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Journal of the Operational Research Society
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Pre-positioning strategies for relief supplies in a
relief supply chain
Yang Liu, Jun Tian & Gengzhong Feng
To cite this article: Yang Liu, Jun Tian & Gengzhong Feng (2022) Pre-positioning strategies
for relief supplies in a relief supply chain, Journal of the Operational Research Society, 73:7,
1457-1473, DOI: 10.1080/01605682.2021.1920343
To link to this article: https://doi.org/10.1080/01605682.2021.1920343
Published online: 26 Jun 2021.
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ORIGINAL ARTICLE
Pre-positioning strategies for relief supplies in a relief supply chain
Yang Liu, Jun Tian and Gengzhong Feng
School of Management, Xi’an Jiaotong University, Xi’an, China
ABSTRACT
It is vital to pre-position a certain amount of relief supplies at pre-disaster. But governments
are often confronted with a dilemma of coordinating inventory cost and stock-out cost in
pre-positioning relief supplies. Pre-positioning strategies for relief supplies depend on the
natural characteristic of relief supplies and the most urgent needs at post-disaster.
Therefore, we classify relief supplies by their natural characteristic and priority, and introduce
an option contract into relief supplies pre-positioning system which consists of a single gov-
ernment and an agreement enterprise. Specially, a consumable and relatively urgent relief
supplies pre-positioning (C-RUP) model and a non-consumable and urgent relief supplies
pre-positioning (NC-UP) model are established. The optimal decisions of the government
and the enterprise are derived, respectively. The relief supply chain coordination is achieved
with the option contract. Under channel coordination, the two models can not only improve
the government’s emergency ability and support capacity, but also reduce the government’s
inventory risk when compared to the government single pre-positioning model. Moreover,
we give out the conditions that the two sides can reach a win-win situation. Lastly, we pro-
pose important managerial insights for pre-positioning strategies related to different types
of relief supplies.
ARTICLE HISTORY
Received 23 September 2019
Accepted 14 April 2021
KEYWORDS
Game theory; government;
inventory; purchasing;
supply chain
1. Introduction
All corners of the whole world are frequently
affected by sudden disaster events such as earth-
quakes, hurricanes and tsunamis and so forth, which
pose disastrous threats and destructions to people’s
health, life and property security. As described in
the 2018 natural disasters report launched by the
Centre for Research on the Epidemiology of
Disasters, there were 315 natural disaster events
recorded with 11,804 deaths, over 68 million people
affected, and 131.7 billion dollars in economic losses
across the whole world. Disaster management as a
hedge against disaster events has drawn great public
attention over the past several decades, including
four sequential stages: mitigation, preparedness,
response and recovery (Altay & Green, 2006).
National governments are the primary relief organi-
zations responsible for pre-disaster mitigation and
preparedness and post-disaster responsiveness and
recovery. Taking China, for example, the Chinese
government has built 24 central warehouses and
more than 1000 local warehouses to reserve a cer-
tain amount of relief supplies before a disaster
strikes. But governments are often confronted with
a dilemma with respect to how many relief supplies
should be pre-positioned (Chakravarty, 2014;Hu
et al., 2019; Liang et al., 2012). In practice, pre-posi-
tioning adequate relief supplies can not only shorten
response time for disaster relief, but also signifi-
cantly improve the efficiency of relief activities
(Galindo & Batta, 2013; Rawls & Turnquist, 2012).
If no disaster occurs before the pre-positioned relief
supplies expire, governments will entail too much
inventory risk. On the other hand, stock-out risk
will occur if the pre-positioned relief supplies can-
not meet the sudden-onset demand. In view of this,
this paper is to present rational strategies of coordi-
nating inventory cost and stock-out cost in pre-posi-
tioning relief supplies.
From literature review, we find that many schol-
ars have proposed various methods to solve the pre-
positioning problem of relief supplies (see Sec. 2).
However, some research shortcomings still exist. In
practice, governments usually buy some kinds of
relief supplies and pre-position them in a warehouse
before a disaster strikes, while retailers in a com-
mercial supply chain usually apply zero inventory
management. But few studies have examined the
influences of regular relief supplies pre-positioned
by governments at pre-disaster, which is the start of
our paper. In addition to pre-position relief supplies
at pre-disaster, governments also can buy the
required relief supplies from agreement enterprises
(Ding & Huang, 2012; Nikkhoo et al., 2018; Wang
et al., 2015) or/and the spot market (Hu et al.,
2019) and receive in-kind donations (Chen et al.,
2017,2018; Holgu
ın-Veras et al., 2014). However,
CONTACT Yang Liu fanxipingly@163.com Xi’an Jiaotong University, Xianning West Road 28, Beilin, Xi’an, Shaanxi, China.
ßOperational Research Society 2021
JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY
2022, VOL. 73, NO. 7, 1457–1473
https://doi.org/10.1080/01605682.2021.1920343
most existing studies only focus on the procurement
of relief supplies from the enterprises or/and spot
market at post-disaster, and ignore the importance
of in-kind donations which are donated by humani-
tarian organizations (HOs) or individuals. This is
mainly because this spontaneous flow of relief sup-
plies may bring much-needed relief and major com-
plications to the operations (Holgu
ın-Veras et al.,
2014). Different from previous research, this paper
considers that if the pre-positioned relief supplies
cannot satisfy the sudden-onset demand at post-dis-
aster, governments can not only buy the needed
relief supplies from the agreement enterprise and
spot market, but also receive in-kind donations. For
example, after the outbreak of the 2008 Wenchuan
earthquake, Pakistan donated more than 22 thou-
sand tents to disaster-affected areas. Under this
hypothesis, we intend to examine the influences of
in-kind donations on pre-positioning strategies for
relief supplies. Last but not least, relief supplies
cover an enormous spectrum, from food, drugs and
clothing to rescue equipment, electric generators
and construction materials. Some scholars divide
relief supplies into two categories: perishable prod-
ucts and non-perishable commodities, and propose
different strategies (outsourcing and insourcing) to
pre-position the two kinds of relief supplies (Wang
et al., 2016; Yao et al., 2018). Many researchers pay
attention to the pre-positioning problem of a certain
type of relief supplies such as food (Zhang et al.,
2019) and vaccine (Shamsi et al., 2018) because
these relief supplies are critical products which are
in great demand in the early post-disaster. It can be
seen that the classification of relief supplies largely
determines pre-positioning strategies for different
types of relief supplies. However, there is no univer-
sal standard for how to classify relief supplies at
present. For instance, relief supplies can be catego-
rized as consumable and non-consumable based on
their natural characteristic (Duran et al., 2012;
Wang et al., 2015; Yao et al., 2018). On the other
hand, they can be classified into three types based
on the most urgent needs (by priority): urgent, rela-
tively urgent, non-urgent (Goyet, 2001). In order to
solve the pre-positioning dilemma of governments,
this paper first intends to classify relief supplies with
the combination of their natural characteristic
and priority.
Since non-urgent relief supplies can be purchased
from the existing spot market at post-disaster, there
is no need to pre-position them at pre-disaster. So
purchasing non-urgent relief supplies is not the
focus of this paper. Considering their natural char-
acteristic and priority, relief supplies are classified
into four types: consumable and urgent (C-U), con-
sumable and relatively urgent (C-RU), non-
consumable and urgent (NC-U), non-consumable
and relatively urgent (NC-RU). C-U relief supplies
are mainly drugs, blood and vaccines and so on,
which require a rather rigorous storage condition
on temperature and moisture as well as high inven-
tory cost (Wang et al., 2016). In practice, we rarely
see blood plasma pre-positioned by commercial
enterprises. NC-RU relief supplies are mainly rescue
equipment, first-aid equipment and communication
equipment and so on, which are usually possessed
by public security fire control institutions, national
hospitals and military organizations. For instance, in
the 2008 Wenchuan earthquake in China, the
needed C-U and C-RU relief supplies were directly
transported to disaster-affected areas without pro-
curement. C-RU relief supplies are mainly drinking
water, fast food and packer milk, and NC-U relief
supplies are mainly tents, blankets and clothing.
Both C-RU and NC-U relief supplies are particularly
suitable for storage in advance of disasters because
they are pre-positioned in a relatively undemanding
storage condition and many enterprises are willing
to store relief supplies for the sake of themselves.
Accordingly, this paper focuses on the pre-position-
ing problem of both C-RU and NC-U
relief supplies.
