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In this paper, we study the impacts of imposing a minimum order quantity (MOQ) on a two-echelon supply chain implementing quick response (QR) and consider the issue of coordination for such a system. By exploring the QR-MOQ supply chain system, we analytically prove that the retailer's expected profit (REP) is nonincreasing in the MOQ. We further find that the MOQ that maximizes the manufacturer's expected profit can significantly reduce the REP and the supply chain's efficiency. Understanding that the static nature of the preagreed MOQ hinders the information updating capability brought about by QR, which, in turn, decreases the supply chain's efficiency, we propose an innovative dynamic MOQ policy and derive the analytical conditions under which channel coordination with Pareto improvement is achieved.
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868 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 42, NO. 4, JULY 2012
Impacts of Minimum Order Quantity on a
Quick Response Supply Chain
Pui-Sze Chow, Student Member, IEEE, Tsan-Ming Choi, Member, IEEE, and T. C. E. Cheng
Abstract—In this paper, we study the impacts of imposing a
minimum order quantity (MOQ) on a two-echelon supply chain
implementing quick response (QR) and consider the issue of co-
ordination for such a system. By exploring the QR-MOQ supply
chain system, we analytically prove that the retailer’s expected
profit (REP) is nonincreasing in the MOQ. We further find that
the MOQ that maximizes the manufacturer’s expected profit can
significantly reduce the REP and the supply chain’s efficiency. Un-
derstanding that the static nature of the preagreed MOQ hinders
the information updating capability brought about by QR, which,
in turn, decreases the supply chain’s efficiency, we propose an in-
novative dynamic MOQ policy and derive the analytical conditions
under which channel coordination with Pareto improvement is
achieved.
Index Terms—Coordination, minimum order quantity (MOQ),
quick response (QR).
I. INTRODUCTION AND LITERATURE REVIEW
B
OTH QUICK response (QR) and minimum order quantity
(MOQ) are common practices in the apparel industry.
Originating in the American apparel industry in the 1980s
to cope with keen overseas competition, QR makes use of
advanced technology to reduce the order lead time [1] and
to improve the forecasting capability of the retailer [2]. As a
result, the retailer has a higher level of ordering flexibility and
can be more responsive to market changes. Both academics
and practitioners assert that QR can help improve the retailer’s
business performance (e.g., [3]–[5]). In contrast, the manu-
facturer is generally believed to be reluctant to QR adoption
owing to the potential reduction in its expected profit (as a
result of potentially smaller order quantity) [5], [6] and the
initial technology and investment required [7]. MOQ, on the
other hand, is a common practice imposed by the manufacturer.
When one browses through various e-commerce business-to-
business sourcing portals in the Internet such as alibaba.com
Manuscript received September 6, 2010; revised June 13, 2011; accepted
October 28, 2011. Date of publication February 17, 2012; date of current
version June 13, 2012. This work was supported in part by the funding provided
by the Hong Kong Polytechnic University (J-BB6U) and the Competitive
Earmarked Research Grant provided by the Hong Kong Research Grant Council
under Grant number PolyU5420/10H. This paper was recommended by Asso-
ciate Editor A. Janiak.
P.-S. Chow and T.-M. Choi are with the Institute of Textiles & Clothing,
Faculty of Applied Science and Textiles, The Hong Kong Polytechnic Univer-
sity, Kowloon, Hong Kong (e-mail: linda.chow@connect.polyu.hk; tcjason@
inet.polyu.edu.hk).
T. C. E. Cheng is with the Department of Logistics and Maritime Studies,
Faculty of Business, The Hong Kong Polytechnic University, Kowloon, Hong
Kong (e-mail: lgtcheng@polyu.edu.hk).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSMCA.2012.2183351
and globalsources.com, it is not difficult to find that many
manufacturers state their MOQ requirement together with their
product information. Like the other forms of quantity commit-
ments, MOQ helps the manufacturer to “reduce the uncertainty
during the ordering process” and “ensure markets” [8]. Making
use of the principle of economy of scale, MOQ can also
justify the production setup cost of the manufacturer, as well
as guarantee its income to be at a certain level. However, the
retailer, in general, does not welcome MOQ as it restricts its
ordering flexibility.
These seemingly conflicting opinions between the retailer
and the manufacturer on these two common practices (i.e.,
on the one hand, QR enhances while MOQ hinders ordering
flexibility for the retailer; on the other hand, MOQ provides
guarantee of income, while QR potentially erodes profit for
the manufacturer) arouse our interest to study the integrated
impacts of the two practices in fashion supply chains. Never-
theless, to the best of our knowledge, there is no prior research
on the channel coordination issue in the presence of both QR
and MOQ. It is this gap in the literature that we would like to
bridge in this paper.
Three aspects of the literature are related to this paper,
namely: 1) supply chain inventory policies with forecast up-
dating; 2) channel coordination issues; and 3) supply chain
inventory management with MOQ. We review them one by one
in the following.
Since its introduction, QR has been a popular topic in the
supply chain management literature (see [9] for a comprehen-
sive review of research on QR). In particular, the study of in-
ventory decision-making problems with information revisions
has drawn the attention of many researchers. For example, [10]
and [11] independently study the two-stage ordering problem
with forecast revisions and uncertain future ordering cost. [10]
employs a bivariate normal model for the information updating
process and explores the expected values of worthless and per-
fect information. Adopting the Bayesian information updating
process, [11] discusses the effect of the optimal dual-ordering
policy on service level and profit uncertainty. Other recently
published works in the literature on inventory decisions with
information updating include [12]–[15].
Channel coordination has always been an important issue in
the supply chain management literature since individual parties’
optimal decisions are suboptimal to the supply chain in general.
