞2/3
1
⎟
⎜
ln = ⎝ ⎠
n (λ1 )2 + (λ12 )2
,
and does not allocate any component at all other times.
Consider any K ∈ {1, 12}. Initially, set AK,n
(0) = 0 and
K,n
B (0) = 0. Then, the process is defined recursively by
n
K,n
(
l n ) − BK,n (
l n ),
AK,n
(
l ) = D
(37)
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Table 1. Summary of comparative statics on optimal replenishment policy
IL (L1 = L2 = L)
Demand-1
Demand-12
Leadtime L
SL (L1 = L, L2 = L + )
CL (L1 = L + , L2 = L)
Component-1
s̄1
Component-2
s̄2
Comp-1
s̃1 (·)
Comp-2
s̃2
Comp-1
s̆1
Comp-2
s̆2 (·)
–
?
?
–
–
?
?
?
?
–
Note: , , –, and ? stands for increasing, decreasing, unaffected, and unknown or nonuniversal, respectively.
where BK,n (
l n ) solves
Let
min b1 B 1 + b12 B 12 + h1 I1 + h2 I2
(38)
B K ≥0
s.t. I1 = In1 (0)+On1 (
ln −L1 ) − D1,n (
ln )
− D12,n (
ln )+B1 +B12 ≥ 0,
(39)
I2 = I2n (0) + O2n (
l n − L2 ) − D12,n (
l n ) + B 12 ≥ 0,
(40)
AK,n ((
− 1)l n ) + B K ≤ DK,n (
l n ).
(41)
K,n
n
For t ∈ [
l n , (
+ 1)l n ), define AK,n
(t) = A (
l ). Then
we know
n
π = (Oin (t), AK,n
(t)), t ≥ 0 ∈ A .
(42)
For ease of exposition, we assume that the variances and
covariance of D1,n ( n1 ) and D12,n ( n1 ) are independent of n, and
denote
1 2
1,n 1
12 2
12,n 1
, (σ ) = V ar D
,
(σ ) = V ar D
n
n
1
1
2
(σ 1,12 ) = Cov D1,n
− λ1 , D12,n
− λ12 .
n
n
2+ε
<
We also assume (in this subsection) that E[DK,n ( n1 )]
∞, K ∈ {1, 12} for ε = 3.
Let (1 , 12 ) be a bivariate normal random vector with
zero-mean and covariance matrix = (ij ) where
2
11 = (σ 1 ) L,
2
22 = (σ 12 ) L,
2
12 = 21 = (σ 1,12 ) L.
(43)
K
Also, let be a normal variable with zero-mean and vari2
ance (σ K ) . Denote 1 = 1 + 12 and 2 = 12 . The
following is an analogous to (21):
F (y1 , y2 ) = h1 E(y1 − 1 ) + h2 E(y2 − 2 )
+ (h1 + h2 + b12 )E(2 − y2 )+
+
+ (h1 + b1 )E(1 − y1 − (2 − y2 )+ ) .
Naval Research Logistics DOI 10.1002/nav
θ1 (y) = min x : Pr(1 + 12 ≥ x,
h1
1
≥ x − y) =
,
h1 + b 1
(44)
F̃ (y) = E[F (θ1 (y − 12 ), y − 12 )],
θ̃2 = argminF̃ (y),
(45)
y
θ2 (y) = min{x : h2 − (h1 + h2 + b12 )Pr(12 > x)
+ (h1 + b1 )Pr(1 ≤ y − x, 12 ≥ x) = 0},
F 1 (y1 , y2 ) = EF (y1 − 1 , y2 ),
F̆ (y) = E[F 1 (y − 12 , θ2 (y − 12 ))],
θ̆1 = argminF̆ (y).
(46)
y
We have
THEOREM 4 (Asymptotically Optimal Policy) Assume
that b1 ≤ h2 + b12 .
(i) (SL) When the common component has a shorter
leadtime, the CBS replenishment policy
√
I P1n (t) = (λ1 + λ12 )nL1 + nθ1
D12,n (t − , t] − λ12 n
θ̃2 −
,
√
n
√
I P2n (t) = λ12 nL2 + nθ̃2 ,
and the discrete-review NHB (myopic) allocation
policy with the PBC rule defined by (37)–(41) are
asymptotically optimal in An .
(ii) (CL) When the common component has a
longer
1,n leadtime and the
product demand
processes
D (t), t ≥ 0 and D12,n (t), t ≥ 0 are independent, the CBS replenishment policy
I P1n (t) = (λ1 + λ12 )nL1 +
√
nθ̆1 ,
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
√
I P2n (t) = λ12 nL2 + nθ2
D12,n (t − , t] − λ12 n
θ̆1 −
,
√
n
and the discrete-review NHB (myopic) policy with
the PBC rule (37)–(41) are asymptotically optimal
in An .
(ii) (CL) When the common component has a
longer
1,n leadtime and the
product demand
processes
D (t), t ≥ 0 and D12,n (t), t ≥ 0 are independent, the CBS replenishment policy
and a discrete-review NHB (myopic) policy with the
PBC rule defined by (37)–(41) are asymptotically
optimal in An .
The above theorem shows that when costs are asymmetric
and demand volume is large, a CBS-policy—a policy type
identified in Section 5—continues to be optimal. For example, under SL, it is optimal to use a base-stock policy to
control the product–specific component-2, and use a realtime CBS-policy, which is updated according to the latest
demand information, to control the common component. The
optimal policy of each component has two parts: the mean of
leadtime demand for the component and a safety–stock which
hedges against the demand volatility. This form of policy is
intuitive and commonly seen in practice.
Next, we look at the case b1 ≥ h2 + b12 . Define
H (y1 , y2 ) = h1 E(y1 − 1 ) + h2 E(y2 − 2 )
631
√
I P1n (t) = (λ1 + λ12 )nL1 + nη̆1 ,
√
I P2n (t) = λ12 nL2 + nη2
D12,n (t − , t] − λ12 n
η̆1 −
,
√
n
and a discrete-review NHB (myopic) policy with
the PBC rule (37)–(41) are asymptotically optimal
in An .
We have, thus, shown that in the high-volume demand
regime, the structure of the asymptotically optimal policy
is essentially the same as that of the exact optimal policy for
the symmetric cost case.
+ (h1 + h2 + b12 )E[(1 − y1 )+
∨ (2 − y2 )+ ],
η1 (y) = min x : Pr(1 + 12 ≥ x,
7.
In this section, we first examine the effectiveness of the
commonly used IBS policies for the symmetric cost case.
Then, based on the optimal and asymptotically optimal policies given in Section 5 and 6, we develop several heuristic
policies for the asymmetric cost case. To compare the effectiveness of these heuristic policies, we also construct bounds
on the system cost.
1 ≥ x − y) =
HEURISTICS
h1
,
h1 + h2 + b12
H̃ (y) = E[H (η1 (y − 12 ), y − 12 )],
η̃2 = argminH̃ (y),
y
η2 (y) = min{x : h2 − (h1 + h2 + b12 )
Pr(1 ≤ y − x, 12 ≥ x) = 0},
7.1.
H (y1 , y2 ) = EH (y1 − , y2 ),
1
1
H̆ (y) = E[H 1 (y − 12 , η2 (y − 12 ))],
η̆1 = argminH̆ (y).
y
(47)
Then, we can show
THEOREM 5 (Asymptotically Optimal Policy) Assume
that b1 ≥ h2 + b12 .
(i) (SL) When the common component has a shorter
leadtime, the CBS replenishment policy
I P1n (t) = (λ1 + λ12 )nL1
√
D12,n (t − , t] − λ12 n
+ nη1 η̃2 −
,
√
n
√
I P2n (t) = λ12 nL2 + nη̃2 ,
Effectiveness of Independent Base-stock Policies
for Symmetric Costs
In this subsection, we assume Condition 1 holds and the
NHB allocation rule is followed. In this case, with the form of
the optimal replenishment policy known to be a CBS policy,
it is interesting to learn under what situations the widely used
IBS policy can perform well.
First, we describe how to compute the optimal IBS levels
for the CL (L1 > L2 ) case. The case of SL (L1 < L2 ) can be
done analogously. Similar to Eq. (21), under the IBS policy
with base-stock levels (y1 , y2 ), the long-run average system
cost denoted by C bs (y1 , y2 ) is
C bs (y1 , y2 ) = Eϕ(y1 − D1 (t), y2 − D2 (t))
= h1 · E(y1 − D1 − D1 ) + h2 · E(y2 − D2 )
+ (c + h1 ) · E((D1 + D1 − y1 )+
∨ (D2 − y2 )+ ).
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7.2.
Heuristic Policies for Asymmetric Costs
In this subsection, we assume asymmetric costs. In this
case, the form of the exact optimal policy is unknown, so we
develop several heuristic policies and examine their effectiveness. Before presenting the heuristic policies, we first
establish upper and lower bounds for the system cost, which
will be used to assess the performance of the heuristic policies. To get these bounds, we construct two bounding systems
that differ from the original one only in their cost structures. For convenience, denote the original system by S =
S(b1 , b12 , h1 , h2 ). Define two systems S = S(b1 , b12 , h1 , h2 )
and S = S(b̄1 , b̄12 , h̄1 , h̄2 ), where
h1 = h1 ,
Figure 3. Optimality loss of IBS policies (b1 = 4).
This is Eq. (21) with D1 replaced by D1 + D1 . As a result,
the optimal value of (y1 , y2 ) can be calculated as in Eqs. (30)
and (31). The corresponding optimal long-run average cost
is denoted by C ∗bs .
Next, we report the results of a numerical study. The
effectiveness of the IBS policy is measured by
err = 100 ×
C ∗bs − C ∗
.
C∗
In this study, we assume that the demand processes of
product-1 and product-12 are Poisson with rates λ1 and
λ12 , and keep h1 = 1, h2 = 1, b12 + h2 = b1 and
(λ1 + λ12 ) × (L + ) = 60. We vary the other parameters as
follows:
b1 ∈ {2, 3, . . . , 10} , λ12 /λ1 ∈ {0.1, 0.5, 1, 2, . . . , 10} ,
/L ∈ {0.1, 0.6, . . . , 4.6, 5} .
Thus, there are a total of 9×12×11 = 1188 scenarios. These
results show that the IBS policy can perform very poorly. In
the 1188 scenarios, compared with the CBS policy, the IBS
policy results in an average 7.65% loss of optimality and a
maximum loss of 33.87%. Fig. 3 demonstrates how the optimality loss of the IBS policy varies with system parameters
such as leadtimes and demand rates. We observe that, the
IBS policy performs fairly well when leadtimes are similar, that is, L, or when product-1 demand dominates,
that is, λ12 λ1 . For example, the average optimality loss
is only 1.57% when /L = 0.1 and is about 1.84% when
λ12 /λ1 = 0.1. However, the optimality loss can be significant when the component leadtimes differ substantially or
when λ12 λ1 . For instance, the average optimality loss is
approximately 10.97% when /L = 5.
Naval Research Logistics DOI 10.1002/nav
h2 = b1 ∧ h2 ,
b1 = b12 + h2
= b1 ∧ (b12 + h2 ),
h̄1 = h1 ,
h̄2 = h2 ,
b̄1 = b̄12 + h̄2 = b1 ∨ (b12 + h2 ).
Clearly, both S and S satisfy Condition 1. From §5, the optimal policies for these systems are characterized by Theorems
2–3, with c = b1 ∧(b12 +h2 ) and c = b1 ∨(b12 +h2 ), respectively. Let the resulting minimum long-run average costs of
S and S be C ∗ and C̄ ∗ . We have
PROPOSITION 5: If b1 = b12 + h2 , the minimum longrun average cost C ∗ is bounded by those of systems S and S.
That is, C ∗ ≤ C ∗ ≤ C̄ ∗ .
In the following, we present five heuristic policies that are
simple and easy to implement. The first three are constructed
from the optimal policies for symmetric costs, and the other
two are generated by the asymptotically optimal policies for
asymmetric costs.