Since option contract has overwhelming advan-
tages in supply elasticity, supply chain coordination
and risk reduction (Barnes-Schuster et al., 2002;
Gastaldi et al., 2006), we introduce an option con-
tract into relief supplies pre-positioning system
which consists of a single government and an enter-
prise, and build a C-RU relief supplies pre-position-
ing (C-RUP) model and a NC-U relief supplies pre-
positioning (NC-UP) model. Next, our research is
intended to address the following critical questions:
1. What are the pre-positioning frameworks for
both C-RU and NC-U relief supplies?
2. What are optimal decision strategies of the gov-
ernment and the enterprise under the two mod-
els, respectively?
3. Under what condition will the government and
the enterprise are willing to conduct the
option contract?
There are several key findings associated with our
analysis. First of all, we propose different pre-posi-
tioning frameworks for both C-RU and NC-U relief
supplies. The main difference is that C-RU relief
supplies are pre-positioned by the enterprise in
form of real products, while NC-U relief supplies
are pre-positioned by real products and production
capacity. Moreover, compared with the government
single pre-positioning model, it is necessary for the
government to develop both the C-RUP model and
1458 Y. LIU ET AL.
the NC-UP model. This is because the two models
can not only improve the government’s emergency
ability and support capacity, but also reduce the
government’s inventory risk. Third, we present the
conditions that the government and the enterprise
can achieve a win-win situation. Lastly, we propose
managerial insights for pre-positioning strategies
related to different types of relief supplies.
The remainder of this paper is organized as fol-
lows. Section 2 presents a brief literature review.
Section 3 is problem descriptions. Section 4 includes
two benchmark models. Section 5 builds the C-RUP
model and Sec. 6 builds the NC-UP model. Section
7concludes this paper and provides important man-
agerial insights. All proofs of Propositions are given
in the Appendix.
2. Literature review
Given that this work intends to build pre-position-
ing models for different types of relief supplies, the
research is related to three streams of literature.
The first stream is with regard to pre-positioning
relief supplies. Pre-positioning is defined as
“stockpiling of equipment and supplies at, or near
the point of planned use”(Lodree et al., 2012). The
mathematical programming methods are often
employed to address the pre-positioning problem of
relief supplies. Balcik and Beamon (2008) proposed
a mixed-integer programming model to determine
the number and location of distribution centers, and
the amount of relief items to be stockpiled at each
distribution center. Rawls and Turnquist (2010)
developed an optimization model that determined
the location and amount of relief supplies to be pre-
positioned. The model was validated by hurricanes
happened in the Gulf Coast area of the US.
Campbell and Jones (2011) established a news-
vendor-based model and examined the decision of
where to pre-position relief supplies prior to a disas-
ter, and how many to pre-position as stock at a cen-
ter. Galindo and Batta (2013) offered an integer
programming model for pre-positioning relief sup-
plies in preparation for disasters. The location and
level of storage for pre-positioned relief supplies
were yielded. Davis et al. (2013) proposed a stochas-
tic programming model that determined how many
global relief supplies from external suppliers, and
how many local relief supplies from internal suppli-
ers. Charles et al. (2016) proposed a deterministic
mixed integer linear programming to help humani-
tarian practitioners improve their supply networks.
The model identified the optimal number and loca-
tion of warehouses on a regional scale. Pacheco and
Batta (2016) put forward a forecast-driven dynamic
model for pre-positioning relief supplies. The
location of supply points and the optimal amount of
pre-positioned relief supplies at each supply point
were determined. The model’s outcomes can be eas-
ily understood by humanitarian practitioners.
Mohammadi et al. (2016) presented a novel multi-
objective stochastic programming model with regard
to pre-position and distribute relief supplies. The
optimal amount and location of supply centers, the
optimal inventory level of relief supplies and the
optimal flow amount of relief supplies were attained.
Klibi et al. (2018) proposed a multi-phase modelling
framework which was applied for humanitarian
relief network design. The framework could deter-
mine the number and location of distribution cen-
ters, each center’s capacity, and the amount of relief
supplies to keep as stock. Ni et al. (2018) proposed
a min-max robust model that can optimize the deci-
sion of facility location and inventory level. The
earthquake happened at Yushu County in China
was used to evaluate the model’s performance.
The second stream is related to pricing relief sup-
plies. The supply chain contracts are often employed
to deal with the pricing problem. Balcik and Ak
(2014) introduced a quantity flexibility contract
(QFC) into the procurement of relief supplies, and
built a stochastic programming model to minimize
the relief organization’s procurement and agreement
costs. Similar to Balcik and Ak (2014), Nikkhoo
et al. (2018) utilized a QFC between a relief organ-
ization and a supplier to coordinate inventory man-
agement. The results showed that the proposed QFC
could improve the demand satisfaction of disaster-
affected areas. Liang et al. (2012) introduced an
option contract into relief supplies management,
and established an option pricing model with
binominal lattice. There was a feasible price interval
in which the members in the relief supply chain
were willing to conduct the option contract.
Rabbani et al. (2015) developed an option pricing
model based on binomial trees. The option and exe-
cution prices in four different conditions were opti-
mized by the proposed model. The results showed
that the members in the relief supply chain were
willing to engage in the option transaction. Wang
et al. (2015) proposed a relief supplies purchasing
mechanism with an option contract, which is super-
ior to both pre-purchasing with a buyback contract
and instant-purchasing with a return policy. Hu
et al. (2019) proposed a relief supplies procurement
model based on a put option contract. The condi-
tions that the government and the supplier con-
ducted transactions and achieved a win-win
situation were derived. On the basis of Hu et al.
(2019), Liu et al. (2019) applied option contracts
into relief supplies management system which con-
sists of a single government and multiple suppliers.
JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1459
The conditions that the government and all suppli-
ers achieved a multi-win situation were
also attained.
The last stream relates to the classification of
relief supplies. As stated by PAHO, relief supplies
can be classified into 10 different categories. From
Duran et al. (2012), they are divided into two types:
consumable and non-consumable. Consumable relief
supplies such as water and fast food are transferred
continuously into disaster-affected areas. Non-con-
sumable relief supplies are delivered to disaster-
affected areas once, and they are further classified as
operational and non-operational. Wang et al. (2015)
divided relief supplies into two categories: imperish-
able commodities and perishable products, and pro-
posed proactive insourcing strategies for pre-
positioning imperishable relief supplies and pro-
active outsourcing strategy for perishable relief sup-
plies. Similar results can be seen in Yao et al.
(2018). Shamsi et al. (2018) presented an option
contract between the buyer and two competing sup-
pliers and determined the optimal amount of the
required vaccine doses provided by the two suppli-
ers. Zhang et al. (2019) built a quantity commitment
contract framework where there was a government
authority and an agreement supplier. The results
showed that the government should actively cooper-
ate with the supplier to co-reserve perishable
relief materials.
We intend to design pre-positioning models for
different types of relief supplies. Such consideration
is different from the existing literature in the follow-
ing ways. First of all, relief supplies are sorted by
their natural characteristic and priority. Specially,
we build a C-RUP model for pre-positioning C-RU
relief supplies and a NC-UP model for NC-U relief
supplies. Second, the government can pre-position a
certain amount of a kind of relief supplies before a
disaster strikes, which is consistent with the charac-
teristic of the relief supply chain. This not only
increases the possibility that the enterprise conducts
the contract, but also timely transports the needed
relief supplies to disaster-affected areas. Third, in
addition to pre-position relief supplies at pre-disas-
ter and buy the needed relief supplies at post-disas-
ter, the government can also receive in-kind
donations which are collected and delivered by the
HO. Therefore, we investigate the influences of in-
kind donations on the interest game between the
government and the enterprise. We give out the
conditions that the relief supply chain coordination
can be achieved and the two sides can reach a win-
win situation. Compared with previous models
developed in the literature, our study is more closely
aligned with reality. And we propose important
managerial insights for the government to adopt a
reasonable pre-positioning model and strategies
related to different types of relief supplies.
3. Problem descriptions
3.1. The pre-positioning framework of
relief supplies
We will propose the pre-positioning frameworks of
both C-RU and NC-U relief supplies in this section.
