Content uploaded by Qing Li
Author content
All content in this area was uploaded by Qing Li on Dec 22, 2020
Content may be subject to copyright.
Received: 8 October 2019 Revised: 14 July 2020 Accepted: 10 October 2020
DOI: 10.1002/nav.21959
RESEARCH ARTICLE
Multiseason production planning under export quotas
Tianxiao Chen1Xiting Gong2Qing Li3He Xu4
1Department of Systems Engineering and
Engineering Management, The Chinese University
of Hong Kong, Shatin, Hong Kong
2Department of Decision Sciences and Managerial
Economics, CUHK Business School, The Chinese
University of Hong Kong, Shatin, Hong Kong
3School of Business and Management, Hong Kong
University of Science and Technology, Kowloon,
Hong Kong
4School of Management, Huazhong University of
Science and Technology, Wuhan, China
Correspondence
Qing Li, School of Business and Management,
Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong.
Email: imqli@ust.hk
Funding information
Hong Kong Research Grants Council, Grant/Award
Number: CUHK14200718. National Natural
Sciences Foundation of China, Grant/Award
Numbers: 71931005, NSFC71871099.
Abstract
An export quota is a direct restriction on the quantity of certain goods that can be
exported and is an important instrument in international trade policy. In this study,
we focus on an exporting firm and examine the implication of export quotas on its
production and inventory policies. In particular, the firm applies for an exporting
license every season (eg, year) and the amount it can sell is unknown for certainty in
advance. The firm must determine its production quantity every period (eg, month)
under a production capacity constraint. The presence of quotas decouples inventory
from sales: the firm can sell its goods if and only if the quota permits, and this
changes the way it manages its production and inventory. We show that the firm
in general benefits from stockpiling—produce up to an inventory level higher than
current quota. Stockpiling happens if and only if the remaining quota for the current
season is smaller than a threshold value. It is more likely to happen when it is closer
to the end of a season or when the quotas in future seasons are larger. Although the
inventory in excess of current quota cannot be used to fill the demand in the current
season, they will be needed in future seasons when new quotas are released and
the firm cannot increase production quickly enough due to capacity constraints. Our
numerical studies show that the cost of limiting production only to current quota can
be substantial.
KEYWORDS
dynamic programming, export quotas, finite capacity, inventory control, optimal
policy, stockpiling
1INTRODUCTION
An export quota is a direct restriction on the quantity of
certain goods that can be exported and is an important instru-
ment in international trade policy. For example, In the early
1980s, there were quotas that limited the Japanese automo-
bile export to the United States. In 2005, the U.S. government
and the government of China signed a memorandum of under-
standing, under which China agreed to establish quotas on its
exports of various types of clothing and textiles to the United
States. The quota for socks in 2006, for example, was 772.8
million pairs (Krugman & Obstfeld, 2009). Other examples
include sugar imports in the United States, beef exports from
Australia to Indonesia, and export restrictions on raw materi-
als for steel production applied by China, India, Russia, and
Ukraine (Krugman & Obstfeld, 2009; Price & Nance, 2010).
Export quotas may be imposed by a country proactively to
protect domestic industries from shortage of raw materials,
to protect advanced technologies and rare resources, and to
protect local population from deficit of foods and essential
goods. Quotas may also be imposed at the request of the
importer and are agreed to by the exporter to forestall other
trade restrictions.
For exporting firms, trade policies are often very important
factors, if not the most important factors, which define the
ways they manage their operations (Dong & Kouvelis, 2020).
Decisions related to, for example, plant location, sourcing,
distribution and logistics, and production and inventory poli-
cies are critically driven by trade policies. While there is a
lot of discussion about quotas in international economics,
there is very little analytical work about its implication on the
operations of exporting firms. In this study, we focus on an
Naval Res Logistics 2020;1–16 wileyonlinelibrary.com/journal/nav © 2020 Wiley Periodicals LLC 1
2CHEN ET AL.
exporting firm and examine the implication of export quotas
on its production and inventory policies. In particular, the firm
applies for an exporting license every season (eg, year) and
the amount it can sell is unknown for certainty in advance.
The firm must determine its production quantity every period
(eg, month) under a production capacity constraint. The pres-
ence of quotas decouples inventory from sales: the firm can
sell its goods if and only if the quota permits, and this changes
the way it manages its production and inventory.
We model the problem as a discrete-time dynamic program.
The total sale in a season is constrained by a known quota,
but the quotas for future seasons are unknown in advance.
Each season consists of multiple production planning peri-
ods. In each period, the firm decides a production quantity
subject to a capacity constraint. Demand in a period is ran-
dom and may be strongly seasonal. Quotas complicate the
planning problems substantially. The dynamic program now
requires a two-dimensional state space: inventory on hand and
remaining quota in a season, which makes analysis and com-
putation more challenging. The key question is whether the
firm should produce more than the remaining quota, given
that sale in the current season is limited by quota, and if so,
when and why. We show that indeed the firm in general bene-
fits from stockpiling—produce up to an inventory level higher
than current quota. Stockpiling happens if and only if the cur-
rent quota is smaller than a threshold value. It is more likely
to happen when it is closer to the end of a season or when
the quotas in future seasons are larger. Although the inven-
tory in excess of current quota cannot be used to fill demand
in the current season, they will be needed in future seasons
when new quotas are released and the firm cannot increase
production quickly enough due to capacity constraints. Tech-
nically, the mathematical property behind the stockpiling
phenomenon is L#-concavity, well known in the inventory
control literature. We provide a practical application of the
property. Through extensive numerical studies, we find that
the value of stockpiling can be very significant, especially
when the firm’s production capacity is moderate, demand is
seasonal with peak demand occurring at the beginning of a
season when a new quota is allocated.
The study will benefit exporting firms which operate under
quotas. The model we have developed can serve as a deci-
sion tool for planning their production and inventory. Our
advice that firms should stockpile—produce and hold inven-
tories in excess of current quota—is particularly worth noting,
as we demonstrate that the cost of limiting production only to
current quota can be substantial. The study may also benefit
policy makers, who either determine the allocation of quo-
tas among firms or negotiate trade terms with foreign trading
partners. We quantify the operational value of maintaining a
consistent and predictable allocation of quotas and the oper-
ating cost that export quotas have on firms. Although these
are by no means the only things that policy makers take into
account, they form the basis for informed and data-driven
policymaking.
The rest of this paper is organized as follows. Section 2
reviews the related literature. In Sections 3 and 4, we present
the model and characterize the structure of the optimal pol-
icy, respectively. In Section 5, we numerically demonstrate the
value of stockpiling and the cost of imposing quotas on the
firm. We conclude the paper in Section 6. All the proofs are
contained in the Appendix A. There is also an Data S1 (Sup-
porting information) which contains several supplementary
tables for Section 5.
2LITERATURE REVIEW
Our work is built on the important literature on stochastic
inventory control that considers a limited production capac-
ity in each period. There has been a lot of progress since the
seminal work by Federgruen and Zipkin (1986). They study
a single-item, periodic-review backlogging inventory model
with independent and identically distributed demands, and a
finite production capacity in each period. They show that a
modified base-stock policy is optimal under a discounted-cost
criterion, the optimal produce-up-to level is nonincreasing
in the production capacity, and it becomes smaller as it
approaches to the end of the planning horizon. Their basic
model has been extended in different directions. For example,
Kapu´
sci´
nski and Sridhar (1998) allow demands to follow a
cyclic pattern. Shaoxiang (2004) extends it to a two-product
system and shows that a hedging point policy is optimal.
Van Mieghem and Rudi (2002) focus on newsvendor net-
works, but also discuss multiperiod settings. In Angelus and
Porteus (2002), capacity can be adjusted up and down at
a cost. In Ozer and Wei (2004), inventory decisions are
made under advance demand information. Ceryan, Sahin, and
Duenyas (2013) consider a joint inventory control and pricing
problem of two substitutable products. Gong and Chao (2013)
study the optimal policy of a capacitated periodic-review
inventory system with remanufacturing.
In all the papers mentioned above, there is a constraint
on production or order quantity in each period and the con-
straint stems from the production capacity of the firm or the
suppliers. The constraint can also be due to reasons other
than production capacity and it may be imposed on the total
quantity in multiple periods. For example, Benjaafar, Chen,
and Wang (2017) study a capacitated production-inventory
system where the total production output quantity over the
planning horizon is upper bounded, which can be due to limit
on the use of scarce natural resources as input or on the
amount of waste or harmful pollution generated by produc-
tion as output. In Gong and Zhou (2013), the limit is on the
amount of emission allowances, and the firm has the options
of trading emission allowances and choosing different pro-
duction technologies, besides planning production. Due to
contractual arrangements with suppliers, the total production
or order quantity over a number of periods might be lower
bounded, as opposed to upper bounded, by a total minimum
CHEN ET AL. 3
order quantity. This stream of research was started by Bassok
and Anupindi (1997) under the assumption of independent
and identically distributed demands. Chen and Krass (2001)
extend their work to allow for nonstationary demands and a
different unit ordering cost for the orders beyond the com-
mitment quantity. Yuan, Chua, Liu, and Chen (2015) study
a lot-sizing inventory system under a similar contract and
partially characterize the optimal inventory policy. Recently,
Wang, Gong, and Zhou (2017) study an inventory system with
a total minimum order quantity constraint and two supply
options. They also consider the system’s asymptotic behavior
with respect to the constraint.