Heuristic CN: CBS Replenishment Policy + NHB Allocation Rule. For the component replenishment, this heuristic
adapts the CBS policy characterized by Theorems 2–3 for
symmetric costs with the modified c = (b1 + b12 + h2 )/2 in
place of (21). This choice of c reflects the desire to balance
the backorder cost of product-1 and the sum of the product-12
backorder cost and the holding cost of the product-specific
component. For component allocation, note that for any given
Ft -adaptive replenishment policy (which includes the CBS
policy), there are two ways to reduce the backorder costs
through a NHB allocation rule, which is also Ft -adaptive
and satisfies Definition 1 (ii–iii): (a) reduce the total number
of backorders or (b) redistribute backorders among different
products such that the product with higher penalty cost has a
small number of backorders. It is shown by [20] that any NHB
allocation rule would achieve (a) if an IBS replenishment
policy is adopted. Furthermore, when there are backorders
of both products, a PBC policy can achieve (b). For this reason, this heuristic uses the combination of NHB and PBC for
component allocation.
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
Heuristic IS: IBS Replenishment Policy + Static Rationing
Allocation Rule. This heuristic uses the simpler IBS replenishment policy with parameters (sIS1 , sIS2 ) given by §7.1.
The allocation rule is a rationing policy with a “static” (i.e.,
fixed) reservation level r for the common component. Consider b1 < b12 + h2 , for example, the common component-1
goes to product-1 only when its on-hand inventory is higher
than r; otherwise, the on-hand inventory of component-1 is
reserved for product-12. Now we demonstrate how to determine r. By an analysis for M/M/1 make-to-stock queue,
[8] (see their paper, P858, the right column, Section 3.3.3
Priority with Reservation Level) prove that when the holding cost is h and penalty cost is b, the optimal reservation
+
level is ln(h)−ln(b)
, where ρ is a ratio between two product
ln(ρ)
demand rates. For N-system, when a common component is
available but being held for satisfying the future product-12
demand while keeping product-1 demand backordered, the
actual holding cost for the common component is h1 + b1 .
Similarly, the backorder cost for product-12 is h1 + h2 + b12 .
Thus, the level r developed by [8] should be modified as
r=
ln(h1 + b1 ) − ln(h1 + h2 + b12 )
ln(ρ)
λ12
where ρ = 1
.
λ + λ12
+
,
(48)
Heuristic CD: CBS Replenishment Policy + Dynamic
Rationing Allocation Rule. The static reservation level r in
Heuristic IS may be too high when the system states such
as Y2 (t) becomes very low. This is because when there are
not many units of product-specific component available, it is
not economical to reserve many common components to wait
for the future product-12 demand arrivals while back ordering
product-1. For this reason, this heuristic dynamically adjusts
the reservation level r̃(t) according to the inventory level of
the product-specific component. As we argued in Section 4.3,
one natural choice is to choose the available on-hand inventory of component-2 (Y2 (t)) as the level. However, this choice
is not optimal for symmetric costs. To overcome this issue,
we assign an asymmetry weight to Y2 (t) and use r and ρ
defined in (48) to set reservation level as follows
r̃(t) = min {αY2 (t), r} ,
where α = (b12 + h2 − b1 ) · ρ ·
b + h2
.
+ h2 + b 1
12
b12
For component replenishment, this heuristic uses the CBS
policy with the same parameters as in CN.
It is not difficult to see that r = 0 and r̃(t) = 0 when
b1 = b12 + h2 . Therefore, the allocation rules of all three
heuristics so far are identical to the optimal policy under
symmetric costs.
633
Heuristic AS: Asymptotically Optimal Replenishment
Policy + Static Rationing Allocation Rule. This heuristic
adopts the asymptotically optimal replenishment policy given
by Theorems 4–5 and the same allocation rule as in Heuristic
IS.
Heuristic AD: Asymptotic Optimal Replenishment Policy +
Dynamic Rationing Allocation Rule. This heuristic employs
the replenishment policy as in Heuristic AS and and the
allocation policy as in Heuristic CD.
Table 2 below summarizes the allocation and replenishment policies of these five heuristics.
We have conducted a numerical experiment to test the performance of these heuristic policies, focusing on the impact
of cost asymmetry and product demand ratio between two
products. The effectiveness of a heuristic k is measured by
Ckh =
Ckh − C ∗
× 100,
C∗
k ∈ {AS, AD, CN, CD, IS} .
Figure 4 reports some results. Here, we assume Poisson
demand processes with demand rates λ1 = 3 and λ12 = 3
(ratio = λ1 /λ12 = 1), fix h1 = 1, h2 = 1, L = 4, = 1
and vary b1 /(b12 + h2 ) ∈ {0.1, 0.2, . . . , 1, 2, . . . , 10}. From
these numerical results, we observe that when cost asymmetry is strong, the performance of both CN and IS are
dominated by the other three. This is not surprising, because
IS uses IBS instead of CBS, and CN does not use rationing
mechanism. For this range of cost parameters, among all
the heuristics, AD is most promising, because it takes into
account the latest information in both replenishment and allocation decisions. Conversely, when cost asymmetry is not
strong, the heuristics CN and CD perform better or reasonably well. We have also tested the robustness of these
observations by varying demand rates, e.g., letting λ1 = 1
and λ12 = 5 (ratio = λ1 /λ12 = 0.2) or λ1 = 5 and λ12 = 1
(ratio = λ1 /λ12 = 5). These additional tests confirm the
above qualitative findings. For space concerns, we do not
report these results here.
8.
CONCLUDING REMARKS
In this article, we have presented the characterization of
the optimal control policy for ATO N- and W-systems with
nonidentical leadtimes and general cost structures. Our main
contributions can be summarized as follows:
• Joint Optimal Inventory Replenishment and Component Allocation Policy. We show that NHB allocation
rule together with a CBS replenishment policy are
optimal control policies when costs are symmetric.
When the cost symmetry is violated, we establish a
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Table 2. Heuristics of allocation and replenishment policies
Replenishment
Allocation
CN
IS
CD
AS
AD
CBS
NHB-PBC
IBS
rationing r
CBS
rationing r̃(t)
Asymptotic CBS
rationing r
Asymptotic CBS
rationing r̃(t)
Figure 4. Comparison of AS, AD, CN, CD, and IS.
cost lower bound and show that our NHB + CBS policy continues to be asymptotically optimal in the highdemand regime. Our policy has an explicit characterization which facilitates implementation: The CBS
policy updates components’ inventory using real-time
information that is closely related to the Rosling’s
well-known BBS policy. The explicit characterization
of the optimal policy also enables sensitivity analysis,
which would not be easy to do otherwise.
• Practical Guidance. We suggest an efficient heuristic which uses a CBS replenishment policy and a
rationing allocation policy whose rationing level is
“dynamically” adjusted by the inventory status. (The
rationing policy reduces to a NHB rule when costs
are symmetric.) Our numerical study shows that the
heuristic outperforms other policies widely adopted
in the literature, such as the IBS replenishment policy
(e.g., [20]) and rationing allocation rules with constant level (e.g., [5, 10, 11]). We find that the IBS
policy can lead to significant optimality loss when
the component leadtimes are very different. These
findings provide a useful guidance for practitioners in
determining control policies of various ATO systems.
• Methodology. The characterization of optimal control policies for ATO systems has long been considered a challenging problem. The breakthrough in this
research is due to a hybrid approach that combines
Naval Research Logistics DOI 10.1002/nav
sample-path analysis, a linear programming lower
bound, and the tower property of conditional expectations. This is very different from the traditional
dynamic programming technique. We hope our
approach can inspire other efforts in related studies.
APPENDIX: W-SYSTEM
In this appendix, we show that our methodology for the N-system is applicable to establish the optimality results for the W-system. Here, I = {1, 2, 3}
and K = {{1, 2} , {1, 3}} . Similar to the the analysis in Section 4.1, for any
admissible policy π , let
• Yi (t, π ): On-hand inventory of product-specific component i not
yet assigned to a demand of product-1i, i = 2, 3;
• Bi1i (t, π ): Backorders of product-1i due to missing product-specific
component i, including backorders missing both components i and
1, i = 2, 3;
• B11i (t, π ): Backorders of product- {1i} due to missing only common
component-1, either component-1 is missing or it is committed to
other products by policy π .
We have that for i = 2, 3,
Ii (t, π ) = Yi (t, π ) + B11i (t, π ),
(49)
B 1i (t, π ) = B11i (t, π ) + Bi1i (t, π ),
(50)
Yi (t, π ) × Bi1i (t, π ) = 0.
(51)
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
Similar to (9), it follows from (49)–(51) that (3) can be written as
635
Similar to the N-system, we pay special attention to the following
condition:
C(t, π ) = h3 · Y3 (t, π ) + h2 · Y2 (t, π ) + b13 · B313 (t, π ) + b12 · B212 (t, π )
+ h1 · I1 (t, π ) + (h3 + b13 ) · B113 (t, π ) + (h2 + b12 ) · B112 (t, π ).
(52)
Analogous to Theorem 1, we have
Moreover, similar to (10)–(11), by (49)–(51), (2) can be expressed as
I1 (t, π ) − B112 (t, π ) − B113 (t, π ) − B212 (t, π ) − B313 (t, π ) = N1 (t, π ),
(53)
Yi (t, π ) − Bi1i (t, π ) = Ni (t, π ),
i = 2, 3.
PROPOSITION 6: If a sample-path optimal admissible policy π̂ exists,
it would yield the following sample-path properties.
i. Under Condition 2, there exists an NHB policy (see
Remark 1) such that the conditional sample-path lower bound
ϕ((N1 (t, π ), N2 (t, π ), N3 (t, π )) given by (58) in Proposition 6 can
be achieved.
ii. When Condition 2 does not hold, no allocation rule can achieve the
conditional sample-path lower bound ϕ((N1 (t, π ), N2 (t, π ), N3 (t, π ))
given by (58) in Proposition 6 [see Proposition 6 (i) and (iii)].
Next, we investigate the form of the optimal replenishment policy. Without
loss of generality, we assume L2 ≥ L3 . We consider three different scenarios: (CS) Common component has the shortest leadtime: L2 ≥ L3 ≥ L1 ;
(CL) Common component has the longest leadtime: L1 > L2 ≥ L3 ; (CM)
Common component has the intermediate leadtime: L2 ≥ L1 ≥ L3 .
Define
G(y1 , y2 , y3 ) = h1 · E(y1 − D1 ) + h2 · E(y2 − D2 ) + h3 · E(y3 − D3 )
i. If b13 + h3 < b12 + h2 , then
−
B113 (t, π̂ ) = (N1 (t, π̂ ) + [N2 (t, π̂ )]− + [N3 (t, π̂ )]− ) ,
(55)
B112 (t, π̂ ) = 0,
C(t, π̂ ) = h1 N1 (t, π̂ ) + h2 N2 (t, π̂ ) + h3 N3 (t, π̂ )
+ (h1 + b
THEOREM 6: [Sample-Path Lower Bound Attainability (W)]
Assume that the component net inventory Ni (t, ·) is given for any time t.
(54)
Similar to Section 4.2, we use a linear programming approach to obtain
a conditional sample-path cost lower bound. In particualar, we formulate a
linear-program in which B112 (t, π ), B113 (t, π ), I1 (t, π ), Y2 (t, π ), B212 (t, π ),
Y3 (t, π ) and B313 (t, π ) are the decision variables, C(t, π ) given by (52) is
the objective function for minimization, and (49)–(51) and (53)–(54) are the
constraints, assuming Ni (t, π ) are fixed. Then, the minimum of the objective function, denote by ϕ((N1 (t, π ), N2 (t, π ), N3 (t, π )) is a lower bound
of C(t, π ).
Solving this linear-program leads to the following results that are similar
to Proposition 1:
13
CONDITION 2: The cost parameters in the W-system satisfy b13 + h3 =
+ h2 = d.
b12
+ (c + h1 ) · E[(y1 − D1 )− ∨ ((y2 − D2 )−
+ (y3 − D3 )− )].
(61)
Similar to Lemma 2, we have
−
+ h3 )[N3 (t, π̂ )]
LEMMA 7: G(y1 , y2 , y3 ) is jointly convex with the unique minimal point
(s̄1 , s̄2 , s̄3 ). Furthermore, s̄2 + s̄3 ≥ s̄1 .
+ (h1 + h2 + b12 )[N2 (t, π̂ )]−
+ (h1 + b13 + h3 )(N1 (t, π̂ ) + [N2 (t, π̂ )]−
+ [N3 (t, π̂ )]− )− .