We introduce an option contract into C-RU relief
supplies pre-positioning system and NC-U relief
supplies pre-positioning system, respectively. Each
system consists of a single government and an
agreement enterprise. It is a Stackelberg game where
the government is a leader and the enterprise is a
follower. The government can pre-position a certain
amount of a kind of relief supplies before a disaster
suddenly strikes. C-RU relief supplies are consum-
able products with high priority such as bottled
water and fast food. The production and inventory
costs of C-RU relief supplies are relatively low and
the sudden-onset demand of C-RU relief supplies in
the aftermath of a disaster is huge. If C-RU relief
supplies are pre-positioned by semi-products, the
shelf-life of such semi-products is relatively short.
So we consider that the enterprise will pre-position
C-RU relief supplies by real products. NC-U relief
supplies are such non-consumable products with
top priority such as tents, blankets and clothing.
The production and inventory costs of NC-U relief
supplies are high. Pre-positioning NC-U relief sup-
plies totally by real products will take up an enor-
mous amount of the enterprise’s money. If the
option contract is not triggered, the enterprise has
to dispose of the unused (unsold) relief supplies,
which leads to the reduction of his profit. So we
consider that the enterprise will pre-position NC-U
relief supplies with the combination of real products
and production capacity. If needed, the enterprise
will convert the prepared production capacity into
real products and deliver them to the government
before the deadline agreed in the contract. This
increases the enterprise’s production flexibility. The
enterprise can certainly convert production capacity
into real products before the default deadline. So
the time of converting production capacity into real
products is not taken into the modelling. In add-
ition to pre-position relief supplies at pre-disaster
and purchase the needed relief supplies from the
enterprise at post-disaster, the government can also
receive in-kind donations. We consider that if the
option contract is triggered, the government first
will buy the need relief supplies from the enterprise.
The enterprise is more likely to accept the option
contract under this schedule. If the sum of the gov-
ernment’s and the enterprise’s pre-positioned relief
1460 Y. LIU ET AL.
supplies cannot meet the sudden-onset demand, the
government will turn to receive in-kind donations
because the collection and disposal of in-kind dona-
tions have been basically completed. If the sum of
the government’s and the enterprise’s pre-positioned
relief supplies plus in-kind donations cannot meet
the sudden-onset demand, the government will buy
the needed relief supplies from spot market. The
remaining unmet demand will lead to shortage cost.
3.2. Assumptions
To ensure that the C-RUP model and the NC-UP
model make sense, some rational assumptions are
presented without loss of generality. The first
assumption is that the government and the enter-
prise are both risk neutral and completely rational
following traditional economics. Next, we assume
that the needed relief supplies cannot be urgently
requisitioned by the government through mandatory
measures. This assumption is to fully depict the
effectiveness of pre-positioning strategies for differ-
ent types of relief supplies in advance of disasters by
an option contract. Similar assumptions could be
found in Zhang et al. (2019) and Liu et al. (2019).
The third assumption is that the government tends
to buy the needed relief supplies at a low price if
the option contract is triggered, which is consistent
with Hu et al. (2019). That is, if the execution price
is less than the spot market price, the government
will buy the needed relief supplies at the execution
price; if not, the government will buy them at the
spot market price. The fourth assumption is that if
the sum of the government’s and the enterprise’s
pre-positioned relief supplies plus in-kind donations
cannot meet the sudden-onset demand, the prob-
ability that the government can buy all the needed
relief supplies from the existing spot market is a
constant (Hu et al., 2019). Lastly, we assume that
after the disaster, the government will commit to
subsidize the HO at a fixed subsidy level. This is
because the HO makes a huge effort in the collec-
tion and delivery of in-kind donations.
3.3. Notations and constraints
The notations are given in Table 1.
4. The benchmark models
This section analyzes two benchmark models: the
government single pre-positioning (GSP) model and
the centralized relief supply chain (CRSC) system.
4.1. The GSP model
We use Qdto denote the government’s pre-posi-
tioned quantity of a kind of relief supplies under
the GSP model. Before a disaster strikes, the govern-
ment buys Qdunits relief supplies at x:Once a dis-
aster happens, if 0 xQd, the government does
not purchase additional relief supplies; if Qd<x
QdþS, the government needs to receive in-kind
donations; if xexceeds QdþS, the government
needs to procure the required relief supplies from
spot market. The remaining unsatisfied demand will
Table 1. Notations.
Variables Explanations
xThe actual demand of a kind of relief supplies that follows a random
distribution. The probability density and cumulative distribution
functions are fðxÞand FðxÞ:FðxÞand F1ðxÞare
increasing functions.
UThe maximal sudden-onset demand of a kind of relief
supplies, FðUÞ¼1:
hThe probability that a disaster occurs.
cUnit production cost.
hUnit inventory cost.
vUnit salvage value, h<v:
MUnit shortage cost.
xUnit wholesale price, x>c:
oOption price, oþv<cþh:
eExecution price, xþh<eþo:
pSpot market price, xþh<p:
uThe probability that the execution price is greater than the spot
market price.
kThe probability that the government can buy all the needed relief
supplies from spot market.
QaThe government’s pre-positioned quantity of C-RU relief supplies.
QbThe government’s pre-positioned quantity of NC-U relief supplies.
qThe amount of C-RU relief supplies that the enterprise stockpiles.
q0The amount of real NC-U relief supplies that the enterprise stockpiles.
q1The amount of production capacity of NC-U relief supplies that the
enterprise stores.
SThe amount of in-kind donations.
mUnit preparation cost of NC-U relief supplies.
c0Unit urgent production cost of NC-U relief supplies, v<c0:
aThe subsidy level that the government offers to the HO.
JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1461
yield stock-out cost. After the disaster, the govern-
ment needs to pay subsidies to the HO. So the gov-
ernment’s expected cost under the GSP model can
be formulated as
Cd¼xþh
ðÞ
Qdv1h
ðÞ
QdþhðQd
0
vQ
dx
ðÞ
fx
ðÞ
dx
þðU
QdþS
kpþ1k
ðÞ
M
xQdS
ðÞ
fx
ðÞ
dxþaS(1)
where the first term is the government’s wholesale
cost and inventory cost, the second term is the sal-
vage profit if no disaster occurs, the last term is the
sum of profit and cost when a disaster occurs. The
first-order and the second-order derivatives of Cd
with respect to Qdare
@Cd
@Qd
¼xþhvhv1FQ
d
ðÞðÞð
kpþ1k
ðÞ
M
1FQ
dþSðÞðÞÞ
(2)
@2Cd
@Q2
d
¼hkpþ1k
ðÞ
M
fQ
dþSðÞvf Qd
ðÞ
(3)
In practice, many studies assume that the sud-
den-onset demand of a kind of relief supplies fol-
lows an uniform distribution (Liu et al., 2019;
Nikkhoo et al., 2018). This paper follows this
assumption and gets @2Cd
@Q2
d
¼hkpþ1k
ðÞ
Mv
ðÞ
U:Owing to
M>p>v, we have @2Cd
@Q2
d
>0:So Cdis convex in
Qd:By the first-order optimality condition, we
obtain the government’s optimal pre-positioned
quantity of a kind of relief supplies under the GSP
model:
Q
d¼UUxþhv
ðÞ
þhSkpþ1k
ðÞ
M
hkp þ1k
ðÞ
Mv
(4)
4.2. The CRSC system
We use Qsto denote the system’s pre-positioned
quantity of a kind of relief supplies under the CRSC
system in which the government is a central deci-
sion-maker (Ding & Huang, 2012). According to the
sudden-onset demand of a kind of relief supplies,
the system’s expected cost under the CRSC system
can be formulated as
Cs¼cþh
ðÞ
Qsv1h
ðÞ
QsþhÐQs
0vQ
sx
ðÞ
fx
ðÞ
dx
þðU
QsþS
kpþ1k
ðÞ
M
xQsS
ðÞ
fx
ðÞ
dxþaS
(5)
where the first term is the system’s production cost
and inventory cost, the second term is the salvage
profit if no disaster occurs, the last term is the sum
of profit and cost when a disaster occurs. The first-
order and the second-order derivatives of Cdwith
respect to Qsare
@Cs
@Qs
¼cþhvhv1FQ
s
ðÞðÞð
kpþ1k
ðÞ
M
1FQ
sþS
ðÞðÞÞ
(6)
@2Cs
@Q2
s
¼hkpþ1k
ðÞ
Mv
U(7)
Owing to M>p>v, we have @2Cs
@Q2
s
>0:So the
first-order optimality condition works. By solving
@Cs
@Qs¼0, we obtain the system’s optimal pre-posi-
tioned quantity of a kind of relief supplies under
the CRSC system:
Q
s¼UUcþhv
ðÞ
þhSkpþ1k
ðÞ
M
hkp þ1k
ðÞ
Mv
(8)
Owing to x>c, we have Q
d<Q
s:It shows that
the GSP model cannot coordinate the relief supply
chain. Therefore, we introduce an option contract
Figure 1. The timing of the interest game under the C-RUP model.