One of the most popular measures to achieve coordination
is by providing incentive alignment contracts (also known as
effective contract design) [16]. A vast amount of research has
been devoted to the design of a coordinating mechanism for
supply chains in which forecast updating is allowed. Iyer and
1083-4427/$31.00 © 2012 IEEE
CHOW et al.: IMPACTS OF MINIMUM ORDER QUANTITY ON A QUICK RESPONSE SUPPLY CHAIN 869
Bergen [1] are among the first to explore such a problem in the
literature. They show that it may not be beneficial to the manu-
facturer if it allows the retailer to postpone its ordering decision
even though such action allows the retailer to update its forecast
and to improve its expected profit. In the light of this, they
suggest adopting commitment policies on various measures
(such as wholesale price, service level, and total volume across
products) that can make both parties better off under QR (Pareto
improving) and hence coordinate the supply chain. Other types
of coordinating mechanisms that have been explored in a simi-
lar context include the following: backup agreement contract in
fashion supply chains [17], two-tier wholesale price contract
with return option for fashion goods [18], advance booking
discount [19], risk-sharing “bidirectional return contract” [20],
return policy with markdown money [21], trade promotion
scan-back rebates [22], and risk and profit sharing contract
under price-dependent demand [23]. Nevertheless, none of the
aforementioned mechanisms considers MOQ in their models.
Traditionally, MOQ appears in the literature on the lot sizing,
economic order quantity, and batch ordering problems. A well-
studied example of the MOQ practice in the literature includes
the U.S. fashion skiwear manufacturer Sport Obermeyer Ltd.
(e.g., [24]–[26]). Recently, [27] explores the performance of
the retailer and the manufacturer under a QR system with dual
ordering flexibility and MOQ imposition only at an earlier
stage. The authors analytically derive the expected profits of
individual supply chain agents under different ranges of the
MOQ, as well as numerically show that MOQ, if set prop-
erly, may be able to enhance the supply chain’s efficiency.
However, how the whole supply chain performs and whether
coordination is feasible remain unknown. The first pieces of
analytical research exploring MOQ models appear in [28] and
[29]
1
, and extensions are reported only recently in [30]–[33].
Most of the aforementioned MOQ-related studies reveal that
the inventory policies (even for single-period problem) with
MOQ are complicated and one needs to resolve this challenging
inventory control problem by providing efficient numerical
heuristics. However, their models do not involve information
updating, and their analyses explore neither the impacts of
MOQ on the supply chain’s performance nor the coordination
issues under the system.
Based on the aforementioned literature works, we study in
this paper the impacts of MOQ on a two-echelon QR supply
chain. We start with deriving the optimal ordering policy under
such a system and then explore how MOQ affects the supply
chain’s efficiency. Afterward, we propose a dynamic form of
the MOQ that can obtain the supply chain’s maximum expected
profit while both channel members are Pareto improving. To
the best of our knowledge, this paper is the first one that
analytically addresses the coordination issue of QR supply
chains with the consideration of MOQ. We believe that our
paper contributes to the literature in the following ways. First,
our work is apparently the first piece of research work that
addresses the coordination issue of a QR supply chain with
MOQ consideration. Second, our analysis illustrates the pos-
1
We sincerely thank a knowledgeable anonymous reviewer for drawing to
our attention these important pioneering works.
TABLE I
T
ABLE OF COMPARISON BETWEEN THIS PAPER AND THE LITERATURE
sible significant inefficiency of such kind of supply chains
owing to the improper value of the fixed MOQ. Third, the
dynamic MOQ policy (DMP) that we propose is innovative yet
applicable in reality. Finally, our work may serve as a starting
point for other researchers to explore this underexplored yet
commonly found supply chain problem.
To provide a clear picture of this paper’s positioning in the
literature, we compare our work with other closely related
works in Table I.
The organization of the rest of this paper is as follows. We
start with formulating the model in Section II. Then, we derive
the optimal ordering policy and investigate numerically the
impacts of MOQ on individual channel members’ performance
and the supply chain’s efficiency in Sections III and IV, respec-
tively. Afterward, we propose a DMP that can achieve channel
coordination in Section V. Finally, we conclude this paper with
managerial insights and suggest topics for future research in
Section VI.
II. M
ODEL
We consider a two-echelon supply chain that comprises a
retailer and a manufacturer of a seasonal “newsvendor-type”
apparel product [34]. Traditionally, since the order lead time is
long, the retailer can only place its order once far in advance
before the season launches (we denote this time point by Stage
0). We call this system the old system. At Stage 0, the retailer
makes its ordering decision based on a preliminary estimate of
the demand. Now with the adoption of QR, the retailer also
has one ordering opportunity only as in the old system, but
it is allowed to postpone its ordering decision to a time point
much closer to the season launch (denoted by Stage 1). At
the same time, the manufacturer requires its order quantity to
be no smaller than the MOQ M
1
. We refer this system as the
QR-MOQ system in the remaining part of this paper. Between
Stage 0 and Stage 1, the retailer can adjust its demand forecast
based on the updated market situation (e.g., from the sales
performance of a closely related preseasonal product) so that
it can have a more accurate demand estimate for its ordering
decision at Stage 1.
Following the approach in [1] and [12], we describe the de-
mand uncertainty structure and the Bayesian updating process
as follows: we model the preliminary demand estimate of the
seasonal product at Stage 0 (x
0
) as normally distributed with
mean θ and variance δ, where θ is also uncertain and follows a
normal distribution with mean μ
0
and variance d
0
. Essentially,
such a formulation reflects the common practice that people
870 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 42, NO. 4, JULY 2012
normally have a better idea regarding the average demand (i.e.,
the distribution of θ). Still, the seasonal product’s real demand
will fluctuate even if we could be certain about the demand
mean. In the light of this, we employ δ to capture such inherent
demand uncertainty. With the previous discussion, the uncondi-
tional prior distribution of x
0
is therefore a normal distribution
with mean μ
0
and variance σ
2
0
=(d
0
+ δ):x
0
N (μ
0
2
0
).
Next, let ˆx
0
denote the observation that the retailer can make
about x
0
between Stage 0 and Stage 1. By Bayesian theory, the
updated distribution of θ is now θ N(μ
1
,d
1
), where μ
1
=
[d
0
/(d
0
+ δ)]ˆx
0
+[δ/(d
0
+ δ)]μ
0
, d
1
= d
0
δ/(d
0
+ δ), and the
unconditional distribution of μ
1
is μ
1
N (μ
0
2
μ
), where
σ
2
μ
= d
2
0
/(d
0
+ δ). As a result, the (conditional) posterior dis-
tribution of the predicted demand of the seasonal product at
Stage 1 (x
1
) is x
1
|ˆx
0
N (μ
1
2
1
), where σ
2
1
=(d
1
+ δ).No-
tice that we employ the Bayesian information updating model
for various reasons. First, this information updating model is
classical and popularly used in the literature (e.g., [35]–[40]).