In all the papers mentioned above, the constraints are all on
the production or order quantity. However, the export quotas
that we focus on impose constraints on sales.
3THE MODEL
Consider an exporting firm producing a single product to sat-
isfy random demands in an overseas market over a planning
horizon of Nseasons, indexed by n=1, …,N.Eachofthe
seasons corresponds to one compliance period (eg, one quar-
ter or year). At the beginning of each season n,thefirmis
allocated with an export quota Qn, which caps the firm’s total
exporting quantity in this season. For n=1, …,N, we assume
that Qnis a random variable with a finite mean before season
n, and it is realized at the beginning of season n. For simplic-
ity, we do not consider the updating of the distribution of Qn
as the time proceeds. We assume that the quotas cannot be
traded, and unused quota at the end of each season expires
and cannot be carried to future seasons, but extensions are
possible and will be discussed in the end. For each season,
there are Tplanning periods (each of which may represent,
for example, 1 week), indexed by t=1, …,T. In each period
tof season n, the firm has a finite production capacity k,and
the production lead time is assumed to be zero. We assume
that the product is nonperishable.
The demand in the overseas market in period tof season
n, denoted by Dn
t, is a random variable with a finite mean.
Denote Fn,t(⋅) as the cumulative distribution function of the
random demand Dn
t. We assume that the demands in different
periods and seasons are independent but can have different
distributions. In our context, a unit of demand in period tof
season ncan be satisfied by one unit of on-hand inventory
only when there is one unit of remaining export quota in that
period. Unmet demand in each period is lost. For simplicity,
we assume zero lost-sales penalty but all of our results can be
easily extended to allow a positive lost-sales penalty. The unit
production cost is cand the unit selling price is p,withp>c.
Leftover inventory in each period is carried over to the next
period with a unit holding cost h. There is also a one-period
discount factor 𝛼,with0<𝛼≤1. The goal of the firm is to
maximize its expected total discounted profit over the whole
planning horizon.
The sequence of events in each period tof each season nis
as follows. First of all, the firm observes the on-hand inven-
tory xand the remaining quota qof season n.Notethatif
t=1, then the remaining quota qequals the newly allocated
quota Qnfor season n. Next, the firm decides on the produc-
tion quantity for this period and raises the after-production
inventory level to y. Since the firm has a finite capacity
k, the constraints x≤y≤x+kmust be satisfied. After that,
the random demand Dn
tis realized and fulfilled as much as
possible, subject to the availability of both the on-hand inven-
tory and the remaining quota. Therefore, the sales quantity
is given by min {y,q,Dn
t}. Consequently, the leftover inven-
tory is y−min {y,q,Dn
t}, and the updated remaining quota
is q−min {y,q,Dn
t}. Finally, the revenue and costs for this
period are calculated.
We next formulate the firm’s optimization problem as a
dynamic program. From the above description, the system
state in each period tof season ncan be represented by (x,
q), where xand qare the starting on-hand inventory level
and remaining quota in that period, respectively. Let Vn
t(x,q)
denote the maximum expected total discounted profit from
period tof season nonwards given the system state (x,q).
Then, the dynamic program can be formulated as: for t=1,
…,Tand n=1, …,N,
Vn
t(x,q)= max
x≤y≤x+k{−c(y−x)+pE[min {y,q,Dn
t}]
−hE[y−min {y,q,Dn
t}]
+𝛼E[Vn
t+1(y−min {y,q,Dn
t},
q−min {y,q,Dn
t})]}.(1)
Note that for n=1, …,N−1, period T+1 of season nis the
same as period 1 of season n+1, and the unused quota at the
end of each season expires. Then, the boundary condition for
each season nis given by
Vn
T+1(x,q)=E[Vn+1
1(x,Qn+1)],
0,
if n=1,…,N−1;
if n=N.
Let yn
t(x,q)be the optimal solution to the optimization
problem (1). That is, yn
t(x,q)is the optimal produce-up-to
level in period tof season nif the firm’s on-hand inventory
level is xand the remaining quota for this season is q.Inthe
next section, we will study the properties of the optimiza-
tion problem (1) and characterize the structure of the firm’s
optimal policy yn
t(x,q).
4STRUCTURE OF THE OPTIMAL POLICY
For convenience, we define
Jn
t(y,q)=−(c+h)y+(p+h)E[min {y,q,Dn
t}]
+𝛼E[Vn
t+1(y−min {y,q,Dn
t},q−min {y,q,Dn
t})].(2)
Then, the optimality Equation (1) can be rewritten as
Vn
t(x,q)=cx +max
x≤y≤x+kJn
t(y,q).(3)
4CHEN ET AL.
For any given q,letyn∗
t(q)denote the ideal inventory level,
that is, yn∗
t(q)=arg max y≥0Jn
t(y,q). The following propo-
sition presents some basic properties of the value functions,
which will facilitate our subsequent analysis.
Proposition 1 For n =1, …,Nandt=1,
…,T,
(i) Vn
t(x,q)and Jn
t(y,q)are increasing in q;
(ii) Vn
t(x−𝜖, q−𝜖)+p𝜖and Jn
t(y−𝜖, q−𝜖)+
(p−c)𝜖are increasing in ϵ.
Proposition 1(i) shows that the firm benefits from a higher
remaining quota in the current season because a higher quota
relaxes the constraint on sales. Having an additional unit of
inventory and quota simultaneously may allow the firm to
generate at most one more unit of sales. So, its benefit is upper
bounded by p, the selling price. Both results are intuitive.
The analysis of structural properties of the optimal pol-
icy relies on the concept of L#-concavity. Let V⊆Rnbe a
polyhedron {v∈Rnai⋅v≥bi,i=1, 2, …,m}thatmeets
the following requirements: each entry of aiis either 0, 1,
or −1, and there can be at most two nonzero entries. When
there are two nonzero entries, they have opposite signs. Let
W⊆Rbeaconvexclosedsetandedenote an n-dimensional
vector of 1s. We say that a function f:V→Ris L#-convex
(L#-concave)if𝜓(x,y)=f(x−ye) is submodular (super-
modular) on S={(x,y)∈Rn×Ry∈W,x−ye∈V}. We
adopt the definition from Li and Yu (2014). There are sev-
eral variations in the literature (Zipkin, 2008). L#-concavity
is a stronger notion of complementarity than supermodular-
ity (Li & Yu, 2014) and has a wide range of applications
(see Chen, 2017 for a review). In what follows, we show that
inventory and remaining quota are economic complements.
That is, the marginal value of inventory is higher if there
is more quota because the more the quota, the more likely
the inventory can be turned into sales. A direct consequence
of L#-concavity in inventory and quota is that the optimal
produce-up-to level is increasing in quota, but the increase in
the optimal produce-up-to level is no greater than the increase
in quota.
Theorem 1 For n =1, …,Nandt=1, …,
T,
(i) Vn
t(x,q)and Jn
t(y,q)are L#-concave func-
tions;
(ii) 0≤yn∗
t(q+𝜖)−yn∗
t(q)≤𝜖for any ϵ≥0.
Because of the concavity of the objective function, the
optimal produce-up-to level is
yn
t(x,q)=
x,if yn∗
t(q)≤x;
yn∗
t(q),if yn∗
t(q)∈(x,x+k];
x+k,if yn∗
t(q)>x+k.
Obviously, Theorem 1(ii) also holds for yn
t(x,q).
4.1 The optimal policy and the stockpiling
phenomenon
As the focus of our study is on the quotas, we are interested
in how the optimal production quantity changes as the quota
in the current season and quotas in future seasons change.
We are particularly interested in when, if at all, the firm pro-
duces up to an inventory level above the current quota. To
facilitate our analysis, we first consider the following auxil-
iary inventory system in which the sale in the current season
is not constrained by quota as a benchmark:
̂
Vn
t(x)=cx +max
x≤y≤x+k
̂
Jn
t(y)(5)
for t=1, …,T;n=1, …,N,where
̂
Jn
t(y)=−(c+h)y+E[(p+h)min {y,Dn
t}
+𝛼̂
Vn
t+1(y−min {y,Dn
t})].(6)
The boundary condition for each season is given by
̂
Vn
T+1(x)=E[Vn+1
1(x,Qn+1)];
̂
VN
T+1(x)=0,
where n<N.Let̂yn
t(x)be the optimal solution to (5) and let
̂yn∗
t=arg max y≥0̂
Jn
t(y). The following lemma compares the
system with a finite quota in the current season and that with
an infinite quota.
Lemma 1 For n =1, …,Nandt=1, …,T,
(i) Vn
t(x,q)≤̂
Vn
t(x)and Jn
t(y,q)≤̂
Jn
t(y);
(ii) Jn
t(y,q)=̂
Jn
t(y)when q ≥y+
T
i=t+1min {̂yn∗
i,k}.