1462 Y. LIU ET AL.
into relief supplies pre-positioning system, and build
the C-RUP model and the NC-UP model in Secs. 5
and 6. Meanwhile, the GSP model is compared with
the two models.
5. The C-RUP model
In this section we intend to build the C-RUP model.
Figure 1 shows the timing of the interest game
between the government and the enterprise under
the C-RUP model. Prior to a disaster, the govern-
ment buys a certain amount of C-RU relief supplies
(the primary order) at the wholesale price, and sim-
ultaneously offers an option contract to the enter-
prise. The enterprise decides whether to accept this
option contract or reject it. If the enterprise accepts
this contract, he delivers the primary order quantity
of C-RU relief supplies to the government, and sim-
ultaneously decides how many real C-RU relief sup-
plies to pre-position at his own warehouse. If a
disaster occurs, according to the sudden-onset
demand of C-RU relief supplies, the government’s
and the enterprise’s response actions include the fol-
lowing situations: if 0 <x<Qa, the government
will deliver his pre-positioned C-RU relief supplies
to disaster-affected areas and the enterprise will dis-
pose of the unsold C-RU relief supplies; if Qa<x<
Qaþq, the government will buy xQaunits C-
RU relief supplies from the enterprise and the enter-
prise will deliver them to the government and dis-
pose of the unsold C-RU relief supplies; if
Qaþq<x<QaþqþS, the government will buy
qunits C-RU relief supplies from the enterprise and
receive in-kind donations from the HO; if Qaþqþ
S<x<U, the government will buy the needed C-
RU relief supplies from spot market and the unmet
demand will lead to stock-out cost.
5.1. The enterprise’s optimal decisions
After accepting the option contract, the enterprise is
to determine his pre-positioned quantity of C-RU
relief supplies so as to maximize his expected profit.
From Figure 1, the enterprise’s expected profit can
be formulated as
Qa¼xc
ðÞ
Qacþho
ðÞ
qþ1h
ðÞ
vq
þhðQa
0
vqf ðxÞdx þðQaþq
Qa
1u
ðÞ
eþup
ðÞ
xQa
ðÞ
þvðQaþqxÞÞfðxÞdx
þðU
Qaþq1u
ðÞ
eþupÞqf ðxÞdx(9)
where the first and second terms are the enterprise’s
fixed profit, the third term is the salvage profit if no
disaster occurs, the last term is the portfolio of
profits when a disaster occurs. The first-order
and the second-order derivatives of Qawith respect
to qare
@Qa
@q¼chþoþv
þh1u
ðÞ
eþupv
ðÞ
1FQ
aþq
ðÞðÞ
(10)
@2Qa
@q2¼h1u
ðÞ
eþupv
ðÞ
fQ
aþq
ðÞ
(11)
Owing to fQ
aþq
ðÞ
>0, p>vand e>v,we
have @2Qa
@q2<0:So Qais concave in q:So we obtain
the enterprise’s optimal pre-positioned quantity of
C-RU relief supplies:
q¼F11cþhov
h1u
ðÞ
eþupv
ðÞ
Qa(12)
5.2. The government’s optimal decisions
Different from the retailer in a commercial supply
chain, the government as the buyer in the relief sup-
ply chain is to reduce the injuries, fatalities, and
damages caused by disasters, as well as to minimize
his expected cost. After all, the government’s input
is always limited. However, the social benefit result-
ing from pre-positioning relief supplies cannot be
easily measured. Therefore, we assume that the
objective of the government is to minimize his
expected cost. Similar assumptions could be found
in Muggy and Jessica (2014), Hu et al. (2019) and
Liu et al. (2019). From Figure 1, the government’s
expected cost can be formulated as
Ca¼xþh
ðÞ
QaþoqvQa1h
ðÞ
þhðQa
0
vðQaxÞfðxÞdx
þðQaþq
Qa
1u
ðÞ
eþup
ðÞ
xQa
ðÞ
fðxÞdx
þÐU
Qaþq1u
ðÞ
eþup
ðÞ
qf x
ðÞ
dx
þðU
QaþqþS
kpþ1u
ðÞ
M
ðÞ
xQaqS
ðÞ
fx
ðÞ
dxþaS
(13)
where the first and second terms are the govern-
ment’s fixed cost, the third term is the salvage profit
if no disaster occurs, the last term is the portfolio of
costs when a disaster occurs. From the definition of
supply chain coordination, when the sum of the
government’s pre-positioned quantity of C-RU relief
supplies plus the enterprise’s pre-positioned quantity
of C-RU relief supplies is equal to the total pre-posi-
tioned quantity of C-RU relief supplies under the
CRSC system (i.e., Qaþq¼Q
s), the coordination of
JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1463
the relief supply chain will be achieved (Liu
et al., 2019;Luoetal.,2015;Zhaoetal.,2013).
By substituting Qaþq¼Q
sinto Eq. (13),weget
the expression of the government’sexpected
cost:
Ca¼xþh
ðÞ
QaþoQ
sQa
ðÞ
þhvQ
aÐQa
0Fx
ðÞ
dx
þ1u
ðÞ
eþup
ðÞ
Q
sQaðQ
s
Qa
Fx
ðÞ
dx
!
þkpþ1k
ðÞ
M
UQ
sSðU
Q
sþS
FðxÞdx
!
þaSÞ(14)
The first-order and the second-order derivatives
of Cawith respect to Qaare
@Ca
@Qa
¼xþhovh1u
ðÞ
eþupv
ðÞ
1FQ
a
ðÞðÞ
(15)
@2Ca
@Q2
a
¼h1u
ðÞ
eþupv
ðÞ
fQ
a
ðÞ (16)
Owing to fQ
a
ðÞ
>0, p>vand e>v, we have
@2Ca
@Q2
a
>0:So Cais convex in Qa:We obtain the gov-
ernment’s optimal pre-positioned quantity of C-RU
relief supplies:
Q
a¼F11xþhov
h1u
ðÞ
eþupv
ðÞ
(17)
Proposition 1. Under the C-RUP model, the govern-
ment’s optimal pre-positioned quantity and the enter-
prise’s optimal pre-positioned quantity of C-RU relief
supplies are
Q
a¼U1xþhov
h1u
ðÞ
eþupv
ðÞ
q¼Uxc
ðÞ
h1u
ðÞ
eþupv
ðÞ
8
>
>
>
<
>
>
>
:
(18)
From Proposition 1, we can see that when h>
xþhov
1u
ðÞ
eþupv, the C-RUP model makes sense. That is,
the occurrence probability of a potential disaster
should be more than a certain value, rather than a
once-in-a-century disaster.
5.3. The relief supply chain coordination
We introduce an option contract into C-RU relief
supplies pre-positioning system and intend to
coordinate the relief supply chain with the
option contract.
Proposition 2. Under the C-RUP model, the relief
supply chain coordination can be achieved
when e ¼1
1uvupþUcþhov
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
:
Proposition 2 shows the condition that the relief
supply chain coordination is achieved. By solving
the first-order derivatives of oþewith respect to o
and S, we get @2oþeðÞ
@o<11
h1u
ðÞ
<0 and @2oþeðÞ
@S<0:
That is, the unit purchasing price of C-RU relief
supplies negatively correlates with the option price.
The higher the option price is, the lower the unit
purchasing price is. It indicates that the enterprise
faces a trade-off: a certain income from premium
and an uncertain gain because of demand uncer-
tainty. The unit purchasing price also negatively
correlates with the amount of in-kind donations.
The increase of in-kind donations leads to the
reduction of the unit purchasing price. When the
unit purchasing price oþeis less than xþh, the
cooperative relationship between the two sides will
be stopped. Therefore, the government should
receive an appropriate quantity of in-kind dona-
tions. To make Proposition 2 clearly, we illustrate
it with a numerical example. The setting of related
parameters is: U¼30, 000, h¼0:5, c¼40, h¼
0:1, v¼20, o¼5, p¼300, M¼500, x¼70,
u¼0:1, k¼0:8, S¼1000, a¼5:After calcula-
tion, when e¼197:2546, the relief supply chain
coordination can be achieved. The government
should receive up to 8885 units C-RU
relief supplies.