Employing it thus allows us to reveal the impacts of MOQ by
directly comparing our new findings with the literature. Second,
this model allows us to generate more analytical insights,
especially related to the impacts brought by demand uncertainty
because we can investigate the related issues by looking at the
standard deviation of the demand distribution.
Similar to [12], we define the cost-revenue parameters as
follows: let r be the unit retail price and c
i
be the unit purchase
cost of the retailer at Stage i (i =0, 1). Denote the salvage
value and the holding cost per product leftover at the end of
the selling season by ν and τ, respectively. Hence, for each unit
of unsold product at the end of the season, it will incur a net cost
of h, where h = τ ν (note that h can be positive or negative).
For the manufacturer, let m
i
be the production cost at Stage i,
(i =0, 1). As a remark, all the supply contract terms, namely,
1) c
0
and c
1
—the unit purchase cost at Stage 0 and Stage 1, res-
pectively, and 2) M
1
—the Stage 1 MOQ, are announced as
early as Stage 0. For other notation adopted in this paper, f (·)
denotes the pdf of its corresponding argument, whereas φ(·) and
Φ(·), respectively, denote the pdf and cdf of the standard normal
distribution. Finally, define s
i
=(r c
i
)/(r + h), s
sc
i
=(r
m
i
)/(r + h),(i =0, 1), and ψ(a)=
a
(z a)φ(z)dz.
III. O
PTIMAL ORDERING DECISION
A. Under the Old System
Under the old system, the retailer’s expected profit (REP) at
Stage 0 for ordering Q
0
is
ER
0
(Q
0
)=r
Q
0
−∞
x
0
f(x
0
)dx
0
+
Q0
Q
0
f(x
0
)dx
0
h
Q
0
−∞
(Q
0
x
0
)f(x
0
)dx
0
c
0
Q
0
=(r + h)μ
0
(h + c
0
)Q
0
(r + h)σ
0
ψ [(Q
0
μ
0
)
0
] . (3.1)
Essentially, the ordering problem for the retailer under the old
system is the classical newsvendor problem. The optimal order
quantity (at Stage 0) and the corresponding retailer’s maximum
expected profit (at Stage 0) can be easily derived as in the
following, respectively:
Q
0,old
= μ
0
+ σ
0
Φ
1
(s
0
) (3.2)
ER
0,old
=(r c
0
)μ
0
(h + c
0
)σ
0
Φ
1
(s
0
)
(r + h)σ
0
ψ
Φ
1
(s
0
)
. (3.3)
Accordingly, the manufacturer’s maximum expected profit (at
Stage 0) under the old system is
EM
0,old
=(c
0
m
0
)Q
0,old
=(c
0
m
0
)
μ
0
+ σ
0
Φ
1
(s
0
)
. (3.4)
B. Under the QR-MOQ System
We start with considering the retailer’s optimal ordering
decision with QR but without MOQ. At Stage 1, after ˆx
0
is
observed and μ
1
is calculated, the REP (at Stage 1) for ordering
Q
1
is given by
ER
1
(Q
1
|μ
1
)=r
Q
1
−∞
x
1
f(x
1
)dx
1
+
Q
1
Q
1
f(x
1
)dx
1
h
Q
1
−∞
(Q
1
x
1
)f(x
1
)dx
1
c
1
Q
1
=(r + h)μ
1
(h + c
1
)Q
1
(r + h)σ
1
ψ [(Q
1
μ
1
)
1
] . (3.5)
It can be easily shown that ER
1
(Q
1
|μ
1
) is a strictly concave
function in Q
1
. Solving its first-order condition gives the opti-
mal order quantity at Stage 1
ˆ
Q
1
for the retailer:
ˆ
Q
1
= μ
1
+ σ
1
Φ
1
(s
1
) (3.6)
and the maximum expected profit that the retailer can obtain
(perceived at Stage 1) is given by
ER
1
= ER
1
(
ˆ
Q
1
|μ
1
)
=(r c
1
)μ
1
(h + c
1
)σ
1
Φ
1
(s
1
)
(r + h)σ
1
ψ
Φ
1
(s
1
)
. (3.7)
Taking expectation on μ
1
, the maximum expected profit of the
retailer at Stage 0 is then
ER
=(r c
1
)μ
0
(h + c
1
)σ
1
Φ
1
(s
1
)
(r + h)σ
1
ψ
Φ
1
(s
1
)
. (3.8)
Now under the QR-MOQ system, the retailer can order
at
ˆ
Q
1
if
ˆ
Q
1
fulfills the MOQ requirement, i.e.,
ˆ
Q
1
M
1
or equivalently μ
1
M
1
σ
1
Φ
1
(s
1
); otherwise, it should
either increase its order quantity up to the MOQ or give up
ordering. To make its decision, the retailer would first calculate
whether its expected profit by ordering up to the MOQ meets
its minimum profit target known as the reservation expected
profitJ
R
0.Thisreservation expected profit refers to the
minimum amount of expected profit with which the retailer will
CHOW et al.: IMPACTS OF MINIMUM ORDER QUANTITY ON A QUICK RESPONSE SUPPLY CHAIN 871
place an order. To avoid the trivial cases, we assume that J
R
is
smaller than ER
1
and ER
(or else the retailer would not order
even without MOQ). Define
ˆμ
1
=arg
μ
1
0
ER
1
(M
1
|μ
1
)=J
R
, where M
1
1
Φ
1
(s
1
)
.
(3.9)
Lemma 3.1 states some properties of ˆμ
1
that facilitate our
further discussion.
Lemma 3.1: 1) ER
1
(Q
1
|μ
1
) is an increasing function in μ
1
;
2) ˆμ
1
is unique; and 3) 0 < ˆμ
1
<M
1
σ
1
Φ
1
(s
1
).
Essentially, ˆμ
1
can be viewed as the threshold of the updated
demand mean for the retailer to decide at Stage 1 whether
ordering at the Stage 1 MOQ exceeds its reservation expected
profit. To be specific, if at Stage 1 the updated demand mean
μ
1
is smaller than ˆμ
1
, then by the increasing property of
ER
1
(Q
1
|μ
1
) in μ
1
, as well as the definition of ˆμ
1
, the retailer
would expect that its profit when ordering at M
1
would be
smaller than J
R
and vice versa.
