OPERATIONS RESEARCH
informs
Vol. 56, No. 5, September–October 2008, pp. 1312–1320
issn 0030-364X eissn 1526-5463 08 5605 1312
®
doi 10.1287/opre.1080.0549
© 2008 INFORMS
TECHNICAL NOTE
A Make-to-Stock System with Multiple
Customer Classes and Batch Ordering
Boray Huang
Department of Industrial and Systems Engineering, National University of Singapore, Singapore 117576, isehb@nus.edu.sg
Seyed M. R. Iravani
Department of Industrial Engineering and Management Sciences, Northwestern University,
Evanston, Illinois 60208, iravani@iems.northwestern.edu
This paper examines the impact of customer order sizes on a make-to-stock system with multiple demand classes. We
first characterize the manufacturer’s optimal production and rationing policies when the demand is nonunitary and lost if
unsatisfied. We also investigate the optimal policies of a backorder system with two demand classes and fixed order sizes.
Through a numerical study, we show the effects of batch orders on the manufacturer’s inventory cost as well as on the
benefit of optimal stock rationing. It is shown that batch ordering may reduce the manufacturer’s overall cost if carefully
introduced in a first-come-first-served (FCFS) system. With the same effective demand rates, the customers’ order sizes
also have a strong impact on the benefit of optimal stock rationing.
Subject classifications: batch ordering; make-to-stock; stock rationing.
Area of review: Manufacturing, Service, and Supply Chain Operations.
History: Received October 2004; revisions received July 2006, June 2007; accepted July 2007.
1. Introduction
stock-rationing problem in a capacitated system with lost
sales. The threshold-type policies are shown to be optimal
for production and rationing decisions. Ha (1997b) then
provides structural results for a two-demand-class system
with backorders. de Véricourt et al. (2001, 2002) extend
Ha’s model in (1997b) to n demand classes and demonstrate that the benefit of inventory pooling can be realized
only if the stock is efficiently allocated. Other extensions
can be found in Deshpande et al. (2003), Gayon et al.
(2004), and Axsäter et al. (2004).
When the customers’ orders are nonunitary, the problem of rationing quantity arises. That is, when an order
of a demand class arrives, the supplier/manufacturer has
to decide whether the entire order should be rejected, partially satisfied, or fully filled with its on-hand inventory.
The impact of customers’ batch ordering behavior on the
benefit of optimal stock rationing is still an open question
to our best knowledge. In this paper, we first characterize the optimal production and rationing decisions for a
lost sale system and a backorder system under nonunitary
demand. Then, through a numerical study, we examine the
effects of customer order sizes on the benefit of optimal
stock rationing. Even without stock rationing (i.e., using the
first-come-first-served (FCFS) policy to fill all customers’
orders), we find that the manufacturer’s overall cost can
sometimes be lowered by slightly encouraging batch ordering from the less-important customers. Our result provides
This paper investigates the impact of customers’ batch
ordering behavior on inventory management with multiple
demand classes. In traditional production/inventory literature, the impact of customer order sizes is usually studied
under a single demand class. When all customers are of the
same importance, it is well known that customers’ batch
ordering behavior has a negative impact on inventory control because of the bullwhip effect (see Lee et al. 1997a, b;
Cachon 1999). When the customers are of different values to the manufacturer, a stock-rationing policy can be
used to improve the manufacturer’s profit margins. That
is, when the manufacturer does not have sufficient inventory on hand, a certain portion of inventory is reserved to
satisfy the orders from more-valuable customers, and the
orders from less-valuable customers may be turned down.
Most existing literature on stock rationing assumes either
an uncapacitated system or unitary customer demands (i.e.,
each customer requests one unit at a time). In uncapacitated systems, the stock-rationing problem is known to
be very complicated, and therefore most existing research
focuses on presumed or heuristic policies (see Aviv and
Federgruen 1998, Chen et al. 2001, Frank et al. 2003).
On the other hand, most studies of capacitated productioninventory systems assume unitary demands. Therefore, the
manufacturer’s rationing decision reduces to accepting or