Lemma 1 (i) shows that relaxing the quota will benefit the
firm, which is expected. In part (ii), min {̂yn∗
i,k}is the opti-
mal produce-up-to level in period iin the system without
quota constraint. Therefore, y+T
i=t+1min {̂yn∗
i,k}represents
the total supply for the rest of the current season. If it is
smaller than the remaining quota, then it is the supply, not the
quota, that limits the sale, and in that case, the ideal inventory
level no longer depends on the remaining quota. Theorem 2
reinforces the same idea by comparing the ideal inventory
levels.
Theorem 2 For n =1, …,Nandt=1, …,
T, yn∗
t(q)=̂yn∗
twhen q ≥Un
t,where
Un
t=̂yn∗
t+
T
i=t+1
min {̂yn∗
i,k}.
Theorems 1 and 2 lead naturally to the following result.
Theorem 3 For n =1, …,Nandt=1, …,
T, there exists a finite quota level, qn
t, such that
yn∗
t(q)≥q if and only if q ≤qn
t.
Theorem 3 shows that when the remaining quota is small
enough, the ideal optimal produce-up-to level is greater than
CHEN ET AL. 5
FIGURE 1 Stockpiling region [Colour figure can be viewed at
wileyonlinelibrary.com]
the quota. The inventory exceeding the remaining quota will
not generate any sales in the current season; instead, it will
be used in future seasons when new quotas become avail-
able. The theorem implies that the firm may produce a strictly
positive amount and raise the inventory level higher than
the remaining quota. We call this phenomenon “stockpil-
ing.” Mathematically, stockpiling happens when yn
t(x,q)>
max (x,q), or equivalently, when the system state (x,q) falls
into the following region {(x,q)∈R2
+q<qn
t,yn∗
t(q)>
x,x+k>q}(Figure 1). Stockpiling may happen even if the
remaining quota in the current season is zero, in which case
any inventory produced in the current season is entirely for
the future seasons. The following theorem provides a simple
expression of yn∗
t(0).
Theorem 4 For n =1, …,Nandt=1, …,
T, yn∗
t(0)=(Sn
t−(T−t)k)+,where
Sn
t=arg max
y≥0{−[c+(1+𝛼+···+𝛼T−t)h]y
+𝛼T−t+1E[Vn+1
1(y,Qn+1)]}.
and, Sn
1≤Sn
2≤···≤Sn
T.
Theorem 4 can be explained as follows. When the quota
is zero for the current season, the firm is unable to sell any
product in this season and the sole purpose of stocking inven-
tory in period tis for future seasons. Note that the marginal
cost of producing the product in period tand holding it until
the end of the season is c+(1 +𝛼+···+𝛼T−t)h. Therefore,
the quantity Sn
t, which is defined in (7), is the ideal inven-
tory level at the end of season nif the firm produces in period
t. In addition, producing and stocking for future seasons in
periods t+1, …,Tis cheaper than doing that in period t.
Consequently, the firm would produce in period tonly if it
also produces with full capacity kin all the subsequent peri-
ods t+1, …,T. As a result, the ideal inventory level yn∗
t(0)
in period tequals (Sn
t−(T−t)k)+.
As we mentioned earlier, stockpiling may happen even
when qis zero and since yn∗
t(0)is increasing in t,foragivenx
and k, stockpiling is more likely to happen when it is closer to
the end of the season. In the inventory literature, a well-known
phenomenon is called horizon effect: more inventory should
be stocked if the remaining planning horizon is longer. This is
because a longer horizon allows one more time to collect pro-
ceeds and as such, more inventory is needed (DeGroot, 2004;
Heyman & Sobel, 1984; Li, Xu, & Zheng, 2008). In our set-
ting, this result no longer holds because the horizon effect and
the stockpiling effect may drive the inventory level toward
opposite directions. However, when the quota for the current
season is zero, the horizon effect disappears and the behavior
of yn∗
tis driven solely by the stockpiling effect.
Theorem 4 implies that if yn∗
t+1(0)≤k,thenyn∗
t(0)=0. It
also implies that if yn∗
t+1(0)>k,thenyn∗
t(0)≤yn∗
t+1(0)−k.
Since there is no possibility of sales in the current season,
production should be postponed to later periods in the season
as much as possible to avoid holding cost and the loss in dis-
counting. Stockpiling happens, but not always. The following
theorem identifies conditions under which it does not happen.
Theorem 5 For t =1, …,T,
(i) yN∗
t(0)=SN
t=0,and0≤yN∗
t(q)≤q;
(ii) For any n, if yn+1,∗
1(Qn+1)≤kforanyreal-
ization of Qn+1,theny
n∗
t(q)=qforq ≤Ln
t,
where
Ln
t=
min F−1
n,t(1−𝛼)(p−c)
(1−𝛼)p+h,Ln
t+1,
if n =1,…,N,t=1,…,T−1;
min F−1
n,tp−c
p+h−𝛼c,yn+1,∗
1(Qn+1),
if n =1,…,N−1,t=T;
F−1
n,tp−c
p+h,if n =N,t=T,
(7)
and yn∗
t(q)≤qforq≥Ln
t.
Because of (4), it is easy to see from part (i) of the above
theorem that yN
t(x,q)cannot be strictly greater than max(x,
q). So stockpiling does not happen in the last season. From
part (ii) of the above theorem, we can see that stockpiling will
not occur when there is no capacity constraint. Furthermore,
because yn+1,∗
1(Qn+1)is increasing in Qn+1, the condition that
yn+1,∗
1(Qn+1)≤kfor any realization of Qn+1is met if Qn+1
is almost surely sufficiently low. Therefore, stockpiling does
not happen when the random initial quota allocated for the
next season is almost surely sufficiently low. This is expected
because when the quota allocated for the next season is almost
surely low, the sales in the next season will be capped by the
quota and having too much inventory has no value.
5NUMERICAL STUDY
Our numerical study consists of three parts. First, we describe
in detail an illustrative example to shed further light on
the structure of the optimal policy. Second, we explore the
6CHEN ET AL.
(A) Season 1 (B) Season 2
FIGURE 2 Optimal policy of the illustrative example
impact of quotas and capacity and their interaction. Third, we
quantify the value of stockpiling.
5.1 An illustrative example
Here we consider an example with two seasons and four
periods in each season. The parameters are as follows. Sell-
ing price p=8, production cost c=4, holding cost h=1,
production capacity k=20, quotas of the two seasons
(Q1,Q2)=(150,150), and the discount factor 𝛼=0.95. The
demands follow discrete uniform distributions on [10, 50]
for all periods in both seasons and are independent across
periods.
Figure 2a shows the unconstrained optimal produce-up-to
level y1∗
t(q)as a function of qin season 1. The x-axis repre-
sents the remaining quota q,andthey-axis represents y1∗
t(q).
We use curves with different line types to distinguish between
the policies y1∗
t(q)in different periods. The solid straight line
represents y=q. For all periods, the slopes of y1∗
t(q)are
between 0 and 1, consistent with Theorem 1. Also for all
periods, y1∗
t(q)no longer changes with qwhen the remain-
ing quota is large enough. This is consistent with Theorem 2.
The stockpiling phenomenon appears only in period 4, the
last period of this season. Figure 2b shows y2∗
t(q)in season
2, the last season in this example. There is no stockpiling
phenomenon here, which confirms Theorem 5.
5.2 Impact of quotas and capacity
In this subsection, we carry out numerical studies to investi-
gate the impact of imposing quotas on the firm’s profit and
the interaction between quotas and capacity.
5.2.1 Quantifying the cost of imposing quotas
The model parameters here are the same as what we used
in the illustrative example unless stated otherwise. In all
cases, the initial inventory is set as zero. For Q1≥0and
Q2≥0, we define V(Q1,Q2) as the optimal expected total dis-
counted profit with Q1units of quota for season 1 and Q2
units of quota for season 2. Both Q1and Q2are determinis-
tic. Similarly, we define V(+∞,+∞) as the optimal expected
total discounted profit without quota constraints. Obviously,
V(+∞,+∞)≥V(Q1,Q2). To quantify the cost of imposing
quotas (Q1,Q2), we define
𝜖(Q1,Q2)= V(+∞,+∞) − V(Q1,Q2)
V(+∞,+∞) ×100%,
as the percentage profit loss due to the quotas. A larger value
of ϵ(Q1,Q2) indicates a higher cost to the firm.
Table 1 reports the values of ϵ(Q1,Q2) under different val-
ues of quotas (Q1,Q2) and capacity level k.Fromthistable,we
can see that the cost decreases when either Q1or Q2increases,
but in all cases, when the quota is large enough, the cost will
not decrease further. This is expected. For the same quota
pair, the cost of imposing quotas is larger when the capacity
kis larger. For example, ϵ(10, 10) equals 73% when k=10
and equals 85.7% when k=20. This indicates that allocating
small quotas to a firm with a large capacity will lead to a large
profit loss, implying that quotas and capacity are economic
complements.