(56)
s̆1 (y2 , y3 ) = argminy1 G(y1 , y2 , y3 ),
ii. If b13 + h3 = b12 + h2 , then
+ [N3 (t, π̂ )]− )− ,
(57)
+ (h1 + b
−
+ h3 )[[N1 (t, π̂ )] ∨ ([N2 (t, π̂ )]
+ [N3 (t, π̂ )]− )].
iii. If
b13
+ h3 >
b12
(58)
+ h2 , then
B113 (t, π̂ ) = 0,
(59)
−
B112 (t, π̂ ) = (N1 (t, π̂ ) + [N2 (t, π̂ )]− + [N3 (t, π̂ )]− ) ,
h1
s1∗ = max y : Pr(D2 + D3 ≥ y) ≥
,
c + h1
vi (x) = max y : Pr(D2 + D3 ≥ y, Di ≥ y − x) ≥
i = 2, 3,
s1 (y2 , y3 ) = max y : Pr(D2 + D3 ≥ y, D3 ≥ y − y2 ,
h1
D 2 ≥ y − y3 ) ≥
.
c + h1
C(t, π̂ ) = h1 N1 (t, π̂ ) + h2 N2 (t, π̂ ) + h3 N3 (t, π̂ )
+ (h1 + b13 + h3 )[N3 (t, π̂ )]−
+ (h1 + b12 + h2 )[N2 (t, π̂ )]−
+ (h1 + h2 + b12 )(N1 (t, π̂ )
+ [N2 (t, π̂ )]− + [N3 (t, π̂ )]− )− .
(s̆2 , s̆3 ) = argmin{y2 ,y3 } Ğ(y2 , y3 ),
(64)
As G(y1 , y2 , y3 ) is jointly convex, s̆1 (y2 , y3 ) and s̆i , i = 2, 3 are unique
and well-defined. To characterize the properties of s̆1 (y2 , y3 ), define
s1∗ , vi (x), i = 2, 3, and s1 (y2 , y3 ) by
C(t, π̂ ) = h1 N1 (t, π̂ ) + h2 N2 (t, π̂ ) + h3 N3 (t, π̂ )
−
(62)
Ğ(y2 , y3 ) = EG(s̆1 (y2 − D2 , y3 − D3 ), y2 − D2 , y3 − D3 ), (63)
B113 (t, π̂ ) + B112 (t, π̂ ) = (N1 (t, π̂ ) + [N2 (t, π̂ )]−
13
We first consider the CS case. Let
(60)
iv. A necessary condition for an allocation rule π̂ to be optimal is
(57).
(65)
h1
,
c + h1
(66)
(67)
Divide the two dimensional space into the following three regions (see
Fig. 5),
1 = (y2 , y3 ) : y3 ≥ s1∗ , y2 ≥ s1∗ ,
Naval Research Logistics DOI 10.1002/nav
636
Naval Research Logistics, Vol. 62 (2015)
REMARK 4: For the identical leadtime case (L1 = L2 = L3 ), from
Theorem 7, we have that an optimal admissible policy is given by the basestock policy with stock level (s̄1 , s̄2 , s̄3 ) and an NHB allocation rule. The
minimum long-run average inventory cost is given by C ∗ = G(s̄1 , s̄2 , s̄3 ).
For the cases CL and CM, we cannot get the optimal long-run average
cost, however, through our approach (linear-programming and the tower
properties of conditional expectations), a lower bound is established. Basically, it tells us that any replenishment policy that would achieve the lower
bound only uses the last demand information over an interval whose length
is the gap between leadtimes, that is, max {Li , i = 1, 2, 3} − L. Denote
{o(1), o(2), o(3)} as a permutation of 1, 2, 3 such that L = Lo(1) ≤ Lo(2) ≤
Lo(3) . Let Ĉ be the minimum value of the following multistage stochastic
programming (SP) problem
Ĉ = minE{minE[minE(ϕ(y1 − D1 (Lo(3) ), y2
yo(3)
yo(2)
yo(1)
− D2 (Lo(3) ), y3 − D3 (Lo(3) ))
|FLo(3) −Lo(1) )|FLo(3) −Lo(2) ]}.
⎧
⎪
miny1 E[miny2 E(Ĝ(y1 − D1 (0, L1 − L3 ],
⎪
⎪
⎪
⎨ y − D (L − L , L − L ])|F
2
3
1
2
1
3
L1 −L2 )], CLCase,
=
⎪
E[min
E(
Ĝ(y
−
D
(L
−
L1 , L2 − L3 ],
min
⎪
y2
y1
1
2
2
⎪
⎪
⎩ y − D (0, L − L ])|F
)],
CM Case,
2
3
2
3
L2 −L1
Figure 5. Optimal replenishment policy for common component
s̆1 (y2 , y3 ).
2 = {(y2 , y3 ) : y2 ≤ v2 (y3 ), y3 ≤ v3 (y2 )} ,
3 = − 1 − 2 .
(69)
where Ĝ(x, y) = minz G(x, y, z).
Then, s̆1 (y2 , y3 ) can be characterized by
⎧
⎪
s1∗ ,
⎪
⎪
⎪
⎨s (y , y ),
1 2 3
s̆1 (y2 , y3 ) =
⎪v2 (y3 ),
⎪
⎪
⎪
⎩v (y ),
3
2
THEOREM 8 (Lower Bound for CL and CM) Assume Condition 2.
The minimum long-run average cost C ∗ under both CL and CM is bounded
from below by Ĉ, namely, Ĉ ≤ C ∗ .
(y2 , y3 ) ∈ 1 ,
(y2 , y3 ) ∈ 2 ,
(y2 , y3 ) ∈ 3 and y2 ≥ y3 ,
(y2 , y3 ) ∈ 3 and y2 ≤ y3 .
(68)
Moreover,
LEMMA 8: For any given y2 and y3 , s̆1 (y2 , y3 ) defined by (68) satisfies
dvi (x)/dx ≤ 1, ∂s1 (y2 , y3 )/∂yi ≤ 1, i = 2, 3, and s̆1 (y2 , y3 ) is continuous
and increasing in yi , i = 2, 3.
We now study the case of asymmetric cost structure: b12 + h2 = b13 + h3 .
Similar to the N-system, we first construct bounding systems for the original system, so that each of the bounding systems has a symmetric cost
structure. For simplicity, we use the same notation as in the N-system, with
the understanding that we are discussing the W-system here. The bounding systems differ from the original one only in their cost structures. Let
S = S(b12 , b13 , h1 , h2 , h3 ) denote the original system. Define two systems
S = S(b12 , b13 , h1 , h2 , h3 ) and S = S(b̄12 , b̄13 , h̄1 , h̄2 , h̄3 ), where
h1 = h1 , h2 = h2 ∧ h3 , h3 = h2 ∧ h3 , b13 + h3 = b12 + h2
We say a replenishment policy for the W-system is a CBS policy with
parameters (s̆1 (·, ·), s̆2 , s̆3 ) if the policy works as follows: at any time t, maintain the inventory positions of component-2 and component-3 at s̆2 and s̆3
respectively, while keeping the inventory position of component-1 at
= (b13 + h3 ) ∧ (b12 + h2 ) (= d),
h̄1 = h1 , h̄2 = h2 ∨ h3 , h̄3 = h2 ∨ h3 , b̄13 + h̄3 = b̄12 + h̄2
= (b13 + h3 ) ∨ (b12 + h2 ) (= d̄).
s̆1 (s̆2 − D2 (t − 2 , t], s̆3 − D3 (t − 3 , t]).
We have
LEMMA 9: Assume I P1 (0) = s̆1 (s̆2 − D2 , s̆3 − D3 ), I P2 (0) =
s̆2 , I P3 (0) = s̆2 , that is, the system starts with zero inventory and backorders.
Then, system is coordinated when π̆ is used, that is, for any t ≥ 0,
Let the resulting minimum long-run average costs of systems S and S be
C ∗ and C̄ ∗ respectively. We have
PROPOSITION 7: If b13 + h3 = b12 + h2 , the minimum long-run average cost C ∗ is bounded from below by C ∗ and from above by C̄ ∗ , that is,
C ∗ ≤ C ∗ ≤ C̄ ∗ .
APPENDIX: PROOFS
IP1 (t, π ) = s̆1 (s̆2 − D2 (t), s̆3 − D3 (t)), IP2 (t, π ) = s̆2 , IP3 (t, π ) = s̆3 .
THEOREM 7 (Optimal Admissible Policy CS(W)) Assume Condition
2. When the common component-1 has the shortest lead time, an optimal
admissible policy constitutes the CBS replenishment policy (s̆1 (·, ·), s̆2 , s̆3 )
and an NHB allocation rule in N . The minimum long-run average cost is
given by C ∗ = Ğ(s̆2 , s̆3 ).
CBS replenishment policy s̆1 (·, ·) is graphically characterized in Fig. 5.
Naval Research Logistics DOI 10.1002/nav
PROOF OF PROPOSITION 2: For Ft -adaptive replenish policy {(O1 (t),
O2 (t)), t ≥ 0}, we define
⎧
1
⎪
⎨D (t),
if D1 (t) + (I2 (0) + O2 (t − L2 ))
A (t) =
∧D12 (t) ≤ I1 (0) + O1 (t − L1 ),
⎪
⎩ 1
1
D (τ (t)) + Ã (t, t), otherwise;
1
(E-1)
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
⎧
12
⎪
⎪
⎪(I2 (0) + O2 (t − L2 )) ∧ D (t),
⎪
⎨ if D1 (t) + (I (0) + O (t − L )) ∧ D12 (t)
2
2
2
A12 (t) =
⎪
(0)
+
O
(t
−
L
≤
I
),
⎪
1
1
1
⎪
⎪
⎩(I (0) + O (τ (t) − L )) ∧ D12 (τ (t)) + Ã12 (t, t), otherwise,
2
2
2
(E-2)
637
2 = {(D 1 , D 12 ) : D 1 ≤ y1 − y2 , D 12 ≥ y2 },
3 = {(D 1 , D 12 ) : D 1 ≥ y1 − y2 , D 1 + D 12 ≥ y1 }.
Noted that when y1 < y2 , 1 , 2 and 3 can be simplified as {(D 1 , D 12 ) :
D 1 + D 12 ≤ y1 }, ∅, and {(D 1 , D 12 ) : D 1 + D 12 ≥ y1 }, respectively.
First, by (21), we have
where
−
−
(y1 − D 1 − D 12 ) ∨ (y2 − D 12 )
⎧
⎪
(D 1 , D 12 ) ∈ 1 ,
⎨0,
12
= D − y2 ,
(D 1 , D 12 ) ∈ 2 ,
⎪
⎩ 1
12
D + D − y1 , (D 1 , D 12 ) ∈ 3 .
τ (t) = sup{s ≤ t : D (s) + (I2 (0) + O2 (s − L2 )) ∧ D (s)
1
12
≤ I1 (0) + O1 (s − L1 )}
and {(Ã1 (t, s), Ã12 (t, s)), s ∈ [τ (t), t]} has the following properties:
Thus, we have
P1 : Ã1 (t, s) and Ã12 (t, s) are increasing in s ∈ [τ (t), t],
P2 : Ã1 (t, s) + Ã12 (t, s) = O1 (s − L1 ) − O1 (τ (t) − L1 ),
P3 : Ã1 (t, s) ≤ D1 (τ (t), s],
+
P4 : Ã12 (t, s) ≤ D12 (τ (t), s] + (D12 (τ (t)) − I2 (0) − O2 (τ (t) − L2 )) ,
+
⎧
⎪h1 − (c + h1 ) · Pr(D 1 + D 12 ≥ y1 ,
∂G(y1 , y2 ) ⎨
=
D 1 ≥ y1 − y2 ),
y1 ≥ y2 ,
⎪
∂y1
⎩
h1 − (c + h1 ) · Pr(D 1 + D 12 ≥ y1 ), y1 < y2 .
Hence,
P5 : Ã12 (t, s) ≤ (I2 (0) + O2 (τ (t) − L2 ) − D12 (τ (t)))
+ O2 (s − L2 ) − O2 (τ (t) − L2 ).
Noting that from the definition of τ (t) we know that for any s ∈ (τ (t), t],
D1 (s) + (I2 (0) + O2 (s − L2 )) ∧ D12 (s) > I1 (0) + O1 (s − L1 ).