rejecting an arriving order. Ha (1997a) first studies the
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Huang and Iravani: A Make-to-Stock System with Multiple Customer Classes and Batch Ordering
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Operations Research 56(5), pp. 1312–1320, © 2008 INFORMS
a different point of view to marketers who usually provide
more-valuable customers with quantity discounts, and to
inventory managers who think of batch ordering as a pure
harm.
This paper is organized as follows. Section 2 introduces
our multiclass lost sale model and characterizes the optimal
production and rationing policies. Section 3 shows some
results for FCFS systems. Section 4 explores a backorder
case with two demand classes and fixed order sizes. The
numerical study is performed in §5, and §6 concludes the
paper.
2. Optimal Policies: The Multiclass Lost
Sale Model
Consider a manufacturer (capacitated supplier) who produces a single product to inventory. The production
capacity is units per unit time. There is a convex,
nondecreasing, and nonnegative holding cost hx per
unit time when the manufacturer’s inventory level is x
(x ∈ Z+ , where Z+ represents the set of nonnegative integers 0 1 2 ). Suppose that there are N different
demand classes, which arrive at the manufacturer as independent Poisson processes with rates i (i = 1 2 N ).
Each customer of class i requests Di (i = 1 2 N ) units
of the product. Let Di Ni=1 be random variables that are
mutually independent among different demand classes and
among different customers in the same class. Without loss
of generality, we assume that there is an upper bound Mi
for the quantity requested by each customer of class i (i =
1 2 N ). The probability that an arriving customer of
demand class i requires k items is PrDi = k
= pi k (k =
1 2 Mi and i = 1 2 N ). The customer’s order can
be partially fulfilled, and the unmet part is lost with a unit
shortfall cost ci for class-i demand. We assume that ci is
nondecreasing in i (i.e., c1 · · · cN ). Note that when
Di = 1 for all i = 1 2 N , our problem reduces to a
special case of unitary demand studied in Ha (1997a).
At any time t, the manufacturer can decide to produce and add the finished product to its on-hand inventory, or not to produce and stay idle. When a customer
demand arrives, the manufacturer has to decide the number of units to be allocated from its on-hand inventory.
Define the state variable xt as the manufacturer’s inventory level at time t. A manufacturer’s control policy describes the action at time t given the system state xt.
We also assume exponentially distributed production times,
and thus, as in de Véricourt et al. (2002), restrict our analysis to Markovian policies because the optimal policy lies
in this class (Bertsekas 1995). Ha (1997b) presents cases
in which the exponential production time is a reasonable
approximation in practice.
M1
MN
Let A x = A0 x A1 k x
k=1
AN k x
k=1
be
the set of actions under policy . A0 x is the control
action associated with the production decision. Ai k x is
the control action when an order in size of k from demand
class i arrives i = 1 2 N . More specifically, when
xt = x x 0, and letting k ∧ x = mink x
, the control actions are
0 not to produce
A0 x =
1 to produce
Ai k x = z to allocate z units from the inventory to an
arriving class-i order in size of k,
0 z k ∧ x
The problem can be formulated as a Markov decision
process (MDP). Let Li t be the accumulated lost sales
(in units) for demand class i up to time t. In addition, let
∈ 0 1 be the discount factor. Our objective is to find the
optimal control policy that minimizes the manufacturer’s
discounted total cost over an infinite horizon. That is, we
try to solve the following problem:
min Ex0
0
e−t hxt dt +
N
ci
i=1
0
e−t dLi t where Ex0
denotes the expectation over time, given the
initial system state x0 and policy .
Without losing
generality, we can scale the parameters
and let + Ni=1 i + = 1. Following Lippman (1975),
the optimality equation of the MDP is
f x = hx + minf x f x + 1
+
N
i=1
i
Mi
k=1
pi k
min f x − z + k − zci (1)
0zk∧x
where f x is the optimal discounted cost under the initial system state x. The first minimization on the righthand side of (1) represents the manufacturer’s production
decision. It is optimal to produce if f x > f x + 1, and
stay idle otherwise. The second minimization corresponds
to the manufacturer’s optimal rationing decisions. When
a customer of class-i arrives and requests k items, it is
optimal for the manufacturer to allocate z∗i x k items and