5.2.2 Interaction between quotas and capacity
We now formally investigate the interaction between quotas
and production capacity. To generate sales, the firm needs the
production capacity to produce the goods and the quotas to
sell the goods. The formal way to examining the economic
relation of two variables is to examine how the marginal profit
with respect to one variable changes when the other variable
changes. To this end, for Q1≥0andQ2≥0, we define
ΔV(Q1,Q2)=V(Q1+1,Q2)−V(Q1,Q2)
as the marginal value of the first season’s quota Q1on
the firm’s optimal profit, where V(Q1,Q2) is defined in
Section 5.2.1. Table 2 reports the values of Δ(Q1,Q2) under
different values of quotas (Q1,Q2) and capacity level k.From
CHEN ET AL. 7
TABLE 1 ϵ(Q1,Q2) under different values of (Q1,Q2)andk
k=10 Q2
Q110 30 50 70 90 110 130 150
10 73% 50.6% 32.6% 30.3% 30.3% 30.3% 30.3% 30.3%
30 45.5% 23.1% 5.1% 5.1% 5.1% 5.1% 5.1% 5.1%
50 32.8% 10.4% 0% 0% 0% 0% 0% 0%
70 32.8% 10.4% 0% 0% 0% 0% 0% 0%
90 32.8% 10.4% 0% 0% 0% 0% 0% 0%
110 32.8% 10.4% 0% 0% 0% 0% 0% 0%
130 32.8% 10.4% 0% 0% 0% 0% 0% 0%
150 32.8% 10.4% 0% 0% 0% 0% 0% 0%
k=20 Q2
Q110 30 50 70 90 110 130 150
10 85.7% 73.5% 62% 51.5% 45.7% 44.8% 44.8% 44.8%
30 70.8% 58.6% 47.1% 36.5% 30.7% 29.9% 29.9% 29.9%
50 56.6% 44.5% 33% 22.4% 16.6% 15.8% 15.8% 15.8%
70 43.6% 31.3% 19.8% 9.3% 3.5% 3.4% 3.4% 3.4%
90 38.7% 26.2% 14.7% 4.2% 0.1% 0% 0% 0%
110 38.7% 26.2% 14.7% 4.2% 0.1% 0% 0% 0%
130 38.7% 26.2% 14.7% 4.2% 0.1% 0% 0% 0%
150 38.7% 26.2% 14.7% 4.2% 0.1% 0% 0% 0%
k=30 Q2
Q110 30 50 70 90 110 130 150
10 88.6% 78.8% 69.6% 61% 54.2% 50.8% 50.5% 50.5%
30 76.6% 66.8% 57.6% 49% 42.2% 38.8% 38.5% 38.5%
50 65.3% 55.5% 46.3% 37.7% 30.9% 27.5% 27.2% 27.2%
70 54.6% 44.9% 35.7% 27.1% 20.2% 16.8% 16.5% 16.5%
90 45.9% 35.9% 26.7% 18.1% 11.2% 7.7% 7.4% 7.4%
110 41.1% 30.2% 20.9% 12.3% 5.5% 1.8% 1.3% 1.3%
130 40.6% 29% 19.8% 11.2% 4.3% 0.6% 0% 0%
150 40.6% 40.6% 19.8% 11.2% 4.3% 0.6% 0% 0%
TABLE 2 ΔV(Q1,Q2) under different values of (Q1,Q2)andk
k=10 k=20 k=30 k=40 k=50 k=60
ΔV(10, 50) 3.800 3.966 3.966 3.966 3.966 3.966
ΔV(30, 50) 1.519 3.687 3.687 3.687 3.687 3.687
ΔV(50, 50) 0 3.485 3.490 3.490 3.490 3.490
ΔV(70, 50) 03.014 3.202 3.202 3.202 3.202
ΔV(90, 50) 0 0 2.419 2.431 2.431 2.431
ΔV(110,50) 0 0 1.069 1.380 1.380 1.380
ΔV(130,50) 0 0 0 0.462 0.463 0.463
ΔV(150,50) 0000.055 0.060 0.060
ΔV(10,100) 3.800 3.966 3.966 3.966 3.966 3.966
ΔV(30,100) 1.519 3.687 3.687 3.687 3.687 3.687
ΔV(50,100) 0 3.469 3.490 3.490 3.490 3.490
ΔV(70,100) 02.298 3.205 3.202 3.202 3.202
ΔV(90,100) 0 0 2.434 2.432 2.432 2.432
ΔV(110,100) 0 0 1.090 1.383 1.384 1.384
ΔV(130,100) 0 0 0 0.466 0.467 0.467
ΔV(150,100) 0000.057 0.062 0.062
8CHEN ET AL.
this table, we observe that ΔV(Q1,Q2) is (weakly) increas-
ing in k, confirming that quotas and production capacity are
economic complements.
5.3 Quantifying the value of stockpiling
The most important feature of the optimal policy is the
stockpiling phenomenon. In this section, we quantify the sig-
nificance of this phenomenon. To this end, we compare the
optimal profit and the profit under the policy of never produc-
ing more than the current quota. For t=1, …,Tand n=1,
…,N,define
Vn
t(x,q)=
max
y≤q,x≤y≤x+kJn
t(x,q,y),if x<q;
Jn
t(x,q,x),if x≥q.
where
Jn
t(x,q,y)=−c(y−x)+E[pmin {y,q,Dn
t}]
−E[h(y−min {y,q,Dt})]
+𝛼E[Vn
t+1(y−min {y,q,Dn
t},
q−min {y,q,Dn
t})],
and the boundary condition for each season nis given by
Vn
T+1(x,q)=Vn+1
1(x,Qn+1);
VN
T+1(x,q)=0.
Here, the produce-up-to level yin each period is upper
bounded by the current quota qwhen the initial inventory
level xis below qand equals x(ie, zero production) oth-
erwise. We consider examples with four seasons and four
periods in each season. The selling price p=8, production
cost c=4, and the discount factor 𝛼=0.95. The demand is
uniformly distributed on [10, 50] for all periods in all sea-
sons and is independent across periods. Let V(Q1,Q2,Q3,Q4)
be the optimal profit of the above system when the expected
quota for the ith seasons is Qi,i=1, …,4.Itisobvi-
ous that V(Q1,Q2,Q3,Q4)≤V(Q1,Q2,Q3,Q4).Weuse
𝜀(Q1,Q2,Q3,Q4) to measure the value of stockpiling, that is,
𝜖(Q1,Q2,Q3,Q4)= V(Q1,Q2,Q3,Q4)−V(Q1,Q2,Q3,Q4)
V(Q1,Q2,Q3,Q4)
×100%.
The higher the value of ϵ(Q1,Q2,Q3,Q4), the more significant
the stockpiling phenomenon.
5.3.1 Without demand seasonality
We first examine the situation when there is no demand sea-
sonality. The second season’s quota is random and equals to
its mean plus or minus 20 with equal probability. The quo-
tas in other seasons are deterministic. We investigate how ϵ
changes with k,h, and the expected quotas. The numerical
results are summarized in Table 3.
In Figure 3, we plot a few cases reported in Table 3. In
Figure 3a, we compare ϵunder different holding costs and
different capacities. In Figure 3b, we compare ϵunder differ-
ent quotas and different capacities. When the holding cost is
small, ϵis first increasing and then decreasing in capacity k.
This implies that the stockpiling phenomenon is most signif-
icant when kis moderate. When the capacity is very small,
the firm does not have the ability to produce extra products
for future seasons; that is, the firm cannot stockpile inventory
aggressively even if doing so can increase profit. When the
capacity is very large, there is no incentive to stockpile inven-
tory, because the firm has enough capacity available in the
future when needed.
In Figure 3a, we compare ϵunder different holding costs.
The lower the holding cost, the higher the ratio. Stockpiling
comes with a cost and its benefit is the highest when the hold-
ing cost is zero. In Figure 3b, we keep the holding cost h
unchanged but vary the quotas. The stockpiling phenomenon
is more significant when Q1is smaller but Q2is larger. A
small Q1and a large Q2create strong incentive for the firm
to produce for the second season, which is what stockpiling is
all about.
5.3.2 With demand seasonality
Many exporting firms such as those in textile industries face
strongly seasonal demand. We model demand seasonality by
using two different demand patterns. In all the numerical stud-
ies in this subsection, the unit holding cost his set to be
0.5.
Demand pattern I:Dn
t=(1−𝛽)̃
Dn
tfor t=1,2,3, and
Dn
4=(1+3𝛽)̃
Dn
4,wherẽ
Dn
tfor all tand nare independent
and follow the same discrete uniform distribution on [10, 50].
Here, 𝛽∈[0, 1] is the seasonality factor. When 𝛽increases,
the mean demand of Dn
tfor t=1,2,3 becomes smaller, the
mean demand of Dn
4larger and hence the demand more sea-
sonal. The numerical results are summarized in Tables S1-S5
in Data S1. In Figure 4, we plot a few cases reported in Table
S1.
Demand pattern II:Dn
1=(1+3𝛽)̃
Dn
1and Dn
t=(1−
𝛽)̃
Dn
tfor t=2,3,4. The rest is the same as in the previous
case. The difference between the two demand patterns is that
in the first demand pattern, the peak is in the last period of a
season, while in the second the peak is in the first period. The
numerical results are summarized in Tables S6 to S10 in Data
S1. We report some of the results in Figure 5.