This implies that the existence of {(Ã1 (t, s), Ã12 (t, s)), s ∈ [τ (t), t]} with
properties (P1)–(P5). Then by (2),
∂G(y1 , y2 )
= h1 − (c + h1 ) · Pr(D 1 + D 12 ≥ y1 , D 1 ≥ y1 − y2 ).
∂y1
(E-4)
Similarly, we can show
∂G(y1 , y2 )
= h2 − (c + h1 ) · Pr(D 1 ≤ y1 − y2 , D 12 ≥ y2 ).
∂y2
(E-5)
Taking further derivatives, we get that
I1 (t) = I1 (0) + O1 (t − L1 ) − A1 (t) − A12 (t),
∞
∂ 2 G(y1 , y2 )
= (c + h1 )
ψ(y1 − y2 , ξ2 )dξ2
2
∂y1
y2
y2
+
ψ(y1 − ξ2 , ξ2 )dξ2 ,
I2 (t) = I2 (0) + O2 (t − L2 ) − A12 (t),
B 1 (t) = D 1 (t) − A1 (t), B 12 (t) = D 12 (t) − A12 (t).
These imply
0
∂ 2 G(y
The Ft -adaptiveness of {(A1 (t), A12 (t)), t ≥ 0} directly follows from its
definition (E-1)–(E-2). Hence, the proposition is proved.
PROOF OF LEMMA 1: Denote v and u as defined in Section 3.1. Then
(20) can be expressed as
(E-3)
(v2 + v5 ) × min{v3 , v2 + v4 } ≥ min{v2 × v3 , (v2 )2 } > 0.
This contradicts (E-3). Hence we have v2 = 0. Note that v3 − v1 − v2 − v5 =
−
u1 and v5 = u−
2 mean v3 − (v1 + v2 ) = u1 + u2 . This, together with (a),
implies (b) and (c).
(E-7)
(E-8)
y2
E-7 implies G(y1 , y2 ) is submodular, proving the second part of (i). Thus,
the minimal point exists. It is trivial to see
∂ 2 G(y1 , y2 )
∂ 2 G(y1 , y2 )
, i = 1, 2.
≤
∂y1 ∂y2
∂yi2
+
We aim to prove (a) v3 × (v1 + v2 ) = 0; (b) v3 = (u1 + u−
2 ) ; and (c)
−
v1 + v2 = (u1 + u−
2) .
We first prove (a). This can be done by proving that if v3 > 0, then
v1 = v2 = 0. From (E-3), we have v1 = 0. If v2 > 0, then
(E-6)
∞
1 , y2 )
= −(c + h1 )
ψ(y1 − y2 , ξ2 )dξ2 ,
∂y1 ∂y2
y2
y1 −y2
∂ 2 G(y1 , y2 )
= (c + h1 )
ψ(ξ1 , y2 )dξ1
∂y22
0
∞
+
ψ(y1 − y2 , ξ2 )dξ2 .
B 1 (t) × I1 (t) = 0, B 12 (t) × min{I1 (t), I2 (t)} = 0.
v1 × v3 = 0, (v2 + v5 ) × min{v3 , v2 + v4 } = 0.
(E-9)
Thus, the Hessian matrix is diagonal dominant, proving (ii). By the contraction theorem, the minimal point (s̄1 , s̄2 ) is unique. Combined with (E-6), and
(E-8)–(E-9) we know that the Hessian matrix is positive definite, and hence
G is jointly convex, proving the first part of (i).
PROOF OF LEMMA 3: First we prove (i). We consider the following
three cases.
∗ (see (25)). By definition, s̃ (y ) = s ∗ < y . Thus, by
Case 1: y2 > s12
1 2
2
12
(E-4), we have
PROOF OF LEMMA 2: Let
1 = {(D 1 , D 12 ) : D 1 + D 12 ≤ y1 , D 12 ≤ y2 },
∂G(y1 , y2 )
∂y1
∗
y1 =s12
∗
∗
= h1 − (c + h1 ) · Pr(D 1 + D 12 ≥ s12
, D 1 ≥ s12
− y2 )
Naval Research Logistics DOI 10.1002/nav
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Naval Research Logistics, Vol. 62 (2015)
∗
= h1 − (c + h1 ) · Pr(D 1 + D 12 ≥ s12
)
By the definition of G̃(·) given by (28) and noting the independence of
D2 (t − L) and Ft−L− , by (E-15)–(E-16), taking a further expectation
yields
= 0.
∗ minimizes G(y , y ) when y > s ∗ .
Hence, s̃1 (y2 ) = s12
1 2
2
12
∗ . By definition, it is not difficult to show s (y ) ≥ y .
Case 2: 0 < y2 ≤ s12
1 2
2
Thus, by the first order condition (E-4), s1 (y2 ), which is the same as s̃1 (y2 )
given by (27), minimizes G(y1 , y2 ).
Case 3: y2 ≤ 0. By (E-4) again,
∂G(y1 , y2 )
∂y1
y1 =y2 +s1∗
= h1 − (c + h1 ) · Pr(D 1 + D 12 ≥ y2 + s1∗ , D 1 ≥ s1∗ )
= h1 − (c + h1 ) · Pr(D 1 ≥ s1∗ )
= 0.
This implies that for given y2 , s̃1 (y2 ) = y2 + s1∗ (see (25)) minimizes
G(y1 , y2 ).
∗ ) = s∗ .
The continuity of s̃1 (·) follows directly from s̃1 (0) = s1∗ and s̃1 (s12
12
∗ ),
Using (E-6)–(E-7), we have that for y2 ∈ (0, s12
ds̃1 (y2 )
=−
dy2
!
∂ 2 G(y
∂ 2 G(y
1 , y2 )
1 , y2 )
∂y2 ∂y1
∂y12
y1 =s1 (y2 )
∞
ψ(s
(y
)
−
y
,
ξ
)dξ
1 2
2 2
2
y2
y2
= ∞
∈ [0, 1].
y2 ψ(s1 (y2 ) − y2 , ξ2 )dξ2 + 0 ψ(s1 (y2 ) − ξ2 , ξ2 )dξ2
(E-10)
ds̃1 (y2 )
= 1,
dy2
ds̃1 (y2 )
∗
If y2 > s12
, then
= 0.
dy2
(E-11)
(E-12)
Combining (E-10)–(E-12) yields the derivative of s̃1 (y2 ) belongs to the
interval [0, 1].
Finally we prove (ii). Note that for any jointly convex function
g(x1 , x2 ), g̃(x2 ) = minx1 g(x1 , x2 ) is convex in x 2 . (See p. 54 of [27]). Therefore, for any realization of D2 , G(s̃1 (y − D2 ), y − D2 ) is convex in
y. Because expectations preserve convexity, G̃(·) is convex.
PROOF OF LEMMA 4: As component-2 uses a base-stock policy s̃2 , the
second equality is trivial. To show that the first equality holds, it suffices to
show that for any τ and t with 0 < τ < t,
s̃1 (s̃2 − D2 (t − , t]) ≥ s̃1 (s̃2 − D2 (τ − , τ ]) − D2 (τ , t].
This directly follows from Lemma 3, 0 ≤ ds̃1 (y2 )/dy2 ≤ 1.
(E-13)
For simplicity, for any admissible policy π ∈ A, in the following we write
IPi = IPi (t − Li , π ), i ∈ I.
(E-14)
PROOF OF THEOREM 2: For any π ∈ A, by (22), we have
≥ E[ϕ(IP1 − D1 (t), IP2 − D2 (t))|Ft−L− ]
(E-15)
As IP1 , IP2 and D2 (t − L) are adapted to Ft−L , by Lemma 3, we obtain
E[G(IP1 , IP2 − D2 (t − L))|Ft−L ]
≥ G(s̃1 (IP2 − D2 (t − L)), IP2 − D2 (t − L)).
Naval Research Logistics DOI 10.1002/nav
(E-17)
The last two inequalities in (E-17) will be equalities, if for all t ≥ 0,
IP1 = s̃1 (s̃2 − D2 (t − L)), IP2 = s̃2 .
By Lemma 4, these two equations hold when the CBS policy (s̃1 (s̃2 −
D2 (t)), s̃2 ) is used for component replenishment. By Theorem 1 (i), Proposition 1 (ii), and Proposition 2, if we employ the CBS policy (s̃1 (s̃2 −
D2 (t)), s̃2 ) and an NHB allocation rule that just follows it, then the
first inequality in (E-17) will be an equality. Hence this policy gives us
EC(t, π ) = G̃(s̃2 ). The theorem is thus proved.
PROOF OF LEMMA 5: First we prove (i). Noting that G1 (y1 , y2 ) differs
from G(y1 , y2 ) only by replacing D1 with D 1 + D 1 , similar to (E-4) and
(E-5), we have
∂G1 (y1 , y2 )
= h1 − (c + h1 ) · Pr(D 1 + D 1 + D 12 ≥ y1 ,
∂y1
D 1 + D 1 ≥ y1 − y2 ),
(E-18)
(E-19)
The result can be argued in the same way as in Lemma 2, so we omit the
details here. Next, we prove (ii) We consider two cases.
Case 1: y1 ≤ s2∗ (see (32)). By definition, s̆2 (y1 ) = y1 − s2∗ ≤ 0. This
implies that
Pr(D 1 + D 1 ≤ y1 − s̆2 (y1 ), D 12 ≥ s̆2 (y1 ))
= Pr(D 1 + D 1 ≤ s2∗ , D 12 ≥ 0) =
h2
,
c + h1
i.e., s̆2 (y1 ) satisfies the first order condition given by (E-19). Hence, s̆2 (y1 )
minimizes G1 (y1 , ·).
Case 2: y1 > s2∗ . It can be verified that, in this case, s2 (y1 ) > 0. Thus,
by the first order condition given by (E-19), s2 (y1 ), which is same as s̆2 (y1 )
given by (33), minimizes G1 (y1 , ·).
The continuity of s̆2 (y1 ) directly follows from s2 (s2∗ ) = 0. Noting that for
y1 > s2∗ ,
ds̆2 (y1 )
∂ 2 G1 (y1 , y2 )
=−
dy1
∂y2 ∂y1
∞
!
∂ 2 G1 (y1 , y2 )
∂y22
y2 =s2 (y1 )
s2 (y1 ) ψ(y1 − s2 (y1 ), ξ2 )dξ2
= y −s (y )
,
∞
1
2 1
ψ(ξ1 , s2 (y1 ))dξ1 + s2 (y1 ) ψ(y1 − s2 (y1 ), ξ2 )dξ2
0
we have 0 ≤ ds̆2 (y1 )/dy1 ≤ 1. If y1 < s2∗ , clearly, ds̆2 (y1 )/dy1 = 1.
Consequently, the derivative of s̆2 (y1 ) always belongs to the interval [0, 1].
The proof of (iii) is similar to that in Lemma 3 (ii).
E[C(t, π )|Ft−L− ]
= E{E[G(IP1 , IP2 − D2 (t − L))|Ft−L ]|Ft−L− }.
≥ G̃(IP2 ) ≥ G̃(s̃2 ).
∂G1 (y1 , y2 )
= h2 − (c + h1 ) · Pr(D 1 ≤ y1 − y2 , D 12 ≥ y2 ).
∂y2
It is straightforward to see that
If y2 < 0, then
E[C(t, π )|Ft−L− ] ≥ E[ϕ(IP1 − D1 (t), IP2 − D2 (t))|Ft−L− ]
(E-16)
PROOF OF LEMMA 6: The proof is similar to that for Lemma 4. Similar
to (E-13), we need to show that for any t and τ with t > τ ,
s̆2 (s̆1 − D2 (t − , t]) ≥ s̆2 (s̆1 − D2 (τ − , τ ]) − D2 (τ , t].
This follows directly from 0 ≤ ds̆2 (y1 )/dy1 ≤ 1 by Lemma 5.
(E-20)
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
639
PROOF OF THEOREM 3: The proof is similar to that of Theorem 2.
Here, we omit the details and sketch only what is unique to this proof.