pay the lost sale cost for the unmet part, where z∗i x k =
arg min0zk∧x f x − z + k − zci . In Ha’s (1997a) unitary demand case, M1 = M2 = k = 1, so z∗i x k = 0 or 1.
Define the difference operator Dx of any function on
Z+ as Dx x = x + 1 − x. The following theorem shows that, when the order sizes are not unitary, the
optimal production and rationing policies are thresholdtype base-stock and stock-reservation policies, respectively.
The proofs of Theorem 1 and all other theorems in this
paper are presented in Online Appendix A, unless otherwise indicated. An electronic companion to this paper is
available as part of the online version that can be found at
http://or.journal.informs.org/.
Theorem 1. (a) There exists an optimal stationary production and stock-rationing policy ∗ .
Huang and Iravani: A Make-to-Stock System with Multiple Customer Classes and Batch Ordering
1314
Operations Research 56(5), pp. 1312–1320, © 2008 INFORMS
(b) A base-stock produce-up-to policy with a threshold base-stock level S is optimal for the manufacturer’s
production decision. That is, there exists a critical stock
level S such that
0 if x S
∗
A0 x =
1 if x < S
S = minx Dx f x 0 x ∈ Z+ (c) A stock-reservation policy is optimal for stock
rationing. There exist reserve-stock levels r1∗ r2∗ rN∗
such that the manufacturer’s optimal rationing decision is
to make the postallocation inventory level as close to the
corresponding reserve-stock level as possible, i.e.,
ri∗ = minx Dx f x −ci x ∈ Z+ where i = 1 2 N , a+ = maxa 0
and k is a natural
number.
(d) The reserve-stock levels are nondecreasing in i, and
S rN∗ · · · r2∗ r1∗ = 0.
Figure 1 shows the optimal stock-rationing decision
when a class-i order in size of k units arrives. If the manufacturer has sufficient inventory (i.e., x ri∗ + k), the order
should be fully satisfied. If the manufacturer has insufficient
inventory (i.e., x ri∗ ), the entire order should be rejected.
When the manufacturer’s inventory is slightly higher than
the reserve-stock level ri∗ , the order should be partially
filled so that at least ri∗ units are still left in the inventory
after allocation.
We would like to note that because it is practically
impossible and never optimal to keep infinite units at
the manufacturer’s inventory, we can set an upper bound
on xt. The long-run average cost optimality can then be
obtained by letting the discount factor go to zero (see
Weber and Stidham 1987, Ha 2000).
The optimal rationing decisions when a
class-i order in size of k units arrives. Left:
x ri∗ + k then z∗i x k = k; Middle: ri∗ + k >
x > ri∗ then z∗i x k = x − ri∗ ; Right: x ri∗ then
z∗i x k = 0.
Inventory
S
Inventory
S
S
Inventory
S
S
S
x
k
x
ri*
ri*
ri*
x – ri*
ri*
ˆ
Ai k x = mink x − ri + Without loss of generality, we let S rˆN · · · rˆ2 rˆ1 = 0.1 The case of Ŝ = 0 is trivial because it means no
inventory would be kept and all demands are lost. When
Ŝ > 0, define
∗
Ai k x = z∗i x k = mink x − ri∗ + Figure 1.
Theorem 1 shows that the optimal policy is a multithreshold policy with thresholds r1∗ r2∗ rN∗ S
. We
develop an algorithm that results in the exact steady-state
probabilities of the system under any multiple-threshold
policy such as the optimal policy. These steady-state probabilities can then be used to obtain the total average costs
of the corresponding multithreshold policy.
Let ˆ be a multithreshold policy with N + 1 thresholds rˆ1 rˆ2 rˆN Ŝ
under which the production and the
rationing decisions are
0 if x Ŝ
ˆ
A0 x =
1 if x < Ŝ
ri*
x
ri*
$i j =
Mi
i pi k
k=j
After
rationing
Before
rationing
After
rationing
Before
rationing
After
rationing
Ŝ
Let qi i=0 be a series of real numbers, and define 'i i=0
as the steady-state probabilities for states 0 1 Ŝ
ˆ Note that any state x > Ŝ is transient
under the policy .
ˆ The following algorithm yields 'i Ŝ .
under .
i=0
2.1. Exact Algorithm for Steady-State Analysis of
the Multiclass Lost Sale Model
Step 1. Let qn = 0 for all n > Ŝ and n ∈ Z+ .
Step 2. Let qŜ be an arbitrary strictly positive real number (e.g., qŜ = 01).
Step 3. Starting from state n = Ŝ, calculate qn−1 for state
n − 1 through the following formula:
Mi
N qn−1 =
i=1 j=1
where
1X =
1
0
1n>rˆi $i j qn−1+j (2)
if X is true
if X is false
As a result, we can find qŜ qŜ−1 q1 q0 recursively.
Step 4. (Normalization) The steady-state probabilities 'n Ŝn=0 can be obtained by
q
'n = n Ŝ
i=0 qi
When the steady-state probabilities 'n Ŝn=0 are obtained
using the above algorithm, the long-run average cost per
unit time TC ˆ under the policy ˆ can then be calculated as
Mi
N
Ŝ TC ˆ =
hn + i ci
pik k − n − ri + + 'n n=0