Based on this, we can derive Proposition 3.1, which states
the optimal ordering policy for the retailer under the QR-MOQ
system.
Proposition 3.1: At Stage 1, the optimal ordering quantity
for the retailer under the QR-MOQ system is given by
Q
1
=
ˆ
Q
1
=μ
1
+σ
1
Φ
1
(s
1
) if μ
1
M
1
σ
1
Φ
1
(s
1
)
M
1
if ˆμ
1
1
<M
1
σ
1
Φ
1
(s
1
)
0 if μ
1
ˆμ
1
.
(3.10)
As shown in Proposition 3.1, the optimal ordering quantity re-
lies on the revised expected demand upon information updating
(i.e., μ
1
) and the size of the MOQ. It is more complicated than
the case without MOQ [1].
As a remark, when finalizing the contract terms with the
manufacturer at Stage 0, the retailer will choose not to accept
MOQ (which takes effect on the ordering at Stage 1) if M
1
is
too large (especially in the presence of J
R
). Let
Q
C
1
=arg
Q
1
>E[
ˆ
Q
1
]
{ER(Q
1
)=J
R
} (3.11)
where
ER(Q
1
)=(r+h)μ
0
(h+c
1
)Q
1
(r+h)σ
1
ψ [(Q
1
μ
0
)
1
]
(3.12)
is the profit that the retailer anticipates at Stage 0 when the order
quantity at Stage 1 is Q
1
and
E[
ˆ
Q
1
]=μ
0
+ σ
1
Φ
1
(s
1
) (3.13)
is the expected quantity at which ER(Q
1
) attains its maximum
(which is equal to ER
).
Q
C
1
can be viewed as the upper limit of the Stage 1 MOQ that
the retailer can accept during contract negotiation at Stage 0.
The rationale is as follows: notice that ER(Q
1
) is strictly con-
cave in Q
1
, with a global maximum attained at E[
ˆ
Q
1
]. There-
fore, ER(Q
1
) is decreasing for any Q
1
>E[
ˆ
Q
1
]. Together
with the definition of Q
C
1
in (3.11), we have ER(M
1
) <J
R
for
all M
1
>Q
C
1
(since Q
C
1
>E[
ˆ
Q
1
]). In other words, the retailer
would anticipate at Stage 0 that its profit would be less than
its reservation expected profit when it needs to order at Stage
1 at the MOQ that is greater than Q
C
1
. Thus, at Stage 0, the
manufacturer should offer an MOQ in the range 0 M
1
Q
C
1
for otherwise the retailer would reject the contract.
IV. I
MPACTS OF MOQ ON SUPPLY CHAIN PERFORMANCE
We explore in this section how MOQ affects the performance
of the supply chain and its agents. More importantly, we also
examine the issue of coordination under a QR supply chain in
the presence of MOQ.
Let EQ
QR-MOQ
(M
1
) and ER
QR-MOQ
(M
1
) be the retailer’s
optimal expected order quantity and the retailer’s maximum
expected profit (both perceived at Stage 0) under the QR-
MOQ system when the Stage 1 MOQ is M
1
, respectively.
Define EM
QR-MOQ
(M
1
) as the corresponding manufacturer’s
expected profit (MEP; perceived at Stage 0) when the retailer
expectedly orders EQ
QR-MOQ
(M
1
) at Stage 1 with M
1
as the
Stage 1 MOQ. Then, by Proposition 3.1, we have
EQ
QR-MOQ
(M
1
)
=
M
1
σ
1
Φ
1
(s
1
)
ˆμ
1
M
1
f(μ
1
)
1
+
M
1
σ
1
Φ
1
(s
1
)
μ
1
+σ
1
Φ
1
(s
1
)
f(μ
1
)
1
=M
1
[1Φ(ρ)]+σ
μ
ψ(ξ) (4.1)
where
ξ =
M
1
σ
1
Φ
1
(s
1
) μ
0
μ
, and
ρ =(ˆμ
1
μ
0
)
μ
. (4.2)
ER
QR-MOQ
(M
1
)=
M
1
σ
1
Φ
1
(s
1
)
ˆμ
1
ER
1
(M
1
|μ
1
)
× f(μ
1
)
1
+
M
1
σ
1
Φ
1
(s
1
)
ER
1
(
ˆ
Q
1
|μ
1
)
× f(μ
1
)
1
(4.3)
EM
QR-MOQ
(M
1
)=(c
1
m
1
)EQ
QR-MOQ
(M
1
)
=(c
1
m
1
){M
1
[1 Φ(ρ)]
+ σ
μ
ψ(ξ)}. (4.4)
A. Definition of Channel Coordination
There are a variety of definitions of the term “channel coor-
dination” in the supply chain management literature. Here, we
incorporate the notions of both [1] and [41] in our definition
of the concept. Specifically, a strategy is said to be able to
coordinate a supply chain if and only if 1) it can maximize the
supply chain’s expected profit and 2) it is Pareto improving [1]
for all the channel members.
872 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 42, NO. 4, JULY 2012
The first condition, namely, supply chain profit maximiza-
tion, implies that the sum of the expected profits of individ-
ual supply chain agents after adopting the strategy should be
equal to the maximum expected profit of the corresponding
centralized supply chain system. In other words, it requires the
decentralized supply chain adopting the strategy to perform as
well as the centralized one [41]. The centralized supply chain
refers to the benchmark system under which all the channel
members are coordinated centrally and the supply chain is
optimal with its expected profit maximized. In this paper, the
maximum expected profit of the centralized system can be
easily verified as
ECSC
=(r m
1
)μ
0
(h + m
1
)σ
1
Φ
1
(s
sc
1
)
(r + h)σ
1
ψ
Φ
1
(s
sc
1
)
. (4.5)
Therefore, to achieve supply chain profit maximization, we
need to have
ER
QR-MOQ
(M
1
)+EM
QR-MOQ
(M
1
)=ECSC
. (4.6)
The second condition, Pareto improvement, refers to the case
where all the channel members have their expected profits after
adopting the strategy no smaller than that when under the old
system, with at least one of them being strictly better off than
before. Now with the expected profits of the retailer and the
manufacturer under the old system given earlier in (3.3) and
(3.4), respectively, to achieve Pareto improvement, we need to
satisfy the following:
ER
QR-MOQ
(M
1
) ER
old
(4.7)
EM
QR-MOQ
(M
1
) EM
old
(4.8)
with at least one of these two inequalities being strict.