It is interesting to compare the impact of stockpiling under
the two demand patterns. First, for almost all cases, the value
of stockpiling is higher when the high demand period is at
the beginning of a season (ie, the second demand pattern).
Under the second demand pattern, the high demand arrives
at the time when a new quota is allocated. With a high initial
quota, the firm can meet a high demand in the first period.
During the subsequent low demand periods (periods 2, 3, and
4), both the remaining quota and demand are low, and the firm
CHEN ET AL. 9
TABLE 3 ϵ(Q1,Q2,Q3,Q4) under different values of kand h
h=0
(Q1,Q2,Q3,Q4)k=10 k=15 k=20 k=25 k=30 k=35 k=40
(20, 140, 20, 140) 22.5% 27.0% 19.7% 11.2% 5.2% 1.9% 0.4%
(30, 130, 30, 130) 11.5% 21.2% 18.0% 10.4% 4.9% 1.6% 0.3%
(40, 120, 40, 120) 0% 14.6% 13.2% 4.5% 1.4% 0.3% 0%
(50, 110, 50, 110) 0% 7.6% 13.5% 7.9% 2.7% 0.5% 0%
(60, 100, 60, 100) 0% 0.1% 9.9% 5.6% 1.1% 0.2% 0%
h=0.5
(Q1,Q2,Q3,Q4)k=10 k=15 k=20 k=25 k=30 k=35 k=40
(20, 140, 20, 140) 17.5% 16.0% 10.2% 5.0% 0.9% 0.1% 0%
(30, 130, 30, 130) 9.8% 14.3% 9.2% 3.9% 0.6% 0% 0%
(40, 120, 40, 120) 0% 11.0% 8.4% 4.3% 1.2% 0.1% 0%
(50, 110, 50, 110) 0% 6.2% 7.7% 3.7% 0.9% 0% 0%
(60, 100, 60, 100) 0% 0.1% 7.0% 2.8% 0.2% 0% 0%
h=1
(Q1,Q2,Q3,Q4)k=10 k=15 k=20 k=25 k=30 k=35 k=40
(20, 140, 20, 140) 11.8% 9.1% 5.6% 1.6% 0.1% 0% 0%
(30, 130, 30, 130) 8.1% 8.0% 5.1% 1.5% 0.1% 0% 0%
(40, 120, 40, 120) 0% 7.2% 4.6% 1.4% 0.1% 0% 0%
(50, 110, 50, 110) 0% 4.8% 3.5% 0.5% 0% 0% 0%
(60, 100, 60, 100) 0% 0.1% 3.8% 0.9% 0% 0% 0%
h=1.5
(Q1,Q2,Q3,Q4)k=10 k=15 k=20 k=25 k=30 k=35 k=40
(20, 140, 20, 140) 7.3% 5.9% 2.2% 0.2% 0% 0% 0%
(30, 130, 30, 130) 6.3% 5.2% 1.9% 0.2% 0% 0% 0%
(40, 120, 40, 120) 0% 4.7% 1.8% 0.1% 0% 0% 0%
(50, 110, 50, 110) 0% 3.3% 1.6% 0.1% 0% 0% 0%
(60, 100, 60, 100) 0% 0.1% 1.5% 0% 0% 0% 0%
capacity k
0
5%
10%
15%
20%
25%
30%
h = 0
h = 0.5
h = 1
h = 1.5
(A) (Q1,Q
2,Q
3,Q
4) = (20 , 140, 20, 140), varying kand h
capacity k
10 15 20 25 30 35 40 10 15 20 25 30 35 40
1,Q2,Q3,Q4)
0
5%
10%
15%
20%
25%
30%
(Q1,Q2,Q3,Q 4) = (20,140,20,140)
(Q1,Q2,Q3,Q 4) = (30,130,30,130)
(Q1,Q2,Q3,Q 4) = (40,120,40,120)
(Q1,Q2,Q3,Q 4) = (50,110,50,110)
(Q1,Q2,Q3,Q 4) = (60,100,60,100)
(B) h=0,varying k,Q1,Q2,Q3and Q4
FIGURE 3 Value of stockpiling: without demand seasonality
will benefit from aggressively stockpiling inventory in antic-
ipation of a new quota and demand in the next season. Under
the first demand pattern, however, a new quota is allocated at
the beginning of a season but the high demand period is at the
end. During the low demand periods (ie, periods 1, 2, and 3),
the quota at the beginning of each period is high, and hence
there is little need to raise the inventory level higher than
the quota (ie, stockpiling). Second, when the demand season-
ality increases, the value of stockpiling decreases under the
first demand pattern, but increases under the second demand
10 CHEN ET AL.
0
2%
4%
6%
8%
10%
12%
14%
16%
18%
k = 10
k = 20
k = 30
k = 40
(A) (Q1,Q
2,Q
3,Q
4) = (20 , 140, 20, 140)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2%
4%
6%
8%
10%
12%
14%
16%
18%
k = 10
k = 20
k = 30
k = 40
(B) (Q1,Q
2,Q
3,Q
4) = (30 , 130, 30, 130)
FIGURE 4 Value of stockpiling: with seasonal demand pattern I
0
5%
10%
15%
20%
25%
30%
25%
40%
k = 10
k = 20
k = 30
k = 40
(A) ( Q1,Q
2,Q
3,Q
4) = (20 , 140, 20, 140)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5%
10%
15%
20%
25%
30%
25%
40%
k = 10
k = 20
k = 30
k = 40
(B) ( Q1,Q
2,Q
3,Q
4) = (30 , 130, 30, 130)
FIGURE 5 Value of stockpiling: with seasonal demand pattern II
pattern. The value of stockpiling is high when high demand
and high quota are synchronized. An increase in the season-
ality factor makes demand and quota less synchronized under
the first demand pattern, but more synchronized under the
second demand pattern. Overall, the value of stockpiling is
highest when capacity is moderate, demand is seasonal and
peak demand arrives at the beginning of a season when a new
quota is allocated.
6CONCLUDING REMARKS
One important assumption in this paper is that unmet demand
in each period is lost. If unmet demand is backlogged, then the
dynamic program formulation requires three state variables,
namely inventory, remaining quota, and backlogged demand,
and the analysis is different and more complex. However, the
firm’s incentive to stockpile remains. While our research is
motivated by export quotas, our analysis and results are appli-
cable to other settings where sales are limited by quotas. One
example is tobacco industry. In many countries, the sales of
tobacco products are regulated. Tobacco manufacturers can
sell their products only when they have been allocated tobacco
codes, and the number of codes are determined by regulators
every season based on a host of factors.
Our model can be easily extended in several dimensions.
First, the unused quota in a season can be carried over to the
next season. Second, the firm may sell to both a domestic mar-
ket and an overseas market and there is no quota limit in the
domestic market. The firm may be operating under a Tariff
CHEN ET AL.11
Rate Quota and it pays a higher tariff if the sale exceeds the
quota. Or, there may be a domestic sale requirement under
which a certain percentage of production output must be sold
in domestic market (Devadoss, 2009). Third, the firm may
actively manage its capacity and adjust its capacity up or down
depending on available quota. The analysis for these exten-
sions is similar. Overall, given the importance of international
trade, we believe that inventory and production management
in exporting firms is a research area that has not been well
studied. We hope that our work will stimulate more interest.
ACKNOWLEDGMENTS
The authors would like to thank the department editor, Xin
Chen, the anonymous associate editor and three referees for
their constructive comments and suggestions, which helped
significantly improve both the content and the exposition
of this paper. The second author was partly supported by a
grant from the Hong Kong Research Grants Council’s General
Research Fund (Grant CUHK14200718). The last author was
partly supported by grants from National Natural Sciences
Foundation of China (Grants NSFC71871099 and 71931005).
ORCID
Xiting Gong https://orcid.org/0000-0002-0941-2218
Qing Li https://orcid.org/0000-0001-8698-014X
REFERENCES
Angelus, A., & Porteus, E. L. (2002). Simultaneous capacity and produc-
tion management of short-life-cycle, produce-to-stock goods under
stochastic demand. Management Science,48(3), 399–413.
Bassok, Y., & Anupindi, R. (1997). Analysis of supply contracts with
total minimum commitment. IIE Transactions,29(5), 373–381.
Benjaafar, S., Chen, D., & Wang, R. (2017). Managing
production-inventory systems with scarce resources. Manufacturing
& Service Operations Management,19(2), 216–229.
Ceryan, O., Sahin, O., & Duenyas, I. (2013). Dynamic pricing of substi-
tutable products in the presence of capacity flexibility. Manufactur-
ing & Service Operations Management,15(1), 86–101.
Chen, F. Y., & Krass, D. (2001). Analysis of supply contracts with mini-
mum total order quantity commitments and non-stationary demands.
European Journal of Operational Research,131(2), 309–323.
Chen, X. (2017). L-Natural-convexity and its applications. Frontiers of
Engineering,4(3), 283–294.
DeGroot, M. H. (2004). Optimal statistical decisions, Wiley Classics
Library Edition. New York: John Wiley & Sons Inc.