Before
rationing
i=1
k=1
The justification of the algorithm is presented in Online
Appendix C.1.
Huang and Iravani: A Make-to-Stock System with Multiple Customer Classes and Batch Ordering
Operations Research 56(5), pp. 1312–1320, © 2008 INFORMS
3. Some Results for the FCFS
Stock-Rationing Policy
We define the FCFS policy by the action set A FCFS x =
A0FCFS x ANFCFS x, where
0 not to produce
FCFS
A0 x =
1 to produce
AiFCFS x = di ∧ x
to allocate di ∧ x units to an
arriving class-i order of size di i = 1 2 N and x is the manufacturer’s on-hand inventory. Letting
zx k = k ∧ x, the optimality equation is
f
FCFS
x = hx + minf
+
FCFS
x f
FCFS
x + 1
Mi
N
i pik f FCFS z−zxk +ci k−zxk i=1
k=1
(3)
It is interesting to note that, with multiclass batch demands,
the optimal cost function f FCFS is not always convex
in x. However, when the demand sizes are identically distributed (i.e., PrDi = k
= pk , k = 1 2 M, for all i =
1 N ), we can rewrite the optimality equation (3) as
f FCFS x = hx + minf FCFS x f FCFS x + 1
+
M
k=1
pk f FCFS x − zx k + ck − zx k (4)
where = Ni=1 i and c = Ni=1 i ci /. The following
theorem shows that the base-stock policy is optimal for the
manufacturer’s production decisions.
Theorem 2. When the customer demand sizes are independently and identically distributed (i.i.d.), and the manufacturer applies an FCFS stock-rationing policy:
(a) There exists an optimal stationary production policy.
(b) The base-stock produce-up-to policy is optimal for
the manufacturer’s production decision.
(c) The lost sale costs ci and the demand rates i affect the optimal cost function and the optimal production
decisions only if c or change. As long as c and remain
the same, any change in ci , or any change in i would
not change the optimal costs and decisions.
In other words, when the order sizes are i.i.d., the different demand classes can be aggregated with an equivalent
(aggregated) arrival rate and an equivalent lost sale cost
rate c. The base-stock policy is also optimal for the manufacturer’s production decision. This simple rule of demand
aggregation may not work when the order sizes are not i.i.d.
Another interesting result in systems under the FCFS
policy is presented in Theorem 3 for a two-demand-class
case with deterministic order sizes. The theorem states that
the FCFS policy and the optimal policy are asymptotically
1315
equivalent in terms of the manufacturer’s long-run average cost when one of the customers’ order sizes becomes
very large (i.e., approaches infinity). The asymptotic analysis and the proof of Theorem 3 are presented in Online
Appendix B.1.
Theorem 3. Suppose that there are two demand classes
with deterministic order sizes Di = di i = 1 2. If the optimal policy has a finite base-stock level and the effective
demand rates eff
i = i di i = 1 2 stay the same, we have
TC opt = TC FCFS when di → (i = 1 or 2).
4. Two Demand Classes with
Fixed Order Sizes and Backorders
The analysis of stock rationing in systems with backorders is usually considered more difficult than its counterpart with lost sales (see Ha 1997b and de Véricourt et al.
2001, 2002). In this section, we explore a simple backorder case with two demand classes and fixed (deterministic)
order sizes d1 and d2 . At any time, the manufacturer can
decide whether to produce and allocate the finished unit to
its on-hand inventory, to produce and allocate the finished
unit to fill the backorders, or not to produce and stay idle.
When an order arrives, the manufacturer must decide how
many on-hand units should be allocated to the order, and
the unsatisfied part of the order will be backordered. There
is a linear holding cost h per unit per unit time for the
manufacturer’s on-hand inventory. There also exist linear
backorder cost rates b1 and b2 per backorder unit per unit
time for class-1 and class-2 demands, respectively. We let
b1 > b2 ; thus, an order from demand class-1 should always
be satisfied if possible. Furthermore, both on-hand inventory and class-1 backorders cannot occur at the same time,
but it is possible that both on-hand inventory and class-2
backorders coexist (see de Véricourt et al. 2001, 2002).
Parallel to Ha (1997b), we can then define the following
system state variables xt and yt:
xt = xt+ − xt− where xt+ is the on-hand
inventory level at time t, and
xt− is the number of class-1
backorders at time t.
yt = number of class-2 backorders at time t,
where xt+ , xt− , and yt are all nonnegative integers. The control actions A,0 x y A,1 x y A,2 x y
associated with a policy , can be defined as follows:

0 not to produce





1 to produce and assign the finished unit




to a class-1 backorder when x < 0, or to




produce and assign the finished unit to

A,0 x y =
on-hand inventory when x 0,



2 to produce and assign the finished unit




to a class-2 backorder when y > 0,





or to produce and assign the finished


unit to on-hand inventory when y = 0
Huang and Iravani: A Make-to-Stock System with Multiple Customer Classes and Batch Ordering
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Operations Research 56(5), pp. 1312–1320, © 2008 INFORMS
A,1 x y = d1 ∧ x+ to allocate (d1 ∧ x+ ) units from
the on-hand inventory to an
arriving class-1 order of size d1 .
A,2 x y = a to allocate a units from the on-hand
inventory to an arriving class-2
order of size d2 , 0 a d2 ∧ x+ ,
where A,0 x y represents the production control under
policy , at state xt yt = x y, and A,i x y (i =
1 2) is the stock-rationing decision under policy , when
a class-i order arrives at state x y. Similar to the previous section, we look for the optimal policy , ∗ to solve the
following minimization problem over an infinite horizon:
,
−t
+
−
min Ex0
e
hxt
+
b
xt
+
b
yt
dt
1
2
y0
,
0
,
Ex0
y0
is the expectation over time, given the initial system state (x0 y0)
and the policy ,.
Again, we let + 2i=1 i + = 1. The new optimality
equation for the backorder case becomes2
gx y = cx y + mingx y gx + 1 y gx y − 1
+ 1 gx − d1 y
+ 2
min
0ad2 ∧x+ gx − a y + d2 − a
(5)
where gx y is the optimal discounted cost with the initial
state x y, and cx y is the instantaneous cost function
at the state x y, which is defined by
+
hx + b1 x− + b2 y when y 0
cx y =
(6)
+
when y < 0
Now define the difference operators Dx , Dy , and Dxy of any
function . on Z2 as
Dx .x y = .x + 1 y − .x y
Dy .x y = .x y + 1 − .x y
Dxy .x y = .x + 1 y − .x y − 1
The following theorem characterizes the optimal production
and stock-rationing decisions:
Theorem 4. (a) There exists an optimal stationary policy , ∗ .
(b) A two-threshold policy is optimal for the manufacturer’s production decision. Specifically, there are two
thresholds S ∗ and R∗ , where
S ∗ = minx0 Dx gx 0 > 0
R∗ = minx0 Dxy gx 1 > 0
Given a system state x y, the manufacturer’s optimal
production decision is

0 if S ∗ x



∗
A,0 x y = 1 if x < R∗ 


2 if R∗ x < S ∗ ∗
∗
where S R 0.
(c) When a class-2 customer arrives and requests d2
items, it is optimal to allocate
∗
A,2 x y = d2 ∧ x − R∗ +
units from the on-hand inventory to satisfy the request. The
∗
unfilled part of the order is backordered. A,2 x y is nondecreasing in x but independent of y.
Theorem 4 shows that we can extend the optimality of
the multilevel rationing (ML) policy of the unitary demand
case (see de Véricourt et al. 2002) to the cases when the
customers’ order sizes are larger than one: there exist two
thresholds S ∗ and R∗ that are independent of the system
state. The optimal production policy is as follows. At any
time, if x S ∗ , it is optimal to stop production and remain
idle. Thus, S ∗ is the manufacturer’s base-stock level under
the optimal production policy. On the other hand, if R∗ x < S ∗ , it is optimal to produce and assign the finished
unit to class-2 backorders if there are any, or to the onhand inventory if there are no class-2 backorders. When
x < R∗ , it is optimal to produce and assign the finished unit
to class-1 backorders if x < 0, or to the on-hand inventory
if 0 x < R∗ . Similar to the lost sale case, the optimal
stock rationing policy is also of a threshold type when a
less valuable order comes: the manufacturer should allocate
the on-hand inventory so that the postallocation inventory
level can be as close to the threshold R∗ as possible.
In the lost sale case, the convexity of the cost function is
the core property that determines the structure of the optimal policy. As shown in the proof of Theorem 4 for the
backorder case, the submodularity/supermodularity of the
cost function are also crucial to the structure of the optimal
policy, which dramatically increases the complexity of the
analysis in the backorder case.
Because the optimal policy in the backorder case is still
of the threshold type, it is interesting to check whether
the recursive algorithm developed by de Véricourt et al.
(2002) for the unitary demand can be extended to systems
with batch demands. When there are two demand classes
with unitary order sizes, de Véricourt et al.’s algorithm
starts with a single-class subproblem that solves an M/M/1
make-to-stock queue with the instantaneous cost function
cx = h + b2 x+ + b1 − b2 x− . The optimal base-stock
level of the subproblem then becomes the optimal stockrationing level R∗ of the main problem. Having R∗ , the
optimal base-stock level S ∗ is then found by applying the
closed-form solutions of M/M/1 queues. We develop a
heuristic algorithm to extend de Véricourt et al.’s idea to
the batch demand case, in which the optimal thresholds are
estimated.
Heuristic Algorithm for Estimating the Optimal
Production and Stock-Rationing Thresholds
Step 1. Solve a single-class M d1 /M/1 make-to-stock
queue with a holding cost rate h + b2 and a backorder cost
Huang and Iravani: A Make-to-Stock System with Multiple Customer Classes and Batch Ordering
Operations Research 56(5), pp. 1312–1320, © 2008 INFORMS
rate b1 − b2 . Find the optimal base-stock level and let it
The problem can be solved by searching for the nonbe R.
negative minimizer of the following cost function g1 R,
where
g1 R = h + b1 1R>0
·
d1 + 1
2
R
i=1
i1i + b1 − b2 1 d1
− 1 d1
−R R ∈ Z+ (7)
1i Ri=− are the steady-state probabilities that can be
obtained in the following recursive method starting with
i = R:

0
if i > R






 1 − 1 d1
if i = R
1i =
(8)


d1


1 

1
if 0 i < R

j=1 i+j
be the
Thus, R = arg ming1 R0 R ∈ Z+ . Let g1∗ = g1 R
optimal long-run average cost of the subproblem.
Step 2. Search for the minimizer of the cost function
g2 S, where
g2 S = 1 − 1S>R
+ b2
S
i=R+1
2i g1∗ + h + b2 1S>R
1 d1 + d12 + 2 d2 + d22 −S 2 − 1 d1 − 2 d2 S
i=R+1
i2i
(9)
Note that in (9), the steady-state probabilities 2i are
When S > R,
probabilities 2i needed only when S > R.
can be obtained in the following recursive method:

0






 1 − 1 d1 + 2 d2
2i =


d1
d2

 1 

2i+j + 2
2

j=1
j=1 i+j
if i > S
if i = S
(10)
if R i < S
S ∈ Z+ . Then, Ŝ R
are
Let Ŝ = arg ming2 S S R
∗
∗
the heuristic’s estimates of S R .
Note that when d1 = d2 = 1, our heuristic algorithm is
exactly the same as the algorithm proposed by de Véricourt
= S ∗ R∗ . For justifications
et al. (2002), and thus Ŝ R
and interpretations of the heuristic algorithm, please refer
to Online Appendix C.2.
We perform an extensive numerical study with 6,400
examples to investigate the effectiveness of the heuristic
algorithm.3 The performance measure PR is defined as
PR =
TC heu − TC opt
× 100%
TC opt
1317
where TC heu is the system’s long-run average cost per unit
and TC opt is
time using the heuristic thresholds Ŝ R
,
the system’s optimal long-run average cost using the optimal thresholds S ∗ R∗ . We find that the heuristic algorithm works well in general, with the average value of
PR around 3%. However, there are some cases where the
heuristic algorithm does not find good estimates of the
thresholds. We observed in 654 (out of 6,400) cases of our
eff
numerical study that, when eff
1 /2 was relatively small
(i.e., smaller than one) and $ was relatively large (i.e.,
larger than 0.9), the error PR was greater than 10%.
Note that the essence of de Véricourt et al.’s (2002)
recursive algorithm lies in the property that, given a
stock-rationing level R, the subset of the system space
4 = xt xt R
can be treated as an independent
single-class make-to-stock M/M/1 queue. When the system enters the subset 4, it starts with the state xt = R.
However, if the order sizes (especially d1 ) are not unitary,
the system may enter the subset 4 with any starting state
of xt = R R − 1 R − d1 + 1+ . The probabilities of the starting state depend on the limiting probabilities of the states xt = R + d1 R + d1 − 1 R + 1
,
which are outside the subset 4. The connection across the
rationing threshold in the batch demand cases makes the
closed-form solutions very complicated. It also makes our
heuristic algorithm, which treats the subset 4 as an independent M d1 /M/1 queue, fail to provide an accurate estimate for the system’s true performance in some cases. In
those cases, the heuristic algorithm assigns a higher probability to the state xt = R because it is the starting state
of the subproblem. As a result, the stockout probability in
the heuristic algorithm is lower than it should be, resulting in an underestimated overall cost of the subset, and
consequently, a lower-than-optimal base-stock level Ŝ. The
eff
error becomes critical when eff
1 /2 is small (i.e., smaller
than one) and $ is relatively large (i.e., larger than 0.9)
because the system would stay in the subset 4 for a longer
period of time.
5. Numerical Study
In this numerical study, we consider a lost sale system
with two demand classes (N = 2). The order size of each
demand class is constant, i.e., Di = di i = 1 2. As
assumed in §2, we let c1 > c2 . We also assume a linear
inventory holding cost where hx = hx. Consider the following system parameters: (i) effective demand rates: eff
i =
i di for i = 1 2; (ii) lost sale cost ratio: 5 = c1 /c2 ; (iii)
eff
traffic intensity of the system $ = eff
1 + 2 /; and (iv)
eff
relative holding cost rate: h = h/1 c1 + eff
2 c2 .
The ranges of the above parameters are chosen to be
consistent with those in Ha (1997a). If a policy is chosen as the benchmark, we evaluate the performance of the
optimal policy and policy through their long-run total
average costs per unit time TC opt and TC , respectively.
Huang and Iravani: A Make-to-Stock System with Multiple Customer Classes and Batch Ordering
1318
Operations Research 56(5), pp. 1312–1320, © 2008 INFORMS