B. Effects of MOQ on Expected Profits of Channel Members
We start with studying the impact of MOQ on the retailer’s
performance.
Proposition 4.1: 1) ER
QR-MOQ
(M
1
) is a nonincreasing
function in M
1
, and 2) the optimal MOQ for the retailer,
denoted by M
R
1
, is given by 0 M
R
1
σ
1
Φ
1
(s
1
).
Proposition 4.1 confirms analytically the reason why the
retailer should not welcome MOQ because a larger MOQ
means a smaller (or nonincreasing) expected profit for it. More
specifically, it is most desirable for the retailer to have the MOQ
as small as possible such that it will not alter its original optimal
ordering decision.
Next, we attempt to explore the effect of MOQ on the manu-
facturer’s performance. Unfortunately, EM
QR-MOQ
is neither
a unimodal function nor a function with nice properties. To be
specific, its first derivative, which is given by
dEM
QR-MOQ
/dM
1
=(c
1
m
1
) {[Φ(ξ) Φ(ρ)]
M
1
φ(ρ) {1 s
1
/Φ[(M
1
ˆμ
1
)
1
]}
μ
} (4.9)
can take very different forms depending on the value of the
MOQ.
Fig. 1. REP function under the QR-MOQ system (r =37.5,MR= 10%).
Fig. 2. MEP function under the QR-MOQ system (r =37.5,MR= 10%).
As a result, we resort to numerical analysis to obtain some
more insights.
C. Numerical Analysis
To have a deeper understanding of the effect of MOQ on
the performance of the QR supply chain, we conducted nu-
merical analysis (including simulations) using the following
parameters: h =0.625, m
0
=10.87, c
0
=12.5, μ
0
=35, d
0
=
100, and δ =25. To further explore whether the cost-revenue
parameters would also have an impact on the QR-MOQ system,
we employed the following sets: r = {75, 37.5}, production
cost ratio: MR (m
1
m
0
)/m
0
= {−10%, 0%, 10%}, and
purchase cost ratio: CR (c
1
c
0
)/c
0
= {−10%, 0%, 10%}
(as a remark, those cases with c
1
<m
1
were excluded from our
experiments as, in reality, the manufacturer would never offer
such terms). For simplicity, we set J
R
=0.
1) Impacts of MOQ on Channel Members’ Performance and
Supply Chain Efficiency: Figs. 1 and 2 show an example of the
appearance of the REP and MEP function under the QR-MOQ
system, respectively. As proved in Proposition 4.1, ER
QR-MOQ
is a nonincreasing function in M
1
. In contrast, EM
QR-MOQ
is
nearly flat at small values of M
1
, then increases until reaching
a maximum, and decreases to zero for sufficiently large M
1
.
However, the wave-look form of dEM
QR-MOQ
/dM
1
against
M
1
(as shown in Fig. 3) suggests that the function is not
concave. Intuitively, the manufacturer may want to impose an
MOQ as large as possible so that it can have a greater guarantee
of the income. However, at the same time, the retailer would
give up ordering anything if the MOQ is too large. This may
explain the crest-like form of the function of EM
QR-MOQ
in
the range of large M
1
and also the existence of an optimal MOQ
that maximizes MEP. We hereafter denote it by M
M
1
.
We proceed to investigate the supply chain’s performance at
M
M
1
as depicted in Table II. Comparing with the case when
there is no MOQ, it seems that the manufacturer enjoys a
CHOW et al.: IMPACTS OF MINIMUM ORDER QUANTITY ON A QUICK RESPONSE SUPPLY CHAIN 873
TABLE II
(a) S
UPPLY CHAIN PERFORMANCE UNDER THE QR-MOQ SYSTEM (M
1
= M
M
1
AND r =75).
(b) S
UPPLY CHAIN PERFORMANCE UNDER THE QR-MOQ SYSTEM (M
1
= M
M
1
AND r =37.5)
874 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 42, NO. 4, JULY 2012
Fig. 3. Derivative of MEP (dEM
QR-MOQ
/dM
1
) under the QR-MOQ sys-
tem (r =37.5,MR= 10%).
significant increase in its expected profit (over 170% increase
for r =75and over 50% increase for r =37.5), whereas the
retailer suffers a drastic fall in its expected profit (over 65%
decrease for r =75and over 54% decrease for r =37.5). From
the supply chain’s perspective, the supply chain’s efficiency is
very low when compared with the corresponding centralized
supply chain (less than 55% in most cases for r =75while
less than 70% for r =37.5). Imagine the situation where the
manufacturer has dominant bargaining power, it would defi-
nitely adopt the QR-MOQ system with the required MOQ set
at M
M
1
to maximize its expected profit. However, this strategy
would reduce the expected profits of the retailer and the whole
supply chain. In other words, imposition of carefully set MOQ
appears to be an effective tool for the manufacturer to offset
the adverse effect of profit reduction owing to QR adoption (as
the strategy normally results in smaller order quantity placed
by the retailer). By contrast, MOQ imposition may significantly
counter-balance the merits of QR for the retailer as it hinders
its ordering flexibility. From the perspective of the whole sup-
ply chain, improper MOQ may be deemed as an undesirable
measure since it may significantly reduce the supply chain’s
efficiency. Therefore, it is of paramount importance for all the
channel members to collaborate on setting a proper MOQ.
Our numerical results show that M
M
1
is the same for the
same r and c
1
. This seems to suggest that the determination of
M
M
1
is related to the retailer’s cost-revenue parameters only.
More interestingly, we find that, for the same r, the REP at
M
M
1
is always the same regardless of the production cost or the
purchase cost [here, ER
QR-MOQ
(M
M
1
) = 670.5 for r =75
and ER
QR-MOQ
(M
M
1
) = 338.0 for r =37.5]. Further inves-
tigation shows that ˆμ
1
at M
1
= M
M
1
is the same for all the
values of r and c
1
under study (and is equal to 27.05 under
our numerical settings). Recall that the ordering decision of the
retailer at Stage 1 solely depends on comparing the revealed
μ
1
with ˆμ
1
and M
1
σ
1
Φ
1
(s
1
). Therefore, REP is the same
when we have the same ˆμ
1
and M
1
. Our numerical findings
suggest that the determination of M
M
1
(and, in turn, ˆμ
1
)may
have some interesting relationship with ER
QR-MOQ
(M
M
1
),
which is independent of m
1
and c
1
.