Proposition 3. Under C-RUP model, the govern-
ment’s optimal pre-positioned quantity of C-RU relief
supplies Q
adecreases with o and S, while the enter-
prise’s optimal pre-positioned quantity of C-RU relief
supplies qincreases with o and S:
Proposition 3 shows that the government can buy
the needed C-RU relief supplies at a low price with
an increasing option price. So the government tends
to reduce his pre-positioned quantity of C-RU relief
supplies. Since the sum of the government’s optimal
pre-positioned quantity plus the enterprise’s optimal
pre-positioned quantity under coordination is a con-
stant, the enterprise’s optimal pre-positioned quan-
tity of C-RU relief supplies will increase with the
option price. As the amount of in-kind donations
increases, the government reduces his pre-positioned
quantity of C-RU relief supplies. Since @Q
a
@S<@Q
s
@S<
0, the enterprise’s optimal pre-positioned quantity
of C-RU relief supplies will increase with in-
kind donations.
Proposition 4. Under C-RUP model, the enterprise’s
expected profit Q
aand the government’s expected
cost C
adecrease with o; the enterprise’s expected
profit Q
adecreases with S; the government’s expected
cost C
adecreases with S when 0<a<
kpþ1k
ðÞ
M
ðÞ
Ucþhv
ðÞ
2þUxþhv
ðÞ
22oU xþhv
ðÞ
þ2hSv cþhov
ðÞ
ðÞ
2hUkpþ1k
ðÞ
Mv
ðÞ
cþhov
ðÞ ,but
increases with S when
a>kpþ1k
ðÞ
M
ðÞ
Ucþhv
ðÞ
2þUxþhv
ðÞ
22oU xþhv
ðÞ
þ2hSv cþhov
ðÞ
ðÞ
2hUkpþ1k
ðÞ
Mv
ðÞ
cþhov
ðÞ :
1464 Y. LIU ET AL.
Proposition 4 shows that the increase of the option
price leads to the reduction of the enterprise’s
expected profit and the government’s expected cost.
The influence of in-kind donations on the enterprise’s
expected profit is different from that on the govern-
ment’s expected cost. This is because after the disas-
ter, the government needs to pay subsidies to the HO.
If ais at a low level, the government’s expected cost
decreases with the amount of in-kind donations.
Once aexcels a certain value, the government’s
expected cost increases with the amount of in-kind
donations. So the government should set a proper
subsidy level for the HO. We illustrate it with the
above example. The government’s maximal subsidy
level is up to 170.4986.
5.4. Comparisons between the C-RUP model and
the GSP model
When the relief supply chain is coordinated, is it
necessary for the government to adopt the C-
RUP model to pre-position C-RU relief supplies?
By the comparisons between the C-RUP model
and the GSP model, Propositions 5 and 6
are presented.
Proposition 5. Compared to the GSP model, the C-
RUP model can improve the total pre-positioned
quantity of C-RU relief supplies (Q
aþq>Q
dÞand
reduce the government’s inventory risk (Q
a<Q
d).
Proposition 5 shows that the C-RUP model is
superior to the GSP model. The increase of the total
pre-positioned quantity of C-RU relief supplies
improves the government’s emergency ability and
support capacity at pre-disaster and in the early
post-disaster. The decrease of the government’s pre-
positioned quantity of C-RU relief supplies means
that the government shifts some of inventory risk to
the enterprise.
From rational decision-making, the government
and the enterprise should be better off under the C-
RUP model than under the GSR model, satisfying
the inequation:
Q
a>Q
d
C
a<C
d
(19)
Proposition 6. Under the C-RUP model, when 0<
o<Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
S
2Uand 0<S<Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
,
the government and the enterprise will be better off.
Proposition 6 presents the condition that the C-
RUP model can achieve Pareto improvement. The
first condition is that the option price should be
controlled in a proper range after bargaining
between the government and the enterprise. The
second condition is that the government should
receive a proper quantity of in-kind donations.
To make Proposition 6 clearly, we illustrate it
with the above example. Since S¼1000 <
Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
¼3547, substituting S¼1000 into
Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
S
2U, we can get o¼7:2167:The
government’s expected cost and the enterprise’s
expected profit are depicted as in Figure 2. From
Figure 2, when 0 <o<7:2167 and 0 <S<3547,
both the government and the enterprise are willing
to conduct the option contract.
Figure 2. The government’s expected cost and the enterprise’s expected profit.
JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1465
6. The NC-UP model
Figure 3 shows the timing of the interest game
between the government and the enterprise under
the NC-UP model. Prior to a disaster, the govern-
ment buys Qbunits NC-U relief supplies at x, and
simultaneously offers an option contract to the
enterprise. The enterprise decides whether to accept
it. If the enterprise accepts it, he delivers the pri-
mary order quantity of NC-U relief supplies to the
government, and decides how many real products
and production capacity to pre-position. If a disaster
occurs, according to the sudden-onset demand of
NC-U relief supplies, the government’s and the
enterprise’s response actions include the following
situations: if 0 <x<Qb, the government will
deliver his pre-positioned NC-U relief supplies to
disaster-affected areas and the enterprise will dispose
of the unsold NC-U relief supplies; if Qb<x<
Qbþq0, the government will buy xQbunits NC-
U relief supplies from the enterprise and the enter-
prise will deliver them and dispose of the unsold
real products; if Qbþq0<x<Qbþq0þq1, the
government will buy q0units real products and the
enterprise will convert xQbq0units production
capacity into real products and deliver them to the
government before the deadline agreed in the con-
tract; if Qbþq0þq1<x<Qbþq0þq1þS, the
government will buy q0þq1units NC-U relief sup-
plies from the enterprise and receive in-kind dona-
tions; if Qbþq0þq1<x<U, the government will
buy the needed NC-U relief supplies from spot mar-
ket and the unmet demand will lead to stock-
out cost.
6.1. The enterprise’s optimal decisions
After accepting the option contract, the enterprise is
to determine how many real products and produc-
tion capacity to pre-position so as to maximize his
expected profit. From Figure 3, the enterprise’s
expected profit can be formulated as
where the first and second terms are the enterprise’s
fixed profit, the third term is the option premium
of real products and preparation cost of production
capacity, the fourth term is the salvage profit if no
disaster occurs, the last term is the portfolio of prof-
its when a disaster occurs. The first-order and the
second-order derivatives of Qbwith respect to q0
and q1are
Figure 3. The timing of the interest game under the NC-UP model.
Qb¼xc
ðÞ
Qbcþho
ðÞ
q0þom
ðÞ
q1þ1h
ðÞ
vq0þhÐQb
0vq0fx
ðÞ
dx
þðQbþq0
Qb
1u
ðÞ
eþup
ðÞ
xQb
ðÞþvQ
bþq0x
ðÞ
fx
ðÞ
dx
þðQbþq0þq1
Qbþq0
1u
ðÞ
eþup
ðÞ
q0þ1u
ðÞ
eþupc0
ðÞ
xQbq0
ðÞ
fx
ðÞ
dx
þðU
Qbþq0þq1
1u
ðÞ
eþup
ðÞ
q0þ1u
ðÞ
eþup
ðÞ
c0
ðÞ
q1Þfðxdx
(20)
1466 Y. LIU ET AL.
We get H¼
@2Qb
@q2
0
@2Qb
@q0@q1
@2Qb
@q1@q0
@2Qb
@q2
1
¼h2c0v
ðÞ
1u
ðÞ
eþupc0
ðÞ
U2>0:
By solving @Qb
@q0¼0 and @Qb
@q1¼0, we attain the
enterprise’s optimal pre-positioned quantities of real
products and production capacity:
q
0¼U1cþhmv
hc0v
ðÞ
!