Devadoss, S. (2009). Domestic sale requirement, price supports, export
quota, and inefficiencies. The Journal of International Trade and
Economic Development,18(2), 297–309.
Dong, L., & Kouvelis, P. (2020). Impact of tariffs on global supply chain
network configuration: Models, predictions, and future research.
Manufacturing & Service Operations Management,22(1), 25–35.
Federgruen, A., & Zipkin, P. (1986). An inventory model with limited
production capacity and uncertain demands. II. The discounted-cost
criterion. Mathematics of Operations Research,11(2),
208–215.
Gong, X., & Chao, X. (2013). Optimal control policy for capaci-
tated inventory systems with remanufacturing. Operations Research,
61(3), 603–611.
Gong, X., & Zhou, S. X. (2013). Optimal production planning with
emissions trading. Operations Research,61(4), 908–924.
Heyman, D., & Sobel, M. (1984). Stochastic models in operations
research (Vol. II). Mineola, NY: Dover Publications.
Kapu´
sci´
nski, R., & Sridhar, T. (1998). A capacitated production-
inventory model with periodic demand. Operations Research,46(6),
899–911.
Krugman, P. R., & Obstfeld, M. (2009). International economics: The-
ory and policy. In The Addison-Wesley Series in Economics (8th
ed.). Boston, MA: Addison-Wesley.
Li, Q., Xu, H., & Zheng, S. (2008). Periodic-review inventory sys-
tems with random yield and demand: Bounds and heuristics. IIE
Transactions,40(4), 434–444.
Li, Q., & Yu, P. (2014). Multimodularity and its applications in three
stochastic dynamic inventory problems. Manufacturing & Service
Operations Management,16(3), 455–463.
Ozer, O., & Wei, W. (2004). Inventory control with limited capac-
ity and advance demand information. Operations Research,52(6),
988–1000.
Porteus, E. L. (2002). Foundations of stochastic inventory theory. Stan-
ford, CA: Stanford University Press.
Price, A. H., & Nance, D. S. (2010). Export barriers and the steel indus-
try. In Chapter 3. The Economic Impact of Export Restrictions on
Raw Materials. Paris: OECD Publishing.
Shaoxiang, C. (2004). The optimality of hedging point policies for
stochastic two-product flexible manufacturing systems. Operations
Research,52(2), 312–322.
Topkis, D. (1998). Supermodularity and Complementarity. Princeton
University Press, Princeton, New Jersey.
Van Mieghem, J. A., & Rudi, N. (2002). Newsvendor newworks:
Inventory management and capacity investment with discretionary
activities. Manfuacturing & Service Operations Management,4(4),
313–335.
Wang, T., Gong, X., & Zhou, S. X. (2017). Dynamic inventory man-
agement with total minimum order commitments and two supply
options. Operations Research,65(5), 1285–1302.
Yuan, Q., Chua, G. A., Liu, X., & Chen, Y. F. (2015). Unsold ver-
sus unbought commitment: Minimum total commitment contracts
with nonzero setup costs. Production and Operations Management,
24(11), 1750–1767.
Zipkin, P. (2008). On the structure of lost-sales inventory models.
Operations Research,56(4), 937–944.
SUPPORTING INFORMATION
Additional supporting information may be found online in the
Supporting Information section at the end of the article.
How to cite this article: Chen T, Gong X, Li Q,
Xu H. Multiseason production planning under
export quotas. Naval Research Logistics 2020;1–16.
https://doi.org/10.1002/nav.21959
12 CHEN ET AL.
APPENDIX A
Proof of Proposition 1 We prove the proposition by induction on nand t.SinceVN
T+1(x,q)=0, we have
JN
T(y,q)=−(c+h)y+(p+h)E[min {y,q,Dn
t}].
Then, it is easy to verify that the proposition holds for JN
T(y,q). Now assume inductively that the proposition holds
for Jn
t(y,q). In what follows, we prove that the proposition holds for Vn
t(x,q)and Jn
t−1(y,q).
We first prove that the proposition holds for Vn
t(x,q).SinceJn
t(y,q)is increasing in qandconcavein(y,q), it
directly follows from the optimality Equation (3) that Vn
t(x,q)is increasing in qandconcavein(x,q). In addition,
(3) can be rewritten as
Vn
t(x−𝜖, q−𝜖)+p𝜖=cx +max
x−𝜖≤y≤x−𝜖+k{Jn
t(y,q−𝜖)+(p−c)𝜖}
=cx +max
x≤̃y≤x+k{Jn
t(̃y−𝜖,q−𝜖)+(p−c)𝜖}.
Since Jn
t(̃y−𝜖,q−𝜖)+(p−c)𝜖is increasing in ϵ,thenVn
t(x−𝜖, q−𝜖)+p𝜖is increasing in ϵ.
We next prove that the proposition holds for Jn
t−1(y,q).SinceVn
t(x−𝜖, q−𝜖)+p𝜖is increasing in ϵ,h≥0and
0<𝛼≤1, we can rewrite (2) as
Jn
t−1(y,q)=−(c+h)y+E[max
w≤y,w≤q,w≤Dn
t
{(p+h)w+𝛼Vn
t(y−w,q−w)}].(A1)
Since Vn
t(y,q)is increasing in qandconcavein(y,q), it is clear that the objective function in (A2) is increasing
in qand concave in (y,q,w). The constraint set {(y,q,w)∶w≤y,w≤q,w≤Dn
t}is a convex set and becomes
less restrictive when qincreases. Then, it follows that Jn
t−1(y,q)is increasing in qandconcavein(y,q). Note also
that (A2) can be rewritten as
Jn
t−1(y−𝜖, q−𝜖)+(p−c)𝜖
=−(c+h)y+E[max
w≤y−𝜖, w≤q−𝜖, w≤Dn
t
{(p+h)(w+𝜖)+𝛼Vn
t(y−w−𝜖, q−w−𝜖)}]
=−(c+h)y+E[max
̃w≤y,̃w≤q,̃w≤Dn
t+𝜖{(p+h)̃w+𝛼Vn
t(y−̃w,q−̃w)}].
Since Vn
t(x−𝜖, q−𝜖)+p𝜖is increasing in ϵ, it is clear that Jn
t−1(y−𝜖, q−𝜖)+(p−c)𝜖is increasing in ϵ.
Note that we have proved that the proposition holds for JN
T(y,q)and for Vn
t(x,q)and Jn
t−1(y,q)given that it holds
for Jn
t(y,q),t=1, …,Tand n=1, …,N. Also note that the boundary condition for each season n(<N)isgiven
by Vn
T+1(x,q)=E[Vn+1
1(x,Qn+1)]. Thus, the proposition holds for Vn
T+1(x,q)given that it holds for Vn+1
1(x,q).
Consequently, one can easily verify by induction that the proposition holds for Vn
t(x,q)and Jn
t(y,q)for every tand
n. The proof is then complete. ▪
Proof of Theorem 1 (i) We prove the theorem by induction on t.SinceVN
T+1(x,q)=0andforn<N, it is easy to
show that Vn
T+1(x,q)=E[Vn+1
1(x,Qn+1)] is concave in xbut independent of q. Therefore, Vn
T+1(x,q)is L#-concave
in (x,q). Now assume that Vn
t+1(x,q)is L#-concave. In what follows, we prove that the theorem holds for period t
of season n, which then completes the proof.
We first prove that Jn
t(y,q)is L#-concave. For convenience, denote
̃
Jn
t(y,q,d)=(p+h)min {y,q,d}+𝛼Vn
t+1(y−min {y,q,d},q−min {y,q,d}).(A2)
Then, Jn
t(y,q)=−(c+h)y+E[̃
Jn
t(y,q,Dn
t)]. To prove the desired result, it suffices to prove that ̃
Jn
t(y,q,d)is
L#-concave in (y,q), or equivalently, ̃
Jn
t(y−𝜉,q−𝜉, d)is supermodular in (y,q,𝜉). Since Vn
t(x−𝜖, q−𝜖)+p𝜖is
increasing in ϵby Lemma 1, h≥0, and 0 <𝛼≤1, we can rewrite (A3) as follows
̃
Jn
t(y−𝜉,q−𝜉, d)= max
w≤y−𝜉, w≤q−𝜉, w≤d{(p+h)w+𝛼Vn
t+1(y−w−𝜉,q−w−𝜉)}.(A3)
Let ̂
𝜉=w+𝜉. Then, (A4) can be rewritten as
̃
Jn
t(y−𝜉,q−𝜉, d)= max
̂
𝜉≤y,̂
𝜉≤q,0≤̂
𝜉−𝜉≤d
{(p+h)(̂
𝜉−𝜉)+𝛼Vn
t+1(y−̂
𝜉,q−̂
𝜉)} (A4)
Since Vn
t+1(x,q)is L#-concave by the inductive assumption, Vn
t+1(y−̂
𝜉,q−̂
𝜉)is supermodular in (y,q,̂
𝜉).Note
that 0 <𝛼≤1. Then, the objective function in (A5) is supermodular in (y,q,̂
𝜉,𝜉). Also note that the constraint set
in (A5) is a lattice. Then, by Theorem 8.2 of Porteus (2002), we have ̃
Jn
t(y−𝜉,q−𝜉, d)is supermodular in (y,q,
𝜉). Therefore, ̃
Jn
t(y,q,d)is L#-concave in (y,q), and consequently, Jn
t(y,q)is L#-concave in (y,q).