2) Impact of MOQ on Variance of Profits: To explore the
effects of MOQ on the variance of profits (which are re-
lated to risk analysis issues; see [42] and [43]) of the chan-
nel members, we simulated 1000 instances of the pairs of
the posterior demand mean and realized demand {μ
1
,x
1
}.
The 95% confidence intervals for the simulated μ
1
and x
1
are 35.41 ± 1.96 9.11 and 35.20 ± 1.96 11.22, respectively.
For each pair of {μ
1
,x
1
}, we determine the corresponding
Stage 1 optimal order quantity under the QR system and the
QR-MOQ system and then calculate the corresponding profits
of the channel members and the whole supply chain. For the
QR-MOQ system, we consider two different values of M
1
,
namely, E[
ˆ
Q
1
]=μ
0
+ σ
1
Φ
1
(s
1
) and M
M
1
.Afterward,we
calculate the mean and the standard deviation of these measures
for these 1000 instances. Table III summarizes the simulated
order quantities under the QR system and QR-MOQ system for
different cost-revenue parameters. We observe that the average
order quantity under the QR-MOQ system with M
1
= E[
ˆ
Q
1
]
is slightly greater than that under the QR system (Q
1
=
ˆ
Q
1
),
but the standard deviation of the order quantity of the former is
smaller than the latter. This demonstrates that MOQ can help
the manufacturer to secure a more stable order quantity, which
is useful for the purchase of raw materials. When M
1
= M
M
1
,
although the average order quantity increases considerably
from that of the case without MOQ, the variance of the order
quantity also increases dramatically. As a result, one can antic-
ipate greater fluctuations in the profits of the channel members.
From Tables IV and V, one can compare the mean and the
standard deviation of the profits of the channel members under
different systems. Our simulation results verify our previous
discussion regarding the impact of MOQ on the REP and MEP.
Specifically, the average profit of the retailer drops while that
of the manufacturer increases with a suitably chosen MOQ.
This seems to suggest that MOQ is a good measure for the
manufacturer. However, if we take the risk aspect into consider-
ation, our simulation results show that the standard deviation of
profits would also increase for both channel members, and such
increase is significant in particular for M
1
= M
M
1
. In other
words, the manufacturer should be more careful in setting the
MOQ if it is more risk concerned.
3) Feasibility of Pareto Improvement Under the QR-MOQ
System: The next issue that we explore is the possibility of
achieving Pareto improvement for both channel members with
MOQ.
Tables VI summarize the range of M
1
within which individ-
ual agents can be better off under the QR-MOQ system and
whether Pareto improvement is feasible in different settings of
the cost-revenue parameters. We observe that the manufacturer
has a wider range of M
1
that can make itself better off than
the retailer in the same cost-revenue parameter settings. On
the other hand, this range changes more vigorously in the
light of changes in the manufacturing cost ratio (i.e., MR).
Comparatively, the range of M
1
that can make the retailer better
off is more stable, and the retailer can always find a range of
M
1
that can make it better off for all the set of cost-revenue
parameters presented here.
When taking both parties’ interests into consideration, we
find that few cases can achieve Pareto improvement. It is
particularly difficult to achieve Pareto improvement at moderate
price levels (i.e., r =37.5) and high production cost ratios (e.g.,
MR = 10%).
CHOW et al.: IMPACTS OF MINIMUM ORDER QUANTITY ON A QUICK RESPONSE SUPPLY CHAIN 875
TABLE III
M
EAN AND STANDARD DEVIATION OF ORDER QUANTITY UNDER DIFFERENT SYSTEMS BASED ON SIMULATED DEMANDS
TABLE IV
M
EAN AND STANDARD DEVIATION OF PROFITS FOR THE RETAILER UNDER DIFFERENT SYSTEMS BASED ON SIMULATED DEMANDS
V. D YNAMIC MOQ POLICY (DMP)
The findings of the numerical analysis suggest that MOQ can
significantly worsen the efficiency of a QR supply chain as well
as the retailer’s performance (even for the manufacturer, it may
not be beneficial to use MOQ all the time). A possible reason
may be due to the opposing effects between MOQ and QR on
the retailer’s ordering decision. To be specific, QR enhances the
retailer’s ordering flexibility by making use of the information
updating capability; however, MOQ limits such ordering flexi-
bility. In other words, MOQ hinders the realization of the full
benefit of the information updating capability provided by QR.
In view of the aforementioned findings, we propose a DMP,
in which the specific MOQ value depends on the updated
information, i.e., μ
1
.
2
To be specific, we define
˜
M
1
= μ
1
+ σ
1
Φ
1
(s
sc
1
) . (5.1)
2
This contract is an innovative one. We believe that it can be implemented
in practice for the case where the manufacturer and the retailer are working
closely together with information sharing measures such as forecast sharing
and the popular collaborative planning, forecasting, and replenishment (CPFR)
scheme.
As a remark,
˜
M
1
is actually the optimal supply chain quantity
with respect to the centralized system at Stage 1. However,
one should note that the resulting supply chain expected profit
perceived at Stage 1 (and subsequently at Stage 0) with its
imposition is different from that of imposing the fixed MOQ
of E(
˜
M
1
)=μ
0
+ σ
1
Φ
1
(s
sc
1
). Specifically, imposing E(
˜
M
1
)
does not necessarily provide the maximum supply chain profit
as expected at Stage 1 because the updated demand mean μ
1
may possibly deviate from its expected value μ
0
. Therefore,
despite being predetermined at Stage 0, the dynamic nature
of
˜
M
1
(or specifically its dependence on the updated demand
mean μ
1
at Stage 1) provides some flexibility for the retailer in
the ordering constraint.
We have Proposition 5.1, which shows that, by imposing
an MOQ of a size of
˜
M
1
, the supply chain’s expected profit
(SCEP) is equal to the centralized benchmark.
Proposition 5.1: The sum of the REP and the MEP under
DMP is the same as the centralized supply chain’s expected
profit ECSC
as defined in (4.5).