12 be the system’s information given by F
12
Let Ft−L
t−L− and D (s) for
s ∈ (t − L − , t − L]. Then D1 (t − L − , t] and D12 (t − L, t] are
12 . Noting that
independent of Ft−L
where the equality is given by the fact that both sides equal to h1 /(c + h1 )
by (30). Similarly, by (31) and (E-23), we get
D1 (t) = D1 (t − L, t] + D 1 (t − L) + D2 (t − L),
From (E-24) we have either z̄1 ≥ s̄1 or z̄1 − z̄2 ≥ s̄1 − s̄2 , and from (E-25)
either z̄2 ≥ s̄2 or z̄1 − z̄2 ≤ s̄1 − s̄2 . Thus, by checking all four combinations
Pr(D 1 ≤ s̄1 − s̄2 , Z 12 ≥ s̄2 ) ≥ Pr(D 1 ≤ s̄1 − s̄2 , D 2 ≥ s̄2 )
= Pr(D 1 ≤ z̄1 − z̄2 , Z 12 ≥ z̄2 ).
12 , we have that for π ∈ A,
and IP1 and D2 (t − L) are adapted to Ft−L
12
E[ϕ(IP1 − D1 (t), IP2 − D2 (t))|Ft−L
]
= E[ϕ(IP1 − D2 (t − L) − D 1 (t
12
IP2 − D2 (t))|Ft−L
]
12
(by Ft−L
⊆
1
•
•
•
•
− L) − D1 (t − L, t],
Ft−L )
12
]
= E[G(IP1 − D2 (t − L) − D (t − L), IP2 )|Ft−L
= G1 (IP1 − D2 (t − L), IP2 )
≥ G1 (IP1 − D2 (t − L), s̆2 (IP1 − D2 (t − L))).
By the definition of Ğ(·) given by (35) and noting that D2 (t − L) has
the same distribution as D2 , taking further expectation yields
12
]|Ft−L− } ≥ Ğ(IP1 ) ≥ Ğ(s̆1 ),
E{E[ϕ(IP1 − D1 (t), IP2 − D2 (t))|Ft−L
where the last inequality follows from (36) and Lemma 5 (ii). Hence, we
have
z̄1 ≥ s̄1
z̄1 ≥ s̄1
z̄1 − z̄2
z̄1 − z̄2
(E-25)
and z̄2 ≥ s̄2 ;
and z̄1 − z̄2 ≤ s̄1 − s̄2 ;
≥ s̄1 − s̄2 and z̄2 ≥ s̄2 ;
≥ s̄1 − s̄2 and z̄1 − z̄2 ≤ s̄1 − s̄2 ,
and using (E-25) for the last combination yields that z̄1 ≥ s̄1 and z̄2 ≥ s̄2 .
Under CL, it is not difficult to show that s̆2 (y1 ) ≤ z̆2 (y1 ) for any given
y1 from (33) and (32), because Pr(D 1 + D 1 ≤ y1 − y2 , Z 12 ≥ y2 ) ≥
Pr(D 1 + D 1 ≤ y1 − y2 , D 12 ≥ y2 ) and both sides are decreasing in y2 .
To show s̆1 ≤ z̆1 , we point out that from the equations (35) and (36) (see
(E-18)), s̆1 is the solution of
Pr(D 1 ≥ y1 − D 1 − D 12 − s̆2 (y1 − D 12 ), D 1 + D 12 ≥ y1 − D1 )
=
h1
.
c + h1
(E-26)
From Lemma 5, y1 − s̆2 (y1 ) is increasing with y1 . Thus,
Pr(y1 − Z 12 − s̆2 (y1 − Z 12 ) > x)
EC(t, π ) ≥ E[ϕ(IP1 − D1 (t), IP2 − D2 (t))]
≤ Pr(y1 − D 12 − s̆2 (y1 − D 12 ) > x).
12
= E{E[E(ϕ(IP1 − D1 (t), IP2 − D2 (t))|Ft−L
)|Ft−L− ]}
≥ Ğ(s̆1 ).
(E-21)
Because s̆2 (y1 ) ≤ z̆2 (y1 ) for any y1 , we have
Pr(y1 − Z 12 − z̆2 (y1 − Z 12 ) > x)
The last inequality will be an equality if for all t ≥ 0,
≤ Pr(y1 − D 12 − s̆2 (y1 − D 12 ) > x).
IP1 = s̆1 , IP2 = s̆2 (s̆1 − D2 (t − L)).
Therefore,
By Theorem 1 (i), Proposition 1 (ii), and Proposition 2, if we use the component replenishment policy (s̆1 , s̆2 (s̆1 − D2 (t))) and an NHB allocation
rule that just follows it, the first inequality in (E-21) will be an equality. The
theorem thus follows from Lemma 6 immediately.
PROOF OF PROPOSITION 3: (i) We consider two N-systems—the
original system and a new system. We use Z(t) for the new demand process,
Z for the counterpart of D, and z for the counterpart of s. Both systems
are identical except that in the new system the product-12 demand process
{Z 12 (t), t ≥ 0} is stochastically larger than {D12 (t), t ≥ 0}. First, we have
that for any x,
Pr(D1 (0, ] + D12 (0, ] > x) ≤ Pr(D1 (0, ] + Z 12 (0, ] > x),
(E-22)
Pr(D (0, ] > x) ≤ Pr(Z (0, ] > x), Pr(D
12
12
12
> x) ≤ Pr(Z
12
> x).
(E-23)
Under IL, from (30) and (E-23), by the independence of D1 and D12 , and
the independence of D1 and Z 12 , we have
Pr(D 1 ≥ s̄1 − s̄2 , D 1 + Z 12 ≥ s̄1 ) ≥ Pr(D 1 ≥ s̄1 − s̄2 , D 1 + D 12 ≥ s̄1 )
= Pr(D 1 ≥ z̄1 − z̄2 , D 1 + Z 12 ≥ z̄1 ),
(E-24)
Pr(D 1 ≥ s̆1 − D 1 − Z 12 − z̆2 (s̆1 − Z 12 ), D 1 + Z 12 ≥ s̆1 − Z1 )
≥ Pr(D 1 ≥ s̆1 − D1 − s̆2 (s̆1 − D 12 ), D 1 + D 12 ≥ s̆1 − D1 )
=
h1
.
c + h1
Thus, s̆1 ≤ z̆1 as the left-hand side of (E-26) is decreasing in y1 .
Finally, the case for SL can be proved in a similar way.
(ii) The proof is similar to the above case with some slight differences. We
consider two N-systems, namely, the original one and the new one. We still
use Z(t) for the new demand process, Z for the counterpart of D, and z for
the counterpart of s. The two systems are identical except that the product-1
demand process in the new system is changed to {Z 1 (t), t ≥ 0}. The process
{Z 1 (t), t ≥ 0} is stochastically larger than {D1 (t), t ≥ 0}. First, we have
that for any x,
Pr(D1 (0, ] + D12 (0, ] > x) ≤ Pr(Z 1 (0, ] + D12 (0, ] > x),
(E-27)
Pr(D 1 > x) ≤ Pr(Z 1 > x).
(E-28)
Under IL, adding (30) and (31), we have
Pr(D 1 + D 12 ≤ s̄1 , D 12 ≤ s̄2 ) = 1 −
h1 + h 2
.
c + h1
(E-29)
Naval Research Logistics DOI 10.1002/nav
640
Naval Research Logistics, Vol. 62 (2015)
≥ Pr(D 1 + D 12 ≤ s̆1 − Z1 , D 12 ≤ z̆2 (s̆1 − Z 12 )).
Similar to (E-24)–(E-25), by (30) and (E-29), we have
Pr(Z 1 ≥ s̄1 − s̄2 , Z 1 + D 12 ≥ s̄1 ) ≥ Pr(Z 1 ≥ z̄1 − z̄2 , Z 1 + D 12 ≥ z̄1 ),
(E-30)
Pr(Z 1 + D 12 ≤ s̄1 , D 12 ≤ s̄2 ) ≤ Pr(Z 1 + D 12 ≤ z̄1 , D 12 ≤ z̄2 ). (E-31)
From (E-30), we have either z̄1 ≥ s̄1 or z̄1 − z̄2 ≥ s̄1 − s̄2 , and from
(E-31), either z̄1 ≥ s̄1 or z̄2 ≥ s̄2 . Assuming z̄1 < s̄1 , then it must be
z̄1 − z̄2 ≥ s̄1 − s̄2 and z̄2 ≥ s̄2 ; This is contradictory, as under this assumption,
we have 0 ≤ z̄2 − s̄2 ≤ z̄1 − s̄1 < 0. Thus, we have s̄1 ≤ z̄1 .
Similar to (i), one can easily show that under CL, s̆2 (y1 ) ≥ z̆2 (y1 ) for any
y1 , and under SL, s̃1 (y2 ) ≤ z̃1 (y2 ) for any y2 . To show under CL, s̆1 ≤ z̆1 ,
we note that (E-26) is equivalent to
Pr(D 1 + D 12 ≤ y1 − D1 , D 12 ≤ s̆2 (y1 − D 12 )) = 1 −
h1 + h 2
.
c + h1
(E-32)
This is because for any realization of D 12 = δ, from (32), we have
Pr(D 12 ≥ s̆2 (y1 − δ), D 1 + D 1 ≤ y1 − δ − s̆2 (y1 − δ)) =
h2
.
c + h1
Thus,
Pr(D 12 ≥ s̆2 (y1 − D1 ),
D + D ≤ y1 − D
1
1
12
h2
− s̆2 (y1 − D )) =
.
c + h1
12
(E-33)
By (E-32), we know that
h1 + h 2
,
c + h1
h1 + h 2
≤ z̆2 (z̆1 − Z 12 )) = 1 −
.
c + h1
Pr(D 1 + D 12 ≤ s̆1 − D1 , D 12 ≤ s̆2 (s̆1 − D 12 )) = 1 −
Pr(D 1 + D 12 ≤ z̆1 − Z1 , D 12
Hence, s̆1 ≤ z̆1 directly follows from (E-33) and the monotonicity of s̆2 (y1 ).
Similarly, we can prove SL case.
(ii) We consider two N-systems, namely, the original one and the new one.
The two systems are identical except that the shorter leadtime in the new
system is changed to L z from L with L < Lz . Then
Pr(D 1 ≥ x) ≤ Pr(Z 1 ≥ x) and Pr(D 12 ≥ x) ≤ Pr(Z 12 ≥ x).
The remaining part of the proof is similar to the proof of Proposition 3 (ii).
The details are therefore omitted.
PROOF OF PROPOSITION 5: First, we establish the lower bound. From
Proposition 1 and the definition of φ(u1 , u2 ) in (P3), letting c = b1 ∧ (b12 +
h2 ), we have
−
ϕ(u1 , u2 ) ≥ h1 u1 + h2 u2 + (h1 + c)(u−
1 ∨ u2 ).
From Theorem 1, therefore, we have,
This, together with (E-26), yields the above equation (E-32). Because
Pr(z̆2 (y1 − D 12 ) ≥ x) ≤ Pr(s̆2 (y1 − D 12 ) ≥ x),
C(t, π ) ≥ ϕ(IP1 − D1 (t), IP2 − D2 (t))
≥ h1 · (IP1 − D1 (t)) + h2 · (IP2 − D2 (t))
+ (h1 + b1 ∧ (h2 + b12 ))
we have
· ((IP1 − D1 (t))− ∨ (IP2 − D2 (t))− )
Pr(Z 1 + D 12 ≤ s̆1 − Z1 , D 12 ≤ z̆2 (s̆1 − D 12 ))
≤ Pr(D 1 + D 12 ≤ s̆1 − D1 , D 12 ≤ s̆2 (s̆1 − D 12 ))
=1−
h1 + h 2
c + h1
= Pr(Z 1 + D 12 ≤ z̆1 − Z1 , D 12 ≤ z̆2 (z̆1 − D 12 )),
where the last two equations follow from (E-32). Because the left-hand side
of (E-32) is increasing in y1 , we obtain s̆1 ≤ z̆1 .