Qb(22)
q
1¼U1u
ðÞ
eþupc0
ðÞ
cþhmv
ðÞ
þc0v
ðÞ
om
ðÞ
hc0v
ðÞ
1u
ðÞ
eþupc0
ðÞ
(23)
6.2. The government’s decision analysis
The government’s expected cost can be formulated as
Cb¼xþh
ðÞ
Qbþoq
0þq1
ðÞ
vQb1h
ðÞ
þhðQb
0
vðQbxÞfðxÞdx
þðQbþq0þq1
Qb
1u
ðÞ
eþup
ðÞ
xQb
ðÞfðxÞdx
þðU
Qbþq0þq1
1u
ðÞ
eþup
ðÞ
q0þq1
ðÞ
fx
ðÞ
dx
þðU
Qbþq0þq1þS
kp þ1u
ðÞ
M
ðÞ
xQbq0
ð
q1SÞfx
ðÞ
dx þaSÞ(24)
where the first term is the government’s wholesale
cost and inventory cost, the second term is the
option cost, the third term is the salvage profit if no
disaster occurs, the last term is the portfolio of costs
when a disaster occurs. When the relief supply chain
coordination is achieved, Qbþq
0þq
1is equal to
Q
s:After transformation, we get
Cb¼xþhv
ðÞ
QbþoQ
sQb
ðÞ
þhvQ
bÐQb
0Fx
ðÞ
dx
þ1u
ðÞ
eþup
ðÞ
Q
sQbðQ
s
Qb
Fx
ðÞ
dx
!
þkpþ1k
ðÞ
M
UQ
sSðU
Q
sþS
FðxÞdx
!
þaS
(25)
The first-order and the second-order derivatives
of Cbwith respect to Qbare
@Cb
@Qb
¼xþhovh1u
ðÞ
eþupv
ðÞ
1FQ
b
ðÞðÞ
(26)
@2Cb
@Q2
b
¼h1u
ðÞ
eþupv
ðÞ
fQ
b
ðÞ (27)
Owing to fQ
b
ðÞ>0, p>vand e>v, we have
@2Cb
@Q2
b
>0:So Cbis convex in Qb:The government’s
optimal pre-positioned quantity of NC-U relief sup-
plies:
Q
b¼U1xþhov
h1u
ðÞ
eþupv
ðÞ
(28)
Proposition 7. Under the NC-UP model, the govern-
ment’s optimal pre-positioned quantity of NC-U relief
supplies and the enterprise’s optimal pre-positioned
quantities of real products and production capacity of
NC-U relief supplies are
@Qb
@q0
¼cþhov
ðÞ
þh1u
ðÞ
eþupvþvc0
ðÞ
FQ
bþq0
ðÞ
1u
ðÞ
eþupc0
ðÞ
FQ
bþq0þq1
ðÞ
!
@2Qb
@q2
0
¼h1u
ðÞ
eþupv
ðÞ
U<0
@2Qb
@q0@q1
¼h1u
ðÞ
eþupc0
ðÞ
U
@Qb
@q1
¼mþoþh1u
ðÞ
eþupc0
ðÞ
1FQþq0þq1
ðÞðÞ
@2Qb
@q2
1
¼h1u
ðÞ
eþupc0
ðÞ
U<0
@2Qb
@q1@q0
¼h1u
ðÞ
eþupc0
ðÞ
U
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(21)
JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1467
Q
b¼U1xþhov
h1u
ðÞ
eþupv
ðÞ
q
0¼Uxþhov
h1u
ðÞ
eþupv
ðÞ
cþhmv
hc0v
ðÞ
!
q
1¼U1u
ðÞ
eþupc0
ðÞ
cþhmv
ðÞ
þc0v
ðÞ
om
ðÞ
hc0v
ðÞ
1u
ðÞ
eþupc0
ðÞ
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:(29)
From Proposition 7, the first condition that the
NC-UP model makes sense is h>xþhov
1u
ðÞ
eþupv:
By solving q
0>0 and q
1>0, the unit urgent pro-
duction cost should meet 1u
ðÞ
eþupv
ðÞ
cþhmv
ðÞ
xþhov
þv<c0<1u
ðÞ
eþupv
ðÞ
cþhmv
ðÞ
cþhovþv:
Proposition 8. Under the RC-UP model, the relief
supply chain coordination can be achieved
when e ¼1
1u
Umo
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
Sþc0up
:
Proposition 8 shows the condition that the relief
supply chain coordination is achieved. By solving the
first-order derivatives of oþewith respect to oand
S, we get @2oþe
ðÞ
@o<11
h1u
ðÞ
<0 and @2oþe
ðÞ
@S<0:The
unit purchasing price of NC-U relief supplies nega-
tively correlates with the option price and the quan-
tity of in-kind donations.
Proposition 9. Under the RC-UP model, the govern-
ment’s optimal pre-positioned quantity of NC-U relief
supplies Q
bdecreases with o and S; the enterprise’s
optimal pre-positioned quantity of real NC-U relief
supplies q
0increases with o and S; the enterprise’s
optimal pre-positioned quantity of production cap-
acity q
1has no concern with o, but decreases with S:
Proposition 9. shows the relationships between Q
b,
q
0,q
1and o,S:To make Proposition 9 clearly, we
illustrate it with a numerical example. The setting of
related parameters is: U¼30, 000, h¼0:5, c¼40,
h¼0:1, v¼20, o¼5, p¼300, M¼500, x¼
70, u¼0:1, k¼0:8, S¼1000, a¼5, m¼18,
c0¼30:First, assume S¼1000 and plot the
changes of Q
b,q
0and q
1with an increasing o
(0 <o<10) (see Figure 4a). Then assume o¼5
and plot the changes of Q
b,q
0and q
1with an
increasing S(0 <S<3000) (see Figure 4b).
Proposition 10. Under the RC-UP model, the enter-
prise’s expected profit Q
bdecreases with o and S; the
government’s expected cost C
bdecreases with o; the
government’s expected cost C
bdecreases with S when
0<a<ab, but increases with S when a>ab:
In Proposition 10,abis the value that makes
@C
b
@S¼0:Proposition 10 shows that as the option
price increases, the government can buy the needed
NC-U relief supplies from the enterprise at a lower
price. So the enterprise’s expected profit and the
government’s expected cost decrease with the
increase of the option price. The increase of in-kind
donations leads to the reduction of the probability
that the government purchases the needed NC-U
relief supplies. So the enterprise’s expected profit
decreases with an increasing quantity of in-kind
donations. Since the government has to pay subsi-
dies to the HO because of his effort in the collection
and delivery of in-kind donations. If the subsidy
level is at a low level, the government’s expected
Figure 4. The pre-positioned quantity of NC-U relief supplies with oand S:
1468 Y. LIU ET AL.
cost decreases with the amount of in-kind donations
because the subsidy cost is relatively low. If the sub-
sidy level excels a certain value, the sharp increase
of the subsidy cost that the government must pay
leads to the increase of the government’s expected
cost. Therefore, the government should set a proper
subsidy level for the HO. We illustrate it with the
above example. The government’s maximal subsidy
level is up to 174.2711.
Proposition 11. Compared to the GSP model, the
NC-UP model also can improve the total pre-posi-
tioned quantity of NC-U relief supplies
(Q
bþq
0þq
1>Q
dÞand reduce the government’s
inventory risk (Q
b<Q
d).
Proposition 11 shows that the NC-UP model
have an advantage over the GSP model because it
can greatly not only improve the government’s
emergency ability and support capacity, but also
shift some of the government’s inventory risk to
the enterprise.
Proposition 12. Under the RC-UP model, when 0<
o<oband 0<S<Sb, the government and the
enterprise will be better off.
In Proposition 12,obis the value that makes
Q
b¼Q
d, which is related with S:Since obshould
be more than 0, we can get the value that makes
ob¼0, which is recorded as Sb:To make
Proposition 12 clearly, we illustrate it with the above
example. The setting of related parameters is the
same as the above. We can get Sb¼2900:Since S¼
1000 <Sb¼2900, we get ob¼5:1915:The govern-
ment’s expected cost and the enterprise’s expected
profit are depicted as in Figure 5. when 0 <o<
5:1915 and 0 <S<2900, the government and the
enterprise will be better off.
7. Conclusions
In order to solve the government’s pre-positioning
dilemma, we first classify relief supplies according to
their natural characteristic and the most urgent
needs at post-disaster, and then introduce an option
contract into relief supplies pre-positioning system.