CHEN ET AL.13
We next prove that Vn
t(x,q)is L#-concave. By definition, we need to prove that
Vn
t(x−𝜉,q−𝜉)=c(x−𝜉)+ max
x−𝜉≤y≤x−𝜉+kJn
t(y,q−𝜉)(A5)
is supermodular in (x,q,𝜉). Let ̂y=y+𝜉. Then, we can rewrite (A6) as
Vn
t(x−𝜉,q−𝜉)=c(x−𝜉)+ max
x≤̂y≤x+k
Jn
t(̂y−𝜉,q−𝜉).
Since Jn
t(̂y−𝜉,q−𝜉)is supermodular in (̂y,q,𝜉)and the constraint set {(x,̂y)∶x≤̂y≤x+k}is a lattice, by
Topkis’ result, it follows that Vn
t(x−𝜉,q−𝜉)is supermodular in (x,q,𝜉). Therefore, Vn
t(x,q)is L#-concave in (x,q).
(ii) The result follows from Lemma 3 of Zipkin (2008). We reproduce the proof here for easy reference. Since
Jn
t(y,q)is L#-concave, it is supermodular. So yn∗
t(q)is increasing (Topkis 1998, Theorem 2.8.1) and the first
inequality holds. Let xrepresents vector (q,y,ϵ). Consider x1=(q+ϵ,y,0)forsomey>yn∗
t(q)+𝜖and x2=
(q+𝜖, yn∗
t(q)+𝜖, 𝜖).Hereϵ≥0. The L#-concavity of Jn
t(y,q)means that Jn
t(y−𝜖, q−𝜖)is supermodular in (y,q,
ϵ). Let us write Jn
t(y−𝜖, q−𝜖)as Jn
t(x). This means
Jn
t(x1∧x2)−Jn
t(x1)≥Jn
t(x2)−Jn
t(x1∨x2),
or
Jn
t(yn∗
t(q)+𝜖, q+𝜖)−Jn
t(y,q+𝜖)≥Jn
t(yn∗
t(q),q)−Jn
t(y−𝜖, q).
The expression on the right hand side is positive because yn∗
t(q)is the optimal solution, but y−ϵis feasible but
not necessarily optimal. Therefore, when the quota is q+ϵ,anyy>yn∗
t(q)+𝜖cannot be the optimal produce-up-to
level. That is, yn∗
t(q+𝜖)≤yn∗
t(q)+𝜖.▪
Proof of Lemma 1 We first prove part (i) by induction on t.NotethatVn
T+1(x,q)=̂
Vn
T+1(x). Now assume
inductively that Vn
t+1(x,q)≤̂
Vn
t+1(x). In what follows, we prove that part (a) holds for period t. By the definition
of Jn
t(y,q)in (2), we have
Jn
t(y,q)≤−(c+h)y+E[(p+h)min {y,q,Dn
t}+𝛼̂
Vn
t+1(y−min {y,q,Dn
t})]
≤−(c+h)y+E[(p+h)min {y,Dn
t}+𝛼̂
Vn
t+1(y−min {y,Dn
t})]
=̂
Jn
t(y),
where the first inequality holds since Vn
t+1(x,q)≤̂
Vn
t+1(x)by the inductive assumption, the second inequality
holds because the maximal value of one unit of inventory is pand hence the function (p+h)z+𝛼̂
Vn
t+1(y−z)is
increasing in z, and the equality follows from the definition of ̂
Jn
t(y). By comparing (1) with (5), we immediately
have Vn
t(x,q)≤̂
Vn
t(x).
We next prove part (ii) by induction on t. First, for period Tof season n,ifq≥y,wehave
Jn
T(y,q)=−(c+h)y+E[(p+h)min {y,Dn
T}+𝛼Vn+1
1(y−min {y,Dn
T},Qn+1)]
=̂
Jn
T(y).
Now assume inductively that part (ii) holds for period t+1, that is, Jn
t+1(y,q)=̂
Jn
t+1(y)when q≥y+
T
i=t+2min {̂yn∗
i,k}. In what follows, we prove that part (ii) holds for period t.
When q≥y+T
i=t+1min {̂yn∗
i,k}, by (2), we can simplify Jn
t(y,q)as
Jn
t(y,q)=−(c+h)y+E[(p+h)min {y,Dn
t}+𝛼Vn
t+1(y−min {y,Dn
t},q−min {y,Dn
t})].(A6)
Subtracting (A6) from (6), we obtain
Jn
t(y,q)−̂
Jn
t(y)=𝛼E[Vn
t+1(y−min {y,Dn
t},q−min {y,Dn
t}) − ̂
Vn
t+1(y−min {y,Dn
t})].(A7)
When q≥y+T
i=t+1min {̂yn∗
i,k}and for any realization of Dn
t,wehave
q−min {y,Dn
t}≥y+
T
i=t+1
min {̂yn∗
i,k}−min {y,Dn
t}
=(y−Dn
t)++
T
i=t+1
min {̂yn∗
i,k}
14 CHEN ET AL.
≥max {(y−Dn
t)+,min {̂yn∗
t+1,(y−Dn
t)++k}} +
T
i=t+2
min {̂yn∗
i,k}
=̂yn
t+1((y−Dn
t)+)+
T
i=t+2
min {̂yn∗
i,k},
where the last equality follows from the definition of ̂yn
t+1(⋅).SinceJn
t+1(y,q)=̂
Jn
t+1(y)when q≥y+
T
i=t+2min {̂yn∗
i,k}by the inductive assumption, we have
Jn
t+1(̂yn
t+1((y−Dn
t)+),q−min {y,Dn
t}) = ̂
Jn
t+1(̂yn
t+1((y−Dn
t)+)).(A8)
Then, it follows that
Vn
t+1(y−min {y,Dn
t},q−min {y,Dn
t})
≥c(y−min {y,Dn
t}) + Jt+1(̂yn
t+1((y−Dn
t)+),q−min {y,Dn
t})
=c(y−min {y,Dn
t}) + ̂
Jn
t+1(̂yn
t+1((y−Dn
t)+))
=̂
Vn
t+1((y−Dn
t)+),
where the first inequality follows from the definition of Vn
t+1(⋅)in (3), the first equality is from (A9), and the second
equality is by the definition of ̂
Vn
t+1(⋅). Therefore, by (A8) we have Jn
t(y,q)≥̂
Jn
t(y).SinceJn
t(y,q)≤̂
Jn
t(y)by part
(i), we obtain that Jn
t(y,q)=̂
Jn
t(y)when q≥y+T
i=t+1min {̂yn∗
i,k}. The proof is then complete. ▪
Proof of Theorem 2 For any q≥Un
t=̂yn∗
t+T
i=t+1min {̂yn∗
i,k},wehave
Jn
t(̂yn∗
t,q)=̂
Jn
t(̂yn∗
t)≥̂
Jn
t(yn∗
t(q)) ≥Jn
t(yn∗
t(q),q),
where the first equality follows from Lemma 1(ii), the first inequality holds since by definition ̂yn∗
t=
arg max y≥0̂
Jn
t(y), and the second inequality follows from Lemma 1(i). Note that Jn
t(̂yn∗
t,q)≤Jn
t(yn∗
t(q),q)by
definition. Then, we have yn∗
t(q)=̂yn∗
t.▪
Proof of Theorem 3 Due to Theorem 1, yn∗
t(q)and qcannot cross more than once and when it does, the former
crosses the latter from above. Because yn∗
t(q)≥0 and Theorem 2, they also must cross. Let qn
t=sup {q≥0∶
yn∗
t(q)≥q}and it must be finite because Un
tin Theorem 2 are all finite. ▪
Proof of Theorem 4 We first prove that Sn
1≤Sn
2≤···≤Sn
T, or equivalently, Sn
t≤Sn
t+1for any t=1, …,T−1.
For convenience, denote
Gn
t(y)=−[c+(1+𝛼+···+𝛼T−t)h]y+𝛼T−t+1E[Vn+1
1(y,Qn+1)].
Then, Gn
t(y)is concave in yand Sn
t=arg maxy≥0Gn
t(y).Notethat
Gn
t(y)=𝛼Gn
t+1(y)−((1−𝛼)c+h)y.
Since 0 <𝛼≤1andh≥0, −((1 −𝛼)c+h)yis decreasing in y. Then, we have
Sn
t=arg max
y≥0Gn
t(y)=arg max
y≥0{𝛼Gn
t+1(y)−((1−𝛼)c+h)y}≤arg max
y≥0𝛼Gn
t+1(y)=Sn
t+1.
We next prove that yn∗
t(0)=[Sn
t−(T−t)k]+by induction on t. First, by the definition of Jn
T(y,q)in (2), we have
Jn
T(y,0)=−(c+h)y+𝛼E[Vn+1
1(y,Qn+1)] = Gn
T(y).(A9)
Then, it follows that
yn∗
T(0)=arg max
y≥0Jn
T(y,0)=arg max
y≥0Gn
T(y)=Sn
T.