PROOF OF PROPOSITION 4: (i) We consider two N-systems, namely,
the original one and the new one. We use Z(t) for the new demand process,
Z for the counterpart of D, and z for the counterpart of s. The two systems are
identical except that the leadtime difference in the new system is changed to
z from with < z . From (27), it is trivial to see that s̃1 (·) has nothing
to do with , and s̆2 (y) ≥ z̆2 (y). Thus, we need to prove that under CL,
s̆1 ≤ z̆1 , and under SL, s̃2 ≤ z̃2 . First, we consider the CL case. As < z ,
then
Pr(D + D
≤ s̆1 − D1 , D
≥ Pr(D + D
1
12
12
≤ s̆1 − Z1 , D
≤ s̆2 (s̆1 − Z ))
Naval Research Logistics DOI 10.1002/nav
−
By (a + b− ) + b− = a − ∨ b− with any real numbers a and b,
C(t, π ) = h1 · (IP1 − D1 (t)) + h2 · (IP2 − D2 (t)) + (h1 + h2 + b12 )
−
· (IP2 − D2 (t))− + h1 · ((IP1 − D1 (t)) + (IP2 − D2 (t))− )
−
· (IP2 − D2 (t))− + h1 · ((IP1 − D1 (t)) + (IP2 − D2 (t))− )
+ c̄ · (B112 (t, π ) + B 1 (t, π ))
= h1 · (IP1 − D1 (t)) + h2 · (IP2 − D2 (t)) + (h1 + h2 + b12 )
· (IP2 − D2 (t))− + (h1 + c̄) · ((IP1 − D1 (t))
≤ s̆2 (s̆1 − D ))
12
12
−
B112 (t, π ) + B 1 (t, π ) = ((IP1 − D1 (t)) + (IP2 − D2 (t))− ) .
≤ h1 · (IP1 − D1 (t)) + h2 · (IP2 − D2 (t)) + (h1 + h2 + b12 )
Recall that s̆2 (y1 ) is increasing with y1 . Noting that (D 1 , D 12 ) and (Z 1 , Z 12 )
have the same distribution, (D 1 , D 12 ) and D1 are independent, and
(Z 1 , Z 12 ) and Z1 are independent, we have that
12
for any feasible π ∈ A. We know that the last expression above is just the
inventory cost at time t of another system which is identical with the original one except that both the common component backorder cost and the
sum of the product-12 backorder cost and the component-2 holding cost are
changed into b1 ∧ (h2 + b12 ). Hence, the lower bound is obtained.
Next, we move to the upper bound. Here, we consider the CL case; the
others can be treated similarly. Let c̄ = b1 ∨ (b12 + h2 ) in (21), and we have
h2 + b12 ≤ c̄. For any π ∈ A,
+ (h2 + b12 ) · B112 (t, π ) + b1 · B 1 (t, π )
Pr(D 12 ≥ x) ≤ Pr(Z 12 ≥ x).
1
(E-34)
12
+ (IP2 − D2 (t))− )−
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
≤ h1 · (IP1 − D1 (t)) + h2 · (IP2 − D2 (t))
X2n (t − L2 ) = I2n (0) + O2n (t − L2 ) − D12,n (t − L2 ) − λ12 nL2 . (E-44)
+ (h1 + c̄) · ((IP1 − D1 (t))− ∨ (IP2 − D2 (t))− ).
Then, similar to the proof for the lower bound, we get
C∗
≤
(E-35)
For any fixed t, we know that
C̄ ∗ .
PROOF OF THEOREM 4: Going along the same lines of Plambeck
and Ward (2006) (see Proposition 4.2 on Page 462), for any π =
{(Oin (t), AK,n (t)), t ≥ 0} ∈ An and T > 0, in view of (38)–(41),
⎛
⎞
T 2
1
n
K
K,n
⎝
hi · Ii (t, π ) +
b · B (t, π )⎠ dt
lim sup √ E
n
n→∞
0
i=1
K=1,12
⎛
⎞
T 2
1
n
K
K,n
⎝
hi · Ii (t, π ) +
b · B (t, π )⎠ dt,
≥ lim sup √ E
n
n→∞
0
((D1,n (t − L, t] − λ1 nL), (D12,n (t − L, t] − λ12 nL))
and
(X1n (t − L), X2n (t − L2 ) − (D12,n (t − L − , t − L] − λ12 n))
are independent. As {(D1,n (t), D12,n (t)), t ≥ 0} is stationary and has inde2+
pendent increments, and E(DK,n ( n1 ))
< ∞, we have that for any
t ≥ L,
K=1,12
i=1
(E-36)
where π is defined in (42). To get the analytic expression of Iin (t, π )
and B K,n (t, π ), let B̂K,n (
l n ) be the solution of (38) only with constraints
(39)–(40). Then we have
⎛
⎞
T 2
1
n
K
K,n
⎝
lim sup √ E
hi · Ii (t, π ) +
b · B (t, π )⎠ dt
n
n→∞
0
i=1
K=1,12
⎛
⎞
T 2
1
n
K
K,n
⎝
hi · Iˆi (t) +
b · B̂ (t)⎠ dt,
≥ lim sup √ E
n
n→∞
0
i=1
K=1,12
converges in distribution to a normal variable 12 (t − L) with zero mean
2
and variance ·(σ 12 ) . By the Skorohod embedding theorem, we can assume
that with probability one,
n
n
Iˆ1
(t) = N̂1n (t) + B̂1,n (t) + B̂12,n (t), Iˆ2
(t) = N̂2n (t) + B̂12,n (t),
N̂1n (t) = I1n (0) + O1n (t − L1 ) − D1n (t),
If ≤
+ h2 , it directly follows from Proposition 1 (i) that the solution
of (38) with constraints (39)–(40) is given by
+
−
+
n
n
Iˆ1
(t) = (N̂1n (t) + (N̂2n (t)) ) , Iˆ2
(t) = (N̂2n (t)) ,
B̂1,n (t)
2
=
(N̂1n (t) + (N̂2n (t))
n
hi · Iˆi
(t) +
i=1
) ,
B̂12,n (t)
=
(E-38)
−
(N̂2n (t))
,
(E-39)
bK · B̂K,n (t)
= h1 · N̂1n (t) + h2 · N̂2n (t) + (h1 + h2 + b12 ) · (N̂2n (t))
− −
+ (h1 + b1 ) · (N̂1n (t) + (N̂2n (t)) ) .
(E-40)
First, we consider the case L1 ≤ L2 . In view of the proof of Theorem 2, we
write
=
I1n (0) + O1n (t
1
12
− L) − D
− (λ + λ )nL − (D
− (D
12,n
(t) − D
12,n
1,n
1,n
(E-45)
(t − L) − D
(t) − D
1,n
12,n
<
converge to (1 (t), 12 (t), 12 (t − L)). In view of E(DK,n ( n1 ))
∞, K = 1, 12 for = 3, thus the random variable (vector) given in (E-45)
also converges to (1 (t), 12 (t), 12 (t − L)) in the sense of L1 .
Define
X1∞ (t − L) = liminf
n→∞
X1n (t − L)
X n (t − L2 )
, X2∞ (t − L2 ) = liminf 2 √
.
√
n→∞
n
n
(E-46)
Thus, by (E-40), in view of L1 -convergence of the random vector given by
(E-45),
K=1,12
−
N̂1n (t)
D1,n (t − L, t] − λ1 nL D12,n (t − L, t] − λ12 nL
,
,
√
√
n
n
D12,n (t − L − , t − L] − λ12 n
√
n
2+
b12
− −
D12,n (t − L − , t − L] − λ12 n
√
n
where
N̂2n (t) = I2n (0) + O2n (t − L2 ) − D2n (t).
D1,n (t − L, t] − λ1 nL D12,n (t − L, t] − λ12 nL
,
√
√
n
n
converges in distribution to a bivariate normal variable (1 (t), 12 (t)) with
zero mean and covariance matrix defined in (43), and
(E-37)
b1
641
⎞
⎞⎞
⎛ ⎛ ⎛
2
K,n
n
(t)
Iˆi
K B̂ (t)
n ⎠ n
⎝
⎝
⎝
Ft−L Ft−L−⎠⎠
lim sup E E E
hi · √ +
b · √
n
n
n→∞
i=1
≥
− (t
(t − L)
(t − L) − λ nL)
K=1,12
E(E(E(F (X1∞ (t
12
− L), X2∞ (t − L2 )
∞
∞
− L))|Ft−L
)|Ft−L−
)).
(E-47)
1
(t − L) − λ nL),
12
Similar to (E-16), by (44), we have
(E-41)
N̂2n (t) = I2n (0) + O2n (t − L2 ) − D12,n (t − L2 ) − λ12 nL2
≥ F (θ1 (X2∞ (t − L2 ) − 12 (t − L)), X2∞ (t − L2 ) − 12 (t − L)).
(E-48)
− (D12,n (t) − D12,n (t − L) − λ12 nL)
− (D12,n (t − L) − D12,n (t − L2 ) − λ12 n(L2 − L)).
∞
E(F (X1∞ (t − L), X2∞ (t − L2 ) − 12 (t − L))|Ft−L
)
(E-42)
Similar to (E-17), by (44)–(45) and (E-48), we have
Let
E(E(F (X1∞ (t − L), X2∞ (t − L2 )
X1n (t − L) = I1n (0) + O1n (t − L) − D1,n (t − L) − D12,n (t − L)
− (λ1 + λ12 )nL,
(E-43)
∞
∞
− 12 (t − L))|Ft−L
)|Ft−L−
) ≥ F̃ (θ̃2 ).
(E-49)
Naval Research Logistics DOI 10.1002/nav
642
Naval Research Logistics, Vol. 62 (2015)
Therefore, in view of (E-37)–(E-40) and (E-47)–(E-49), we know that
− (D1,n (t − L) − D1,n (t − L1 ) − λ1 n)
⎞
⎛
T 2
1
hi · Iin (t, π ) +
bK · B K,n (t, π )⎠dt ≥ F̃ (θ̃2 ).
lim sup √ E ⎝
n 0
n→∞
− (D12,n (t − L) − D12,n (t − L1 ) − λ12 n)
i=1
N̂2n (t) = I2n (0) + O2n (t − L) − D12,n (t − L) − λ12 nL
K=1,12
− (D12,n (t) − D12,n (t − L) − λ12 nL).
(E-50)
Similar to (E-43)–(E-44), let
By again by (44)–(45), the inequality of (E-49) would be equality if
X1n (t − L1 ) = I1n (0) + O1n (t − L1 ) − D1,n (t − L1 )
X1∞ (t − L) = θ1 (θ̃2 − 12 (t − L)), X2∞ (t − L2 ) = θ̃2 .
− D12,n (t − L1 ) − (λ1 + λ12 )nL1 ,
Hence, by the definition of X1∞ (t − L) and X2∞ (t − L2 ) given by (E-46),
we know that if
√
D12,n (t − L − , t − L]
n · θ1 θ̃2 −
√
n
√
X2n (t − L2 ) = nθ̃2 ,
X1n (t − L) =
For any fixed t, we also know that
and
This implies that (E-47) becomes equality with (X1n (t−L), X2n (t−L2 )) given
by (E-51). Consequently, the inequality of (E-50) would become equality
if (X1n (t − L), X2n (t − L2 )) is chosen according to (E-51). It follows from
(E-43)–(E-44) that the inequality of (E-50) would become equality if
O1n (t − L1 ) = D1,n (t − L1 ) + D12,n (t − L1 ) + (λ1 + λ12 )nL1
√
D12,n (t − L1 − , t − L1 ] − λ12 n
,
+ n · θ1 θ̃2 −
√
n
(E-52)
√
O1n (t − L2 ) = D12,n (t − L2 ) + λ12 nL2 + nθ̃2 .
(E-53)
Therefore, to finish the proof of (i), in view of (E-36)–(E-37), it suffices to
prove that for the replenishment process given by (E-52)–(E-53), the solution of (38) only with constraints (39)–(40) should be asymptotic same as
BK,n (
l n ) (see below (E-54)). That is, in high-volume, the constraint (41)
2+
does not matter. By E[DK,n ( n1 )]
< ∞, K ∈ {1, 12} for = 3, we know
that, by Ata and Kumar (2006), for any finite constant α > 0, there exists a
constant β > 0 such that
"
#
max max DK,n ((
+ 1)l n ) − DK,n (
l n ) − λK nl n < αn1/3
=0,···l n K=1,12
≥ 1 − βn−1/6 .
n
(t) and B̂K,n (t) given by (E-38)–(E-39) are the Lipschitz conNote that Iˆi
tinuous in (N̂1n (t), N̂2n (t)). Hence, following the proof of Proposition 4.1 in
Plambeck and Ward (2006), we can show that there exists a constant β > 0
such that
$
%
Pr(B̂K,n (
l n ) = BK,n (
l n ), = 1, · · · , 1/l n ) ≥ 1 − βn−1/6 .
(E-54)
With (E-54) in hand, we obtain that the replenishment policy defined by
(E-52)–(E-53), and the discrete-review allocation rule defined by (38)–(41)
are asymptotically optimal in An .
