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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

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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. ▪