It remains to show that DMP can also lead to Pareto im-
provement for both the retailer and the manufacturer. Before
876 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 42, NO. 4, JULY 2012
TABLE V
M
EAN AND STANDARD DEVIATION OF PROFITS FOR THE MANUFACTURER UNDER DIFFERENT SYSTEMS BASED ON SIMULATED DEMANDS
doing so, define (5.2) and (5.3), shown at the bottom of the
page. Then, Lemma 5.1 states some necessary and sufficient
conditions for achieving Pareto improvement.
Lemma 5.1: With DMP, 1) the manufacturer is not worse
off under QR with MOQ if and only if c
1
c
M
1
, and 2) the
retailer is not worse off under QR with MOQ if and only if
c
1
c
R
1
.
Lemma 5.1 provides guidelines on the situation in which
the manufacturer and the retailer will not be worse off. This,
together with Proposition 5.1, can help derive how DMP can
achieve coordination. Define
g(m
1
)=(m
1
m
0
)μ
0
+(h + m
1
)σ
1
Φ
1
(s
sc
1
)
(h + m
0
)σ
0
Φ
1
(s
0
)+(r + h)
σ
1
ψ
Φ
1
(s
sc
1
)
σ
0
ψ
Φ
1
(s
0
)

. (5.4)
Proposition 5.2 gives a necessary and sufficient condition for
employing DMP to coordinate the QR supply chain.
Proposition 5.2: Channel coordination can be achieved by
DMP with
˜
M
1
(μ
1
)=μ
1
+ σ
1
Φ
1
(s
sc
1
) for any c
M
1
<c
1
<c
R
1
if and only if m
1
∈{m
1
: g(m
1
) < 0}.
VI. C
ONCLUSION
In this paper, we have investigated the impacts of MOQ on
a QR system. We have shown that the REP under such system
is nonincreasing in the MOQ, while the MEP function does not
possess nice analytical properties. We have found from our nu-
merical analysis that the manufacturer can obtain a substantial
gain in its expected profit by imposing its optimal MOQ, yet
this measure would deteriorate REP significantly as well as the
overall supply chain’s efficiency. Moreover, MOQ may not be
c
M
1
= m
1
+(c
0
m
0
)
μ
0
+ σ
0
Φ
1
(s
0
)
/
μ
0
+ σ
0
Φ
1
(s
sc
1
)
(5.2)
c
R
1
=
c
0
μ
0
+ σ
0
Φ
1
(s
0
)
+ h
σ
0
Φ
1
(s
0
) σ
1
Φ
1
(s
sc
1
)
+(r + h)
σ
0
ψ
Φ
1
(s
0
)
σ
1
ψ
Φ
1
(s
sc
1
)

[μ
0
+ σ
1
Φ
1
(s
sc
1
)]
(5.3)
CHOW et al.: IMPACTS OF MINIMUM ORDER QUANTITY ON A QUICK RESPONSE SUPPLY CHAIN 877
TABLE VI
(a) R
ANGE OF MOQ FOR PARETO IMPROVEMENT UNDER THE QR-MOQ SYSTEM(r = 75).
(b) R
ANGE OF MOQ FOR PARETO IMPROVEMENT UNDER THE QR-MOQ SYSTEM(r =37.5)
too helpful in achieving Pareto improvement for the channel
members. We have speculated that the static nature of the prea-
greed MOQ may hinder the full use of the information updating
capability provided by QR. To rectify this shortcoming, we have
proposed the use of a DMP, in which the value of MOQ depends
on the updated demand information. We have proven analyti-
cally that such a DMP can achieve channel coordination in the
sense that the supply chain can attain the maximum expected
profit with Pareto improvement enjoyed by supply chain agents.
We believe that this innovative contract can be implemented in
practice for the case where the manufacturer and the retailer
are working closely together with information sharing measures
such as forecast sharing and the popular CPFR scheme.
In this paper, we have modeled demand as normally
distributed. Being one of the most commonly known
probability distributions, normal distribution has been widely
used in the literature, which includes both recent and classical
papers published in leading journals (e.g., [1], [17], and [39]). It
also helps us explore some important issues such as the impacts
brought by demand uncertainty. We have to admit that some
of our findings cannot be treated as a conclusive generalization
for the integrated effects of QR and MOQ. However, being
apparently the first piece of research work on the coordination
issue of the commonly found QR-MOQ system (e.g., in the
apparel industry), this paper makes another contribution that
we provide some scientific evidence that QR adoption with an
improper MOQ can be a very disastrous practice to the retailer
and the supply chain. We believe that our work may arouse the
interests of other researchers and may provide some starting
point for them to further investigate the related issues.
We have also examined by a mean-variance approach the
risk issue associated with MOQ imposition under QR through
numerical analysis. It appears that different optimal decisions
will result depending on the risk attitudes of the channel
members. If all of the channel members are risk neutral and
are concerned only about expected profit maximization, then
we have shown that our proposed DMP is an effective tool to
coordinate the supply chain. For future research directions, one
promising area is to study a QR supply chain with a price-
setting retailer [43], [44]. This extension is an interesting and
challenging one because the resulting supply chain will be
much more difficult to coordinate and the imposition of MOQ
878 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 42, NO. 4, JULY 2012
will affect both the optimal ordering quantity and the optimal
retail price [45]. In addition, exploring the case with risk-
averse supply chain agents [46]–[48] is another challenging and
important future research direction.
A
PPENDIX
(A1) ALL PROOFS
Proof of Lemma 3.1: 1) By direct manipulation, we have
dER
1
(M
1
|μ
1
)/dμ
1
=(r+h)Φ[(Q
1
μ
1
)
1
]>0,soER
1
(M
1
|
μ
1
) is a strictly increasing function in μ
1
. 2) and 3) Let
η(μ
1
)=ER
1
(M
1
|μ
1
)J
R
. In other words, we have ημ
1
)=0.
By 1), η(μ
1
) is also a strictly increasing function in μ
1
,soˆμ
1
is unique. Now, lim
μ
1
0
η(μ
1
)=(h+c
1
)M
1
(r+h)σ
1
ψ(M
1
/
σ
1
)J
R
<0, and η[M
1
σ
1
Φ
1
(s
1
)]=ER
1
J
R
>0.Bythe
definition of ˆμ
1
and the strictly increasing property of η(μ
1
),
we have 0 < ˆμ
1
<M
1
σ
1
Φ
1
(s
1
). (Q.E.D.)