Thus, the desired result holds when t=T. Now assume inductively that the desired results hold for t+1, t+2,
…,T. In what follows, we prove that it holds for t, which then completes the proof of this part.
We consider two cases. First, suppose yn∗
t+1(0)≤k. In this case, it is easy to show that yn∗
t(0)=0. On the other
hand, since Sn
t≤Sn
t+1and yn∗
t+1(0)=[Sn
t+1−(T−t−1)k]+by the inductive assumption, we have
[Sn
t−(T−t)k]+≤[Sn
t+1−(T−t)k]+≤[(Sn
t+1−(T−t−1)k)+−k]+=0.
Thus, yn∗
t(0)=[Sn
t−(T−t)k]+when yn∗
t+1(0)≤k.
CHEN ET AL.15
Next, suppose yn∗
t+1(0)>k. In this case, since Sn
t+1≤··· ≤Sn
Tand by the inductive assumption, we have
yn∗
t+i(0)≤yn∗
t+i+1(0)−k,i=1, …,T−t. In addition, we can show that yn∗
t(0)≤yn∗
t+1(0)−k. Then, by the concavity
of Jn
t(y,0),wehave
yn∗
t(0)=arg max
y≥0Jn
t(y,0)=arg max
0≤y≤yn∗
t+1(0)−kJn
t(y,0).
Thus, to prove the desired result, it remains to show that
arg max
0≤y≤yn∗
t+1(0)−kJn
t(y,0)=[Sn
t−(T−t)k]+.
Suppose 0 ≤y≤yn∗
t+1(0)−k. Then, the inventory level after production in period t,y,isatmostyn∗
t+1(0)−k.
Since the remaining quota for season nis zero, the starting inventory level in period t+1, which is also y,isat
most yn∗
t+1(0)−k. By the definition of yn∗
t+1(0), it is optimal to produce in period t+1 with full capacity k. Then,
the inventory level after production in period t+1, which is also the starting inventory level in period t+2, is
at most yn∗
t+1(0).Sinceyn∗
t+1(0)≤yn∗
t+2(0)−k, it is optimal to produce in period t+2 with full capacity k.Since
yn∗
t+i(0)≤yn∗
t+i+1(0)−kfor all i=1, …,T−t, by similar arguments it is optimal to produce with capacity kfor all
the periods t+1, t+2, …,T. Consequently, when 0 ≤y≤yn∗
t+1(0)−k,wehave
Jn
t(y,0)=−(c+h)y+𝛼Vn
t+1(y,0)
=−(c+h)y+𝛼cy +𝛼Jn
t+1(y+k,0)
=−(c+h)y+𝛼cy +𝛼(−(c+h)(y+k)+𝛼c(y+k)) + 𝛼2Jn
t+2(y+2k,0)
=
T−t−1
i=0
𝛼i(−(c+h)(y+ik)+𝛼c(y+ik)) + 𝛼T−tJn
T(y+(T−t)k,0),(A10)
where the first equality follows from (2), the last three equalities follow from (3), the optimal production quantities
for periods t+1, …,Tare all k, and a recursive argument. Therefore,
arg max
0≤y≤yn∗
t+1(0)−kJn
t(y,0)
=arg max
0≤y≤yn∗
t+1(0)−kT−t−1
i=0
𝛼i(−(c+h)y+𝛼cy)+𝛼T−tJn
T(y+(T−t)k,0)
=arg max
0≤y≤yn∗
t+1(0)−k{−(c+(1+𝛼+···+𝛼T−t)h)y+𝛼T−t+1E[Vn+1
1(y+(T−t)k,Qn+1)]}
=−(T−t)k+arg max
(T−t)k≤̂y≤Sn
t+1
Gn
t(̂y)
=[Sn
t−(T−t)k]+,
where the first equality follows from (A11), the second equality follows from (A10), the third equality follows
from the change of variable from yto ̂y=y+(T−t)kand that Sn
t+1=yn∗
t+1+(T−t−1)k,and the last equality follows
from the concavity of Gn
t(⋅), the definition of Sn
t,andSn
t≤Sn
t+1. Thus, yn∗
t(0)=[Sn
t−(T−t)k]+when yn∗
t+1(0)>k.
▪
Proof of Theorem 5 We first prove part (i). yN∗
t(0)=SN
t=0 follows directly from Theorem 4. Theorem 1 (ii)
implies that yn∗
t(q)is increasing in qand the slope is no greater than 1. It follows that 0 ≤yN∗
t(q)≤q.
Next we prove part (ii). From Theorem 4, the condition that yn+1,∗
1(Qn+1)≤kfor any realization of Qn+1implies
yn∗
t(0)=0fort=1, …,T, and from Theorem 3, yn∗
t(q)≤qfor any q≥0. We will prove that 𝜕Jn
t(y,q)∕𝜕y≥0
for any y≤q≤Ln
t,whereLn
tis given in (7), which then completes the proof.
In period Tof season n,wheny≤q,wehave
Jn
T(y,q)=−(c+h)y+E[(p+h)min {y,Dn
T}+𝛼Vn+1
1(y−min {y,Dn
T},Qn+1)].
Taking the partial derivative of Jn
T(y,q)over y, we obtain
𝜕Jn
T(y,q)
𝜕y=−(c+h)+(p+h)Pr (y≤DN
T)+𝛼E𝜕Vn+1
1(y−Dn
T,Qn+1)
𝜕x1{y≥Dn
T}.(A11)
When n=N,sinceVN+1
1(x,QN+1)=0, (A12) can be simplified as
𝜕Jn
T(y,q)
𝜕y=−(c+h)+(p+h)Pr (y≤DN
T).
When q≤LN
T, one can easily verify that 𝜕Jn
T(y,q)
𝜕y≥0foranyy≤q. Thus, yN∗
T(q)=qwhen q≤LN
T.
16 CHEN ET AL.
Now suppose n<N.SinceVn+1
1(x,Qn+1)=cx +maxx≤y≤x+kJn+1
1(y,Qn+1)and yn+1,∗
1(Qn+1)is the maximizer
of Jn+1
1(y,Qn+1)over y≥0, Vn+1
1(x,Qn+1)−cx is increasing in xwhen x≤yn+1,∗
1(Qn+1). Consequently, when
y≤q≤Ln
T=min F−1
n,Tp−c
p+h−𝛼c,yn+1,∗
1(Qn+1),wehave
𝜕Jn
T(y,q)
𝜕y≥−(c+h)+(p+h)Pr (y≤Dn
T)+𝛼cPr (y≥Dn
T)
=p−c−(p+h−𝛼c)Pr (y≥Dn
T)
≥0.
Therefore, yn∗
T(q)=qwhen q≤Ln
T,n=1, …,N−1.
Now we assume inductively that yn∗
t+1(q)=qwhen q≤Ln
t+1for period t+1 and season n.Wheny≤q≤Ln
t+1,
we have
Jn
t(y,q)=−(h+c)y+E[(p+h)min {y,Dn
t}+𝛼Vn
t+1(y−min {y,Dn
t},q−min {y,Dn
t})],(A12)
Jn
t(q,q)=−(h+c)q+E[(p+h)min {q,Dn
t}+𝛼Vn
t+1(q−min {q,Dn
t},q−min {q,Dn
t})].(A13)
Let 𝜖n
t=min {q,Dn
t}−min {y,Dn
t}.SinceVn
t+1(x−𝜖, q−𝜖)+p𝜖is increasing in ϵby Lemma 1, we have
Vn
t+1(q−min {y,Dn
t},q−min {y,Dn
t}) ≤Vn
t+1(q−min {q,Dn
t},q−min {q,Dn
t}) + p𝜖n
t.(A14)
Since y≤q≤Ln
t+1,wehave
y−min {y,Dn
t}≤q−min {q,Dn
t}≤q−min {y,Dn
t}≤Ln
t+1.
Then, by the inductive assumption, we have
Vn
t+1(y−min {y,Dn
t},q−min {y,Dn
t}) = c(y−min {y,Dn
t}) + Jn
t+1(q−min {y,Dn
t},q−min {y,Dn
t});
Vn
t+1(q−min {y,Dn
t},q−min {y,Dn
t}) = c(q−min {y,Dn
t}) + Jn
t+1(q−min {y,Dn
t},q−min {y,Dn
t}).
Combining the above two equations with (A13), (A14) and (22), we obtain
Jn
t(y,q)−Jn
t(q,q)≤((1−𝛼)c+h)(q−y)−((1−𝛼)p+h)E[𝜀n
t]
≤((1−𝛼)c+h)(q−y)−((1−𝛼)p+h)(q−y)Pr(q≤Dn
t)
≤[(−(1−𝛼)(p−c)+((1−𝛼)p+h)Pr (Dn
t≤q)](q−y).
Therefore, if q≤Ln
t=min Ln
t+1,F−1
n,t(1−𝛼)(p−c)
(1−𝛼)p+h,thenJn
t(y,q)≤Jn
t(q,q). The proof is complete. ▪