Specially, we build the C-RUP model for pre-posi-
tioning consumable, relatively urgent relief supplies,
determining the decision of how many C-RU relief
supplies to be pre-positioned by the government
and the enterprise, and simultaneously build the
NC-UP model for non-consumable, urgent relief
supplies which is used to determine the decision of
how many NC-U relief supplies to be pre-positioned
by the government, and how many real products
and production capacity to be pre-positioned by the
enterprise. The relief supply chain coordination
could be achieved with the option contract. Under
channel coordination, the C-RUP model and the
NC-UP model are superior to the GSP model
because they cannot only improve the government’s
emergency ability and support capacity, but also
reduce the government’s inventory risk. Moreover,
we give out the conditions that the government and
the enterprise are willing to conduct the option con-
tract. Lastly, we propose important managerial
insights with numerical examples and sensitivity
analysis, as follows: First of all, the C-RUP model
and the NC-UP model are superior to the GSP
Figure 5. The government’s expected cost and the enterprise’s expected profit.
JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1469
model. So it is beneficial for the decision-makers to
adopt the two models to pre-position a kind of relief
supplies. Moreover, pre-positioning a kind of relief
supplies depends on the natural characteristic of
relief supplies and the most urgent needs at post-
disaster. Specifically, the C-RUP model is mainly
designed to pre-position consumable, relatively
urgent relief supplies such as bottled water, while
the NC-UP model is mainly designed to pre-pos-
ition non-consumable, urgent relief supplies such as
tents. Third, the government should negotiate with
the enterprise for an appropriate option price, not
too big. Eventually, the government should receive
an appropriate quantity of in-kind donations, and
set a proper subsidy level for the HO.
This paper also has such disadvantages. First, the
quantity of in-kind donations is still regarded as an
exogenous variable. In the future research we will
extend our model by considering it as an endogen-
ous variable. Second, the proposed models are sin-
gle-period. We will furthermore extend them to
multiple-period models.
Acknowledgements
The authors would like to thank the associate editor and
referees for their valuable suggestions and comments,
which help us to improve this paper greatly.
Disclosure statement
No potential conflict of interest was reported by
the authors.
Funding
This work was supported by the Humanities and Social
Sciences Foundation of Ministry of Education of China
under Grant 19YJA630068 and the Key Soft Science
Research Projects in Shaanxi Province of China under
Grant 2019KRZ012.
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Appendix
Proof of Proposition 1. See Secs. 5.1 and 5.2.
Proof of Proposition 2. From Proposition 1, we get
qþQ
a¼U1xþhov
h1u
ðÞ
eþupv
ðÞ
and Q
s¼U
Ucþhv
ðÞ
þhSkpþ1k
ðÞ
M
ðÞ
hkpþ1k
ðÞ
Mv
ðÞ
. By solving qþQ
a¼Q
s, we can
obtain e¼1
1uvupþUcþhov
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
:
Proof of Proposition 3. By substituting einto Eq.
(18), the first-order derivatives of Q
aand qwith respect
to oand Sare: @Q
a
@o¼ xc
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
ðÞ
hcþhov
ðÞ
2kpþ1k
ðÞ
Mv
ðÞ
<0,
@q
@o¼xc
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
ðÞ
hcþhoc
ðÞ
2kpþ1k
ðÞ
Mv
ðÞ
>0, @Q
a
@S¼
xþhov
ðÞ
kpþ1k
ðÞ
M
ðÞ
cþhov
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
<0, @q
@S¼xc
ðÞ
kpþ1k
ðÞ
M
ðÞ
cþhov
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
>0.
Proof of Proposition 4. The enterprise’s expected
profit is
Q
a¼xc
ðÞ
Q
acþhov
ðÞ
q
þh1u
ðÞ
eþupv
ðÞ
qÐQ
aþq
Q
aFðxÞdx
(A.1)
The first-order derivative of Q
awith respect to ois
@Q
a
@o¼qþh1u
ðÞ
@e
@oqðQ
aþq
Q
a
FðxÞdx
!
(A.2)
We have qþh1u
ðÞ
@e
@oq
ðÐQ
aþq
Q
aFðxÞdxÞ<
qþh1u
ðÞ
@e
@oq1FQ
aþq
¼q1þh1u
ðÞð
@e
@o1
ðFQ
s
ðÞ
ÞÞ ¼ 0, i.e., @Q
a
@o<0. Due to C
s¼C
aQ
a,
we have @C
a
@o¼@Q
a
@o<0.
JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1471
The first-order derivative of Q
awith respect to Sis
@Q
a
@S¼h1u
ðÞ
@e
@SqðQ
aþq
Q
a
FðxÞdx
!
(A.3)
We have h1u
ðÞ
@e
@SqÐQ
aþq
Q
aFðxÞdx
<h1
ð
uÞ@e
@Sq1FQ
aþq
ðÞðÞ
¼h1u
ðÞ
@e
@Sq1FQ
s
ðÞðÞ
<0,
i.e., @Q
a
@S<0.
The first-order derivative of C
awith respect to Sis
@C
a
@S¼ xc
ðÞ
kp þ1k
ðÞ
M
xþcþ2h2o2v
ðÞ
2hcþhov
ðÞ
kp þ1k
ðÞ
Mv
Ucþhv
ðÞ
þhvS
ðÞ
kp þ1k
ðÞ
M
hUakp þ1k
ðÞ
Mv
Ukpþ1k
ðÞ
Mv
(A.4)
We get a¼kpþ1k
ðÞ
M
ðÞ
Ucþhv
ðÞ
2þUxþhv
ðÞ
22oU xþhv
ðÞ
þ2hSv cþhov
ðÞ
ðÞ
2hUkpþ1k
ðÞ
Mv
ðÞ
cþhov
ðÞ by
@C
a
@S¼0. When 0<a<kpþ1k
ðÞ
M
ðÞ
Ucþhv
ðÞ
2þUxþhv
ðÞ
22oU xþhv
ðÞ
þ2hSv cþhov
ðÞ
ðÞ
2hUkpþ1k
ðÞ
Mv
ðÞ
cþhov
ðÞ ,
the government’s expected cost decreases with the donated
quantity of relief supplies S; The government’s expected
cost increases with the donated quantity of relief supplies
Swhen akpþ1k
ðÞ
M
ðÞ
Ucþhv
ðÞ
2þUxþhv
ðÞ
22oU xþhv
ðÞ
þ2hSv cþhov
ðÞ
ðÞ
2hUkpþ1k
ðÞ
Mv
ðÞ
cþhov
ðÞ .
Proof of Proposition 5. The total pre-positioned quan-
tity of C-RU relief supplies under the C-RUP model is
Q
aþq¼U1Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
hUkpþ1k
ðÞ
Mv
ðÞ
; the pre-positioned
quantity of C-RU relief supplies under the GSP model is
Q
d¼U1Uxþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
hUkpþ1k
ðÞ
Mv
ðÞ
. Since x>cand Fx
ðÞ
and F1x
ðÞare increasing functions, we get Q
aþq>Q
d.
The government’s pre-positioned quantity of C-RU
relief supplies under the C-RUP model is
Q
a¼U1xþhov
h1u
ðÞ
eþupv
ðÞ
. Due to Q
dQ
a¼
xc
ðÞ
hkpþ1k
ðÞ
M
ðÞ
SþoU
ðÞ
hcþhov
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
>0, i.e., Q
a<Q
d.
Proof of Proposition 6. Assume Q
ad ¼Q
aQ
dand
we obtain:
Q
ad ¼xc
ðÞ
Q
aQ
d
ðÞ
cþhov
ðÞ
q
þh1u
ðÞ
eþupv
ðÞ
qÐQ
aþq
Q
aFðxÞdx
(A.5)
By solving Q
ad >0, we have
o<Ucþhv
ðÞ
hkp þ1k
ðÞ
M
S
2U(A.6)
Since Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
S
2Ushould be more than 0, we
have 0 <S<Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
.If0<o<Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
S
2U
and 0 <S<Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
, the enterprise’s expected profit
will be better off.
Assume C
ad ¼C
aC
dand we obtain:
C
ad ¼xþhv
ðÞ
Q
aQ
d
ðÞ
þoq
þhvQ
aQ
dðQ
a
Q
d
Fx
ðÞ
dx
!
þkp þ1k
ðÞ
M
ðQ
sþS
Q
dþS
Fx
ðÞ
dx Q
sþQ
d
!
þ1u
ðÞ
eþup
ðÞ
ðqðQ
s
Q
a
Fx
ðÞ
dxÞÞ (A.7)
The first-order derivative of C
ad with respect to ois
@C
ad
@o¼qþh1u
ðÞ
@e
@oqðQ
s
Q
a
Fx
ðÞ
dx
!