Next, consider (ii), L1 > L2 . We write (E-41)–(E-42) as
N̂1n (t) = I1n (0) + O1n (t − L1 ) − D1,n (t − L1 ) − D12,n (t − L1 )
− (λ1 + λ12 )nL1 − (D1,n (t) − D1,n (t − L) − λ1 nL)
− (D12,n (t) − D12,n (t − L) − λ12 nL)
Naval Research Logistics DOI 10.1002/nav
((D1,n (t − L, t] − λ1 nL), (D12,n (t − L, t] − λ12 nL))
(E-51)
then
n
X1 (t − L) X2n (t − L2 )
,
converges to (θ1 (θ̃2 − 12 (t − L)), θ̃2 ) in L1 .
√
√
n
n
Pr
X2n (t − L2 ) = I2n (0) + O2n (t − L) − D12,n (t − L) − λ12 nL.
and
(X1n (t − L1 ) − (D1,n (t − L1 , t − L] − λ1 n)
− (D12,n (t − L1 , t − L] − λ12 n), X2n (t − L))
are independent. Then as in the proof for L1 ≤ L2 , we can prove (ii).
PROOF OF THEOREM 5: In the proof of Theorem 4, we use Proposition 1 (i) to get the solution of (38) with constraints (39)–(40) (see (E-38)–
(E-40)). Similarly we first use Proposition 1 (iii) to get the solution of (38)
with constraints (39)–(40) for the case b1 ≥ h2 +b12 , then follow the exactly
same procedure to get the theorem.
PROOF OF LEMMA 7: First note that
−
−
a4 u1 + a5 u2 + a3 u3 + (a1 + a3 + a4 )(u−
3 ∨ [u1 + u2 ])
−
−
is jointly convex over (u1 , u2 , u3 ) because u−
1 + u2 and u3 are both jointly
−
−
−
convex over (u1 , u2 , u3 ), and thus, (u1 + u2 ) ∨ u3 is also jointly convex
over (u1 , u2 , u3 ). Consequently, G(y1 , y2 , y3 ) defined by
h1 · E(y1 − D1 ) + h2 · E(y2 − D2 ) + h3 · E(y3 − D3 )
+ (c + h1 ) · E[[y1 − D1 ]− ∨ ([y2 − D2 ]− + [y3 − D3 ]− )]
is also jointly convex over (y1 , y2 , y3 ).
To show that the optimal point (s̄1 , s̄2 , s̄3 ) satisfies s̄2 + s̄3 ≥ s̄1 . Assume,
otherwise, s̄2 + s̄3 < s̄1 is optimal. For any demand realization, we
have s̄1 − D2 − D3 > s̄2 + s̄3 − D2 − D3 , thus, (s̄1 − D2 − D3 )− ≤
(s̄2 + s̄3 − D2 − D3 )− ≤ (s̄3 − D3 )− + (s̄2 − D2 )− . Consequently,
G(s̄1 , s̄2 , s̄3 ) = h1 s̄1 + h2 s̄2 + h3 s̄3 − 2ED2 − 2ED3
+ (c + h1 )E[(s̄2 − D2 )− + (s̄3 − D3 )− ].
This means that we can get a lower value by decreasing s̄1 to s̄2 + s̄3 , which
contradicts the optimality of (s̄1 , s̄2 , s̄3 ).
PROOF OF LEMMA 8: As G(y1 , y2 , y3 ) is convex in y1 for any (y2 , y3 ),
the first order condition is guaranteed to find the minimal point. Taking
derivatives, we get
∂G(y1 , y2 , y3 )
= h1 − (c + h1 ) · Pr(D2 + D3 ≥ y1 ,
∂y1
D3 ≥ y1 − y2 , D2 ≥ y1 − y3 ).
(E-55)
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
We need to consider the following three cases.
Case I: (y2 , y3 ) ∈ 1 . In this case, y2 , y3 ≥ s1∗ , thus, we have
∂G(y1 , y2 , y3 )
∂y1
y1 =s1∗
= h1 − (c + h1 ) · Pr(D2 + D3 ≥ s1∗ ) = 0,
where the first equality follows from the nonnegativity of the demand, and
the last equality follows from definition of s1∗ in (65). Thus, s̆1 (y2 , y3 ) = s1∗ .
In this region, it is trivial that s̆1 (y2 , y3 ) is increasing and continuous.
Case II: (y2 , y3 ) ∈ 2 . 2 is partitioned into two regions: (R1 ) y3 ≤ y2 ≤
v2 (y3 ), y3 < s1∗ and (R2 ) y2 ≤ y3 ≤ v3 (y2 ), y2 < s1∗ . We only consider
region (R1 ), the region (R2 ) can be proved similarly. Because y2 ≤ v2 (y3 )
(defined in (66)), we have
∂G(y1 , y2 , y3 )
∂y1
y1 =y2
= Pr(D2 + D3 ≥ y2 , D2 ≥ y2 − y3 ) ≥
h1
,
c + h1
where the inequality is because the left-hand-side is decreasing in y2 and
y2 ≤ v2 (y3 ). By the definition of s1 (y2 , y3 ) in (67), we have s1 (y2 , y3 ) ≥ y2 .
Thus, by (E-55), s̆1 (y2 , y3 ) = s1 (y2 , y3 ) minimizes G(·, y2 , y3 ) for any y2
and y3 .
To prove the other parts of the lemma, note that
Pr(D2 + D3 ≥ y1 , D3 ≥ y1 − y2 , D2 ≥ y1 − y3 )
⎧
⎪
⎨Pr(D3 ≥ y3 , D2 ≥ y1 − y3 ) + Pr(D2 + D3 ≥ y1 ,
=
y1 − y2 ≤ D3 < y3 ), if y1 ≤ y2 + y3 ,
⎪
⎩
Pr(D2 ≥ y1 − y3 , D3 ≥ y1 − y2 ), if y1 < y2 + y3 .
643
PROOF OF LEMMA 9: As under the coordinated base-stock policy
(s̆1 (·, ·), s̆2 , s̆3 ), the component-2 and component-3 use a base-stock policy
with levels s̆2 and s̆3 , respectively, the first two equalities of (4) are trivial.
We only need to show that the third one holds. To show this, we first show
that for any τ and t with 0 ≤ τ < t,
s̆1 (s̆2 − D2 (t − 2 , t], s̆3 − D3 (t − 3 , t])
≥ s̆1 (s̆2 − D2 (τ − 2 , t], s̆3 − D3 (τ − 3 , t])
≥ s̆1 (s̆2 − D2 (τ − 2 , τ ], s̆3 − D3 (τ − 3 , t]) − D3 (τ , t]
≥ s̆1 (s̆2 − D2 (τ − 2 , τ ], s̆3 − D3 (τ − 3 , τ ]) − D2 (τ , t] − D3 (τ , t],
where the first inequality is because ∂ s̆1 (y2 , y3 )/∂yi ≥ 0, i = 2, 3 by Lemma
8, while the second and third inequalities are because ∂ s̆1 (y2 , y3 )/∂yi ≤
1, i = 2, 3 by Lemma 8 and the fact that
Di (τ − i , t] = Di (τ − i , τ ] + Di (τ , t].
Thus, under the above CBS policy, for any τ ≥ 0 such that IP1 (τ ) =
s̆1 (s̆2 − D2 (τ ), s̆3 − D3 (τ )), for any t > τ such that no order for
component-1 is placed during (τ , t), we have
IP−
1 (t) = IP1 (τ ) − D1 (τ , t]
= s̆1 (s̆2 − D2 (τ − 2 , τ ], s̆3 − D3 (τ − 3 , τ ])
− D2 (τ , t] − D3 (τ , t]
≤ s̆1 (s̆2 − D2 (t − 2 , t], s̆3 − D3 (t − 3 , t]).
As a result, IP1 (t) = s̆1 (s̆2 − D2 (t), s̆3 − D3 (t)) is achievable.
Thus, taking a further derivatives in (E-55), we get that if y1 ≤ y2 + y3 ,
∞
∂ 2 G(y1 , y2 , y3 )
=
(c
+
h
)
ψ(ξ1 , y1 − y3 )dξ1
1
∂y12
y3
∞
y3
+
ψ(y1 − y2 , ξ2 )dξ2 +
ψ(ξ1 , y1 − ξ1 )dξ1 ,
y2
∂ 2 G(y1 , y2 , y3 )
= −(c + h1 )
∂y1 ∂y3
∂ 2 G(y1 , y2 , y3 )
= −(c + h1 )
∂y1 ∂y2
y1 −y2
∞
∞
As IPi − Di (t − L) and IP1 are adapted to Ft−L , by equations (62), we
have
ψ(y1 − y2 , ξ2 )dξ2 .
y2
E[G(IP1 , IP2 − D2 (t − L), IP3 − D3 (t − L))|Ft−L ]
By the Implicit Function Theorem, we have that if y1 ≤ y2 + y3 ,
∂s1 (y2 , y3 )
=−
∂yi
∂ 2 G(y
1 , y2 , y3 )
∂y1 ∂yi
!
∂ 2 G(y
1 , y2 , y3 )
∂y12
∈ [0, 1], i = 2, 3.
Along the same lines, we can show that if y1 > y2 +y3 , the above relationship
still holds.
Case III: (y2 , y3 ) ∈ 3 . We only prove the case for y2 > v2 (y3 ) and
y3 < s1∗ (see Figure 5), the other case y3 > v3 (y2 ) and y2 < s1∗ can be
similarly proved. It is not difficult to see that v2 (y3 ) ≥ y3 when y3 < s1∗ .
Thus, by (E-55) and (66), we obtain
∂G(y1 , y2 , y3 )
∂y1
y1 =v2 (y3 )
E[C(t, π )|Ft−L2 ] = E{E[G(IP1 , IP2 − D2 (t − L),
IP3 − D3 (t − L))|Ft−L ]|Ft−L2 }.
ψ(ξ1 , y1 − y3 )dξ1 ,
y3
PROOF OF THEOREM 7: In view of Theorem 6, we can assume that
an NHB policy π ∈ N is used. By the tower properties of conditional
expectations, we have
= h1 − (c + h1 ) · Pr(D2 + D3 ≥ v2 (y3 ),
D2 ≥ v2 (y3 ) − y3 ) = 0.
Thus, s̆1 (y2 , y3 ) = v2 (y3 ) minimizes G(·, y2 , y3 ). Applying a similar
methodology as in Case II, v2 (y3 ) can be shown to be increasing and
0 ≤ dv2 (y3 )/dy3 ≤ 1. The details are omitted.
The global continuity of s̆1 (y2 , y3 ) can be easily verified by checking the
boundary of each region.
≥ G(s̆1 (IP2 − D2 (t − L), IP3 − D3 (t − L)),
IP2 − D2 (t − L), IP3 − D3 (t − L)).
Taking a further expectation, by the fact that IPi is adapted to Ft−L and
Di (t − L) is independent to Ft−L2 , together with equations (63) and (64),
we have
E{E[G(IP1 , IP2 − D2 (t − L), IP3 − D3 (t − L))|Ft−L ]|Ft−L2 }
≥ Ğ(IP2 , IP3 ) ≥ Ğ(s̆2 , s̆3 ).
As a result, EC(t, π ) ≥ Ğ(s̆2 , s̆3 ). The inequality will be equality if and only
if for any t ≥ 0,
IP1 (t − L, π ) = s̆1 (s̆2 − D2 (t − L), s̆3 − D3 (t − L)),
IP2 (t − L − , π ) = s̆2 , IP3 (t − L − , π ) = s̆3 ,
this follows immediately from Lemma 9.
PROOF OF THEOREM 8: The proof is similar to that of Theorem 7. We
only present the proof for the case CL, the case CM can be done in the
Naval Research Logistics DOI 10.1002/nav
644
Naval Research Logistics, Vol. 62 (2015)
same way. According to Theorem 6, we can assume that an NHB admissible
policy π is used. By Proposition 6, for any π ∈ N , we have
C(t, π ) = Eϕ(IP1 − D1 (t), IP2 − D2 (t), IP3 − D3 (t)).