Proof of Proposition 3.1: As discussed in the text, the re-
tailer can order at its optimal order quantity
ˆ
Q
1
for μ
1
M
1
σ
1
Φ
1
(s
1
). Now, with the monotonic property of ER
1
(M
1
|μ
1
)
and the definition of ˆμ
1
,wehaveER
1
(M
1
|μ
1
)J
R
for ˆμ
1
μ
1
<M
1
σ
1
Φ
1
(s
1
), so the retailer is willing to order up to
M
1
in this case. If 0μ
1
< ˆμ
1
, then ER
1
(M
1
|μ
1
)<J
R
; there-
fore, the retailer will not order anything in this case. (Q.E.D.)
Proof of Proposition 4.1: 1) For 0 M
1
σ
1
Φ
1
(s
1
),
MOQ would have no effect on the retailer’s ordering decision
[as μ
1
0 M
1
σ
1
Φ
1
(s
1
)]. Thus, the expected profit it
anticipates at Stage 0 within this range of M
1
is always equal
to ER(
ˆ
Q
1
).ForM
1
1
Φ
1
(s
1
), according to (4.3), the REP
(anticipated at Stage 0) with the MOQ being M
1
is given by
ER
QR-MOQ
(M
1
)=
M
1
σ
1
Φ
1
(s
1
)
ˆμ
1
ER
1
(M
1
|μ
1
)f(μ
1
)
1
+
M
1
σ
1
Φ
1
(s
1
)
ER
1
(
ˆ
Q
1
|μ
1
)f(μ
1
)
1
.
Differentiating ER
QR-MOQ
(M
1
) w.r.t. M
1
gives
dER
QR-MOQ
/dM
1
=(r c
1
)[Φ(ξ) Φ(ρ)] (r + h)
×
ξ
ρ
Φ[(M
1
μ
0
σ
μ
z)
1
] φ(z)dz
where
ρ =(ˆμ
1
μ
0
)
μ
ξ =
M
1
σ
1
Φ
1
(s
1
) μ
0
μ
.
For z (−∞),wehaveΦ[(M
1
μ
0
σ
μ
z)
1
] >s
1
.
Therefore
ξ
ρ
Φ[(M
1
μ
0
σ
μ
z)
1
] φ(z)dz > s
1
[Φ(ξ) Φ(ρ)] .
Then
dER
QR-MOQ
/dM
1
< (r c
1
)[Φ(ξ) Φ(ρ)]
(r c
1
)[Φ(ξ) Φ(ρ)] = 0.
In other words, ER
QR-MOQ
is strictly decreasing in M
1
for
M
1
1
Φ
1
(s
1
). Hence, we have the nonincreasing property
for the retailer’s expected function under the QR-MOQ system.
2) The retailer would want to have M
1
as small as possible
owing to the result in 1). On the other hand, notice that, for 0
M
1
σ
1
Φ
1
(s
1
), the retailer can always order at its optimal
quantity (because μ
1
>M
1
σ
1
Φ
1
(s
1
) for all μ
1
0 within
this range of M
1
). Hence, it is most desirable for the retailer to
have 0 M
R
1
σ
1
Φ
1
(s
1
). (Q.E.D.)
Proof of Proposition 5.1: At Stage 1, REP and MEP with
˜
M
1
are, respectively, given by
ER
1
(
˜
M
1
|μ
1
)=(r c
1
)μ
1
(h + c
1
)σ
1
Φ
1
(s
sc
1
)
(r + h)σ
1
ψ
Φ
1
(s
sc
1
)
EM
1
(
˜
M
1
|μ
1
)=(c
1
m
1
)
μ
1
+ σ
1
Φ
1
(s
sc
1
)
.
Correspondingly, their expected profits anticipated at Stage 0
are, respectively
ER(
˜
M
1
)=(r c
1
)μ
0
(h + c
1
)σ
1
Φ
1
(s
sc
1
)
(r + h)σ
1
ψ
Φ
1
(s
sc
1
)
EM(
˜
M
1
)=(c
1
m
1
)
μ
0
+ σ
1
Φ
1
(s
sc
1
)
.
Summation of the two equations yields the result. (Q.E.D.)
Proof of Lemma 5.1: 1) MEP under the old system and DMP
are, respectively
EM
Old
=(c
0
m
0
)
μ
0
+ σ
1
Φ
1
(s
0
)
EM
DMP
=(c
1
m
1
)
μ
0
+ σ
1
Φ
1
(s
sc
1
)
.
The difference in MEP after adopting DMP is given by
ΔEM = EM
DMP
EM
old
=(c
1
m
1
)
μ
0
+ σ
1
Φ
1
(s
sc
1
)
(c
0
m
0
)
μ
0
+ σ
1
Φ
1
(s
0
)
.
After some manipulation, one can show that ΔEM 0
c
1
c
M
1
. 2) The difference in REP after adopting DMP is
given by
ΔER = ER
DMP
ER
old
= c
0
μ
0
+ σ
0
Φ
1
(s
0
)
c
1
μ
0
+ σ
1
Φ
1
(s
sc
1
)
+ h
σ
0
Φ
1
(s
0
) σ
1
Φ
1
(s
sc
1
)
(r + h)
σ
0
ψ
Φ
1
(s
0
)
σ
1
ψ
Φ
1
(s
sc
1
)

.
Then, with some manipulation, we have ΔER0c
1
c
R
1
.
(Q.E.D.)
Proof of Proposition 5.2: For Pareto improvement to be
feasible under DMP, we need to have c
M
1
<c
R
1
. By direct ma-
nipulation, c
R
1
c
M
1
> 0 g(m
1
) < 0, and thus, Proposition
5.2 is resulted. (Q.E.D.)
A
CKNOWLEDGMENT
The authors would like to thank the chief editor, the associate
editor, and the anonymous reviewers for their constructive
comments which led to a substantial improvement of this paper.
CHOW et al.: IMPACTS OF MINIMUM ORDER QUANTITY ON A QUICK RESPONSE SUPPLY CHAIN 879
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Authors’ photographs and biographies not available at the time of
publication.
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