Figure 2.
The effect of order sizes. Left: on the total average cost, and Right: on the benefit of the optimal policy
eff
CR FCFS , compared with the FCFS policy (eff
1 = 01, 2 = 10, 5 = 60, $ = 18, h = 001).
35
70
Average cost per unit time
65
Cost reduction CR (%)
Cost reduction by
optimal rationing
55
50
45
40
Cost reduction by
optimal rationing
35
30
25
20
d1 = n, d2 = 1
d1 = 1, d2 = n
30
60
0
5
10
15
d1 = n, d2 = 1, FCFS
d1 = n, d2 = 1, optimal
d1 = 1, d2 = n, FCFS
d1 = 1, d2 = n, optimal
20
25
25
20
15
10
5
0
30
0
5
10
Order size n
The values of TC opt and TC are obtained from the successive approximation with an error bound 0.01%. The convergence of the successive approximation is shown in Sennott
(1999). Define cost reduction CR that measures the cost
effectiveness of the optimal policy compared with that of
the benchmark policy as
CR =
The larger CR is, the more beneficial it is to apply the
optimal policy instead of the benchmark policy .
Our first benchmark policy is the optimal base-stock policy with FCFS stock allocation defined in §3. Let S FCFS
be the optimal base-stock level. S FCFS can be obtained
from the successive approximation. Under batch demands,
we observe that the impacts of the effective demand rates
eff
i , the lost sale cost ratio 5, and traffic intensity $
are similar to the results in the unitary demand case of
Ha (1997a). Therefore, in this paper we only focus on the
impact of order sizes. A representative example is chosen
60
30
25
d1 = n, d2 = 1
d1 = 1, d2 = n
50
40
d1 = n, d2 = 1, FCFS (S )
d1 = n, d2 = 1, optimal (S )
d1 = n, d2 = 1, optimal (r)
d1 = 1, d2 = n, FCFS (S )
d1 = 1, d2 = n, optimal (S )
d1 = 1, d2 = n, optimal (r)
30
20
10
0
25
The effect of order sizes. Left: on the base-stock levels and the reserve-stock levels, and Right: on the benefit
of the optimal stock rationing policy, compared with Ha’s (1997a) policy.
Cost reduction CR (%)
Base-stock level S and reserve-stock level r
20
in Figure 2, which shows the impact of the order sizes on
the total average costs TC opt and TC FCFS , and on the benefit
of optimal stock rationing CR FCFS . In addition, the left part
of Figure 3 shows the impact on S, S FCFS and r2∗ = r. Note
that when we increase one of the order sizes (di ) in these
figures, we keep eff
the same and the other order size dj
i
equals one (i = j).
In Figure 2, we observe that both TC opt and TC FCFS
increase with d1 . The benefit of implementing the optimal rationing policy over the FCFS policy (CR FCFS ) may
first increase, but eventually decreases with d1 as d1 /d2
becomes very large. This is because the manufacturer usually has to raise its inventory to deal with large orders
from more valuable customers. Under the FCFS policy, the
manufacturer does not reserve stock for class-1 demands,
so to avoid a large lost sale penalty a much higher basestock level is needed. As a result, the total inventory cost
is higher under the FCFS policy and CR FCFS may first
increase with d1 . However, when d1 /d2 is very large,
the class-1 orders seldom arrive (because eff
1 remains the
TC − TC opt
× 100%
TC Figure 3.
15
Order size n
0
5
10
20
15
10
5
15
Order size n
20
25
30
0
0
5
10
15
Order size n
20
25
30
Huang and Iravani: A Make-to-Stock System with Multiple Customer Classes and Batch Ordering
Operations Research 56(5), pp. 1312–1320, © 2008 INFORMS
same). It may not be optimal to raise the inventory without
a limit. As Theorem 3 depicts, CR FCFS converges to zero
when d1 → with a finite base-stock level.
When d2 increases, the class-2 customers order less frequently, but with more units in each order. It is interesting
to note from the left part of Figure 2 that, under the FCFS
policy, the total average cost may decrease with d2 , especially when 5 and $ are large. Our explanation is that, with
infrequent orders from demand class-2, class-1 orders are
easier to be filled with the same base-stock level, resulting in an overall saving in lost sale penalty. The base-stock
level can even be lowered (see the left part of Figure 3)
without significantly deteriorating the fulfillment of class-1
orders. As a result, the total cost TC FCFS may decrease
with d2 when d2 /d1 is not very large. We have also proved