(E-56)
Next, we move to the upper bound. Similar to (E-35) in the proof of
Proposition 5, we have
C(t, π ) = h1 · (IP1 − D1 (t)) + h2 · (IP2 − D2 (t)) + h3 · (IP3 − D3 (t))
+ (h1 + h3 + b13 ) · (IP3 − D3 (t))− + (h1 + h2 + b12 )
Let
· (IP2 − D2 (t))− + h1 · ((IP1 − D1 (t)) + (IP2 − D2 (t))−
Ĝ1 (x, y) = EĜ(x − D1 (t − L1 , t − L3 ], y − D2 (t − L3 )),
Ĝ2 (x) = minĜ1 (x, y),
(E-57)
+ (IP3 − D3 (t))− )− + (h3 + b13 ) · B113 (t, π ) + (h2 + b12 )
(E-58)
· B112 (t, π )
y
Ĉ = minEĜ2 (x − D1 (t − L1 , t − L2 ]),
≤ h1 · (IP1 − D1 (t)) + h2 · (IP2 − D2 (t)) + h3 · (IP3 − D3 (t))
(E-59)
x
As L3 ≤ L2 ≤ L1 , we have Ft−L1 ⊆ Ft−L2 ⊆ Ft−L3 , by the tower
property of conditional expectations, we have
+ (h1 + h3 + b13 ) · (IP3 − D3 (t))− + (h1 + h2 + b12 )
· (IP2 − D2 (t))− + h1 · ((IP1 − D1 (t)) + (IP3 − D3 (t))−
+ (IP2 − D2 (t))− )− + d̄ · (B113 (t, π ) + B112 (t, π ))
= h1 · (IP1 − D1 (t)) + h2 · (IP2 − D2 (t)) + h3 · (IP3 − D3 (t))
E[C(t, π )|Ft−L1 ] = E{E[E(ϕ(IP1 − D1 (t), IP2 − D2 (t),
+ (h1 + h3 + b13 ) · (IP3 − D3 (t))− + (h1 + h2 + b12 )
IP3 − D3 (t))|Ft−L3 )|Ft−L2 ]|Ft−L1 }.
· (IP2 − D2 (t))− + (h1 + d̄) · ((IP1 − D1 (t))
As IP3 , IP2 − D2 (t − L3 ) and IP1 − D1 (t − L3 ) are adapted to Ft−L3 ,
and D3 (t), D2 (t) and D1 (t) are independent of Ft−L3 , we have
+ (IP2 − D2 (t))− + (IP3 − D3 (t))− )−
≤ h1 · (IP1 − D1 (t)) + h̄2 · (IP2 − D2 (t)) + h̄3 · (IP3 − D3 (t))
+ (h1 + d̄) · ((IP1 − D1 (t))− ∨ [(IP2 − D2 (t))−
E(ϕ(IP1 − D1 (t), IP2 − D2 (t), IP3 − D3 (t))|Ft−L3 )
+ (IP3 − D3 (t))− ]).
= G(IP1 − D1 (t − L3 ), IP2 − D2 (t − L3 ), IP3 )
≥ minG(IP1 − D1 (t − L3 ), IP2 − D2 (t − L3 ), y3 )
Then, similar to the proof for the lower bound, we get C ∗ ≤ C̄ ∗ .
y3
= Ĝ(IP1 − D1 (t − L3 ), IP2 − D2 (t − L3 )),
in which the first equation is from the equation (61) and the fact that Di (t)
is a stationary process, and the last equation is from the definition of Ĝ(·).
Taking a further expectation, conditioning on Ft−L2 , yields
E[Ĝ(IP1 − D1 (t − L3 ), IP2 − D2 (t − L3 ))|Ft−L2 ]
= Ĝ1 (IP1 − D1 (t − L1 , t − L2 ], IP2 )
ACKNOWLEDGMENT
We would like to thank the Associate Editor and three anonymous referees
for their valuable suggestions and corrections. We would also like to thank
Awi Federgruen and Paul Zipkin for their helpful comments. This research
was supported in part by the National Natural Science Foundation of China
under award 70328001 and 70731003.
≥ minĜ1 (IP1 − D1 (t − L1 , t − L2 ], y2 )
y2
REFERENCES
= Ĝ2 (IP1 − D1 (t − L1 , t − L2 ]),
where the first equation is because IP2 and IP1 − D1 (t − L1 , t − L2 ] are
adapted to Ft−L2 , D2 (t − L3 ) and D1 (t − L2 , t − L3 ] are independent of
Ft−L2 , and (E-57). The last equation is from (E-58). Similarly, we get
E[Ĝ2 (IP1 − D1 (t − L1 , t − L2 ])|Ft−L1 ] ≥ Ĉ.
Thus, we have that for t ≥ 0 and π ,
C(t, π ) = E[E(C(t, π )|Ft−L1 )] ≥ Ĉ.
From the above, it is easily to see that Ĉ is exactly the same as (69).
PROOF OF PROPOSITION 7: Similar to (E-34) in the proof of Proposition 5, we have that for any feasible π ∈ A,
C(t, π ) ≥ h1 · (IP1 − D1 (t)) + h2 · (IP2 − D2 (t))
+ h3 · (IP3 − D3 (t)) + (h1 + d) · ((IP1 − D1 (t))−
∨ [(IP2 − D2 (t))− + (IP3 − D3 (t))− ]).
The last expression above is just the inventory cost at time t of another system which is identical with symmetric costs with b12 + h2 = b13 + h3 = d.
Hence, similar to Theorem 8, the lower bound is obtained.
Naval Research Logistics DOI 10.1002/nav
[1] H. Arslan, S. Graves, and T. Roemer, A single-product inventory model for multiple demand classes, Manag Sci 53 (2007),
1486–1500.
[2] B. Ata and S. Kumar, Heavy traffic analysis of open processing
networks with complete resource pooling: Asymptotic optimality of discrete review policies, Ann Appl Probab 15 (2005),
331–391.
[3] N. Agrawal and M. Cohen, Optimal material control and
performance evaluation in an assembly environment with
component commonality, Nav Res Logist 48 (2001), 409–429.
[4] Y. Akcay and S. H. Xu, Joint inventory replenishment and
component allocation optimization in an assemble-to-order
system, Manag Sci 50 (2004), 99–116.
[5] S. Benjaafar and M. Elhafsi, Production and inventory control
of a single product assemble-to-order system with multiple
customer classes, Manag Sci 52 (2006), 1896–1912.
[6] F. Chen and Y.S. Zheng, Lower bounds for multi-echelon stochastic inventory systems, Manag Sci 40 (1994), 1426–1443.
[7] V. Deshpande, M.A. Cohen, and K. Donohue, A threshold
inventory rationing policy for service-differentiated demand
classes, Manag Sci 49 (2003), 683–703.
[8] M. Dogru, M. Reiman, and Q. Wang, A stochastic programming based inventory policy for assemble-to-order systems
with application to the W model, Oper Res 58 (2010), 849–864.
Lu, Song, and Zhang: Optimal and Asymptotically Optimal Policies
[9] M. El Hafsi, H. Camus, and E. Craye, Optimal control of a
nested-multiple-product assemble-to-order system, Int J Prod
Res 46 (2008), 5367–5392.
[10] A. Ha, Inventory rationing in a make-to-stock production system with several demand classes and lost sales, Manag Sci 43
(1997a), 1093–1103.
[11] A. Ha, Stock rationing policy for a make-to-stock production
system with two priority classes and backordering, Nav Res
Logist 43 (1997b), 458–472.
[12] W.H. Hausman, H.L. Lee, and A.X. Zhang, Joint demand
fulfillment probability in a multi-item inventory system with
independent order-up-to policies, Eur J Oper Res 109 (1998),
646–659.
[13] D.P. Heyman and M.J. Sobel, Stochastic models in operations
research, Vol. 2, Dover Publications, Mineola, NY, 1984.
[14] K. Huang and T. de Kok, Cost minimization schemes of
component allocation in an assemble-to-order system, Naval
Research Logistics 62 (2015), 158–169.
[15] R. Kapuscinski, R. Zhang, P. Carbonneau, R. Moore, and B.
Reeves, Inventory decisions in Dell’s supply chain, Interfaces
34 (2004), 191–205.
[16] L. Lu, J. Song, and H. Zhang, Optimal and near-optimal policies for assemble-to-order systems, Fuqua School of Business,
Duke University, 2010a.
[17] Y. Lu and J. Song, Order-based cost optimization in assembleto-order systems, Oper Res 53 (2005), 151–169.
[18] Y. Lu, J. Song, and D.D. Yao, Order fill rate, lead-time variability and advance demand information in an assemble-to-order
system, Oper Res 51 (2003), 292–308.
[19] Y. Lu, J. Song, and D.D. Yao, Backorder minimization in
multiproduct assemble-to-order systems, IIE Trans 37 (2005),
763–774.
[20] Y. Lu, J. Song, and Y. Zhao, No-holdback allocation rules
for continuous-time assemble-to-order systems, Oper Res 58
(2010b), 691–705.
[21] E. Nadar, M. Akan, and A. Scheller-Wolf, Technical note:
Optimal structural results for assemble-to-order generalized
M-systems, Oper Res 62 (2014), 571–579.
[22] E. Plambeck, Asymptotically optimal control for an
assemble-to-order system with capacitated component production and fixed transport costs, Oper Res 56 (2008), 1158–
1171.
[23] E. Plambeck and A. Ward, Optimal control of a highvolume assemble-to-order system, Math Oper Res 31 (2006),
453–471.
[24] E. Plambeck and A. Ward, Note: A separation principle for a
class of assemble-to-order systems with expediting, Oper Res
55 (2007), 603–609.
[25] M. Reiman and Q. Wang, A stochatic program based lower
bound for assemble-to-order systems, Oper Res Lett 40 (2012),
89–95.
645
[26] M. Reiman and Q. Wang, Asymptotically optimal inventory control for assemble-to-order systems with identical lead
times, Oper Res 63 (2015), 716–732.
[27] R. Rockafellar, Convex analysis, Princeton University Press,
Princeton, NY, 1968.
[28] K. Rosling, Optimal inventory policies for assembly systems
under random demands, Oper Res 37 (1989), 565–579.
[29] J. Song, The effect of lead time uncertainty in a simple
stochastic inventory model, Manag Sci 40 (1994), 603–612.
[30] J. Song, On the order fill rate in a multi-item, base-stock
inventory system, Oper Res 46 (1998), 831–845.
[31] J. Song, Order-based backorders and their implications
in multi-item inventory systems, Manag Sci 48 (2002),
499–516.
[32] J. Song, S. Xu, and B. Liu, Order-fulfillment performance
measures in an assemble-to-order system with stochastic lead
times, Oper Res 47 (1999), 131–149.
[33] J. Song and D.D. Yao, Performance analysis and optimization
in assemble-to-order systems with random lead-times, Oper
Res 50 (2002), 889–903.
[34] J. Song and P. Zipkin, “Supply chain operations: Assemble-toorder systems,” in: T. de Kok and S. Graves (Editors), Handbooks in operations research and management science, Chapter 11, Vol. 11: Supply Chain Management, North-Holland
Publisher, 2003.
[35] J. Song and Y. Zhao, The value of component commonality in
a dynamic inventory system with lead times, Manuf Serv Oper
Manag 11 (2009), 493–508.
[36] D. Topkis, Optimal ordering and rationing policies in a nonstationary dynamic inventory model with n demand classes,
Manag Sci 15 (1968), 160–176.
[37] W. van Jaarsveld and A. Scheller-Wolf, Optimization of
industrial-scale assemble-to-order systems, Journal of Computing 27 (2015), 544–560.
[38] F. de Vericourt, F. Karaesmen, and Y. Dallery, Optimal stock
allocation for a capacitated supply system, Management Sci
48 (2002), 1486–1501.
[39] H. Wan and Q. Wang, Asymptotically-optimal component
allocation for assemble-to-order production inventory systems, Oper Res Lett 43 (2015), 304–310.
[40] P. Xu, R. Allgor, and S. Graves, Benefits of reevaluating realtime order fulfillment decisions, Manuf Serv Oper Manag 11
(2009), 340–355.
[41] A. Zhang, Demand fulfillment rates in an assemble-to-order
system with multiple products and dependent demands, Prod
Oper Manag 6 (1997), 309–323.
[42] Y. Zhao and D. Simchi-Levi, Performance analysis and evaluation of assemble-to-order systems with stochastic sequential
lead times, Oper Res 54 (2006), 706–724.
Naval Research Logistics DOI 10.1002/nav