in Theorem 3 that CR FCFS will decrease to zero when the
class-2 order size d2 goes to infinity.
With the same effective demand rates, customers’ order
sizes have a strong impact on the benefit of optimal stock
rationing. For example, in the right part of Figure 2, when
d1 increases from 1 to 5, CR FCFS increases from 25.68%
to 30.07%. It then decreases dramatically to 1.39% when
d1 = 30. We also study the impact of the order size variability on CR FCFS in Online Appendix B.2, which shows
that a large variability in class-1 order sizes will deteriorate the benefit of optimal stock rationing, but the order
size variability in class-2 demands has little impact on the
benefit of optimal stock rationing.
The right part of Figure 3 investigates the impact of
the order sizes on CRHa , where the benchmark is the policy proposed by Ha (1997a). Specifically speaking, we
first use Ha’s model of unitary demands with arrival rates
i = eff
i (i = 1 2), and obtain the optimal base-stock level
SHa , as well as the optimal reserve-stock level rHa . We
then computed the manufacturer’s average cost if SHa and
rHa are used as the base-stock level and the reserve-stock
level in the corresponding batch demand systems under the
same effective demand rates eff
i = i di . We find that when
d1 = 1 but d2 > 1, the performance of Ha’s policy remains
close to optimal regardless of the batch size d2 (note that
eff
2 stays the same). However, when d2 = 1 and class-1
customers order in batches, the manufacturer may save up
to 21% in cost if it uses the optimal policy instead of Ha’s
policy.
6. Conclusions
This paper studies the optimal production and rationing
policies in a make-to-stock system where different classes
of customers order in batches. A lost sale case with multiple demand classes and a backorder case with two demand
classes are analyzed. We characterized the optimal policies
as threshold-type policies. Through the numerical study on
the lost sale case, we have also illustrated that the customers’ order sizes can significantly affect the benefit of
optimal stock rationing.
1319
As indicated in the introduction, batch ordering is usually regarded as a negative factor in inventory management.
This is true when all customers are of the same economical
importance to the manufacturer. However, when the customers are of different importance and stock rationing is
not allowed by regulations or by contracts (which is common in many retail businesses), encouraging less-valuable
customers to order in large batches may lower the manufacturer’s overall cost. When suppliers are allowed to
ration their inventory for different demand classes, they
should also be aware of the impact of batch demands on
the cost effectiveness of optimal stock rationing. When
less-valuable customers have larger order sizes but order
less frequently, the need for stock rationing decreases. On
the other hand, when a quantity discount is introduced
to more valuable customers, and therefore they order in
large batches, the optimal stock rationing may become
more beneficial. Nevertheless, the benefit of optimal stock
rationing decreases rapidly if either order size becomes
very large. As a result, when adopting stock rationing as an
operations strategy, managers should take the customers’
ordering behaviors and the company’s marketing strategies
into consideration to correctly identify the benefit of stock
rationing. Future research on the integrated solutions for
both inventory management and marketing strategies is thus
highly expected.
7. Electronic Companion
An electronic companion to this paper is available as part
of the online version that can be found at http://or.journal.
informs.org/.
Endnotes
1. If rˆ1 rˆ2 rˆN are not monotonically increasing, we
can rearrange the demand classes to make rˆ1 rˆ2 rˆN nondecreasing.
2. Note that the original system state space has a boundary y = 0. We extend the system state space to Z2 and let
cx y = + when y < 0. The operators in this section
and in Online Appendix A.3 are also defined on Z2 . For a
detailed discussion, see Ha (1997b) and Glassman and Yao
(1994).
3. Please refer to Online Appendix C.3 for the parameter
settings of the test.
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