(A.8)
Since qþh1u
ðÞ
@e
@oq
ðÐQ
s
Q
aFx
ðÞ
dxÞ<q1þh1
ðð
uÞ@e
@o1FQ
s
ðÞðÞ
Þ¼ 0, we have @C
ad
@o<0. Hence when
o¼0, the government’s expected cost is maximal, as follows:
C
ad ¼xþhv
ðÞ
Q
aQ
d
ðÞ
þoq
þhvQ
aQ
dðQ
a
Q
d
Fx
ðÞ
dx
!
þkp þ1k
ðÞ
M
ðQ
sþS
Q
dþS
Fx
ðÞ
dx Q
sþQ
d
!
þvþUcþhv
ðÞ
kp þ1k
ðÞ
Mv
Ucþhv
ðÞ
þhkp þ1k
ðÞ
M
S
!
qðQ
s
Q
a
FðxÞdxÞ
! (A.9)
Due to @C
admax
@S<0, S¼0 maximizes C
admax. So we get
C
admax <xþhv
ðÞ
Q
aQ
d
ðÞ
þhvkp 1k
ðÞ
M
Q
aQ
dþÐQ
d
Q
aFx
ðÞ
dx
¼0, i.e., C
admax <0. To sum up,
when 0 <o<Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
S
2Uand 0 <S<
Ucþhv
ðÞ
hkpþ1k
ðÞ
M
ðÞ
, the enterprise and the government will be
better off.
Proof of Proposition 7. See Secs. 6.1 and 6.2.
Proof of Proposition 8. Given q
0þq
1þQ
b¼
U1mo
h1u
ðÞ
eþupc0
ðÞ
and Q
s¼UUcþhv
ðÞ
þhSkpþ1k
ðÞ
M
ðÞ
hkpþ1k
ðÞ
Mv
ðÞ
.By
solving q
0þq
1þQ
b¼Q
s, we can obtain
e¼1
1u
ðÞ
Umo
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
Sþc0up
:
Proof of Proposition 9. The first-order derivatives of
Q
b,q
0, and q
1with respect to oand Sare: @Q
b
@o¼
Uc
0v
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
ðÞ
U2kpþ1k
ðÞ
Mv
ðÞ
xþhmv
ðÞ
hvc0Umo
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
2
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
ðÞ
<
U2kpþ1k
ðÞ
Mv
ðÞ
xc
ðÞ
hvc0Umo
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
2
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
ðÞ
<0,
@q
0
@o¼
@Q
b
@o>0, @Q
b
@S¼ Ukpþ1k
ðÞ
M
ðÞ
mo
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
xþhov
ðÞ
hvc0Umo
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
2
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
ðÞ
<0,
@q
0
@S¼
@Q
b
@S>0, @q
1
@o¼0, @q
1
@S¼ kpþ1k
ðÞ
M
kpþ1k
ðÞ
Mv<0.
Proof of Proposition 10. The enterprise’s expected
profit is
Y
b¼xc
ðÞ
Q
bcþhov
ðÞ
q
0þom
ðÞ
q
1
þh1u
ðÞ
eþupv
ðÞ
q
0ðQ
bþq
0
Q
b
Fx
ðÞ
dx
!
þ1u
ðÞ
eþupc0
ðÞ
q
1ðQ
bþq
0þq
1
Q
bþq
0
FðxÞdx
!
(A.10)
1472 Y. LIU ET AL.
The first-order derivative of Q
bwith respect to ois
@Q
b
@o¼q
0þq
1þh1u
ðÞ
@e
@oq
0þq
1ðQ
bþq
0þq
1
Q
b
FðxÞdx
!
(A.11)
We then have q
0þq
1þh1u
ðÞ
@e
@oq
0þq
1
ð
ÐQ
bþq
0þq
1
Q
bFðxÞdxÞ<q
0þq
1þh1u
ðÞ
@e
@oq
0þq
1
ðÞ
1
ð
FQ
bþq
0þq
1
ðÞ
Þ¼ q
0þq
1þh1u
ðÞ
@e
@oq
0þq
1
ðÞ
1FQ
s
ðÞðÞ
¼0 i.e., @Q
b
@o<0.
The first-order derivative of Q
bwith respect to Sis
@Q
b
@S¼h1u
ðÞ
@e
@Sq
0þq
1ðQ
bþq
0þq
1
Q
b
FðxÞdx
!
(A.12)
Similarly, we have h1u
ðÞ
@e
@Sq
0þq
1ÐQ
bþq
0þq
1
Q
b
FðxÞdxÞ<h1u
ðÞ
@e
@Sq
0þq
1
ðÞ
1FQ
bþq
0þq
1
ðÞðÞ
¼h1u
ðÞ
@e
@Sq
0þq
1
ðÞ
1FQ
s
ðÞðÞ
<0, i.e., @Q
b
@S<0.
The first-order derivative of C
bwith respect to ois
@C
b
@o¼q
0þq
1þh1u
ðÞ
@e
@oq
0þq
1ðQ
bþq
0þq
1
Q
b
FðxÞdx
!
(A.13)
We then have q
0þq
1þh1u
ðÞ
@e
@oq
0þq
1
ð
ÐQ
bþq
0þq
1
Q
bFðxÞdxÞ<q
0þq
1þh1u
ðÞ
@e
@oq
0þq
1
ðÞ
1
ð
FQ
b
ðþq
0þq
1ÞÞ¼ 0, i.e., @C
b
@o<0.
The first-order derivative of C
bwith respect to Sis @Cb
@S.
By solving @Cb
@S¼0, we can get the analytic expression of
a. Since it is extremely complicated, we use abto denote
it. When 0 <a<ab,@Cb
@S<0, the government’s expected
cost decreases with S. When a>ab,@Cb
@S>0, the govern-
ment’s expected cost increases with S.
Proof of Proposition 11. The total pre-positioned
quantity of NC-U relief supplies under the NC-UP model
is Q
bþq
0þq
1¼U1Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
hUkpþ1k
ðÞ
Mv
ðÞ
; the pre-posi-
tioned quantity of NC-U relief supplies under the GSP
model is Q
d¼U1Uxþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
hUkpþ1k
ðÞ
Mv
ðÞ
. Since x>c
and Fx
ðÞ and F1x
ðÞ are increasing functions, we get
Q
bþq
0þq
1>Q
d. The government’s pre-positioned
quantity under the NC-UP model is
Q
b¼U1xþhov
h1u
ðÞ
eþupv
ðÞ
. Due to Q
dQ
b¼
xþhov
h1u
ðÞ
eþupv
ðÞ
Uxþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
S
hUkpþ1k
ðÞ
Mv
ðÞ
>xc
ðÞ
hkpþ1k
ðÞ
M
ðÞ
SþoU
ðÞ
hcþhov
ðÞ
kpþ1k
ðÞ
Mv
ðÞ
>0,
i.e., Q
b<Q
d.
Proof of Proposition 12. Assume Q
bd ¼Q
bQ
d
and we obtain:
Q
bd ¼xc
ðÞ
Q
aQ
d
ðÞ
cþhov
ðÞ
q
0þom
ðÞ
q
1
þh1u
ðÞ
eþupv
ðÞ
q
0ðQ
bþq
0
Q
b
Fx
ðÞ
dx
!
þ1u
ðÞ
eþupc0
ðÞ
q
1ðQ
bþq
0þq
1
Q
bþq
0
FðxÞdx
!
(A.14)
By solving Q
bd >0, we can get the analytic expression
of o. Since it is extremely complicated, we use obto
denote it. Since obshould be more than 0, we can get the
value that makes ob¼0, which is recorded as Sb.Soif
0<o<oband 0 <S<Sb, we have Q
bd >0.
AssumeC
bd ¼C
bC
dand we obtain:
C
bd ¼xþhv
ðÞ
Q
bQ
d
ðÞ
þoq
0þq
1
ðÞ
þhvQ
bQ
dðQ
b
Q
d
Fx
ðÞ
dx
!
þkp þ1k
ðÞ
M
ðQ
sþS
Q
dþS
Fx
ðÞ
dx Q
sþQ
d
!
þ1u
ðÞ
eþup
ðÞ
q
0þq
1ðQ
s
Q
b
FðxÞdx
!
(A.15)
The first-order derivative of C
bd with respect to ois
@C
bd
@o¼Q
sQ
bþh1u
ðÞ
@e
@oQ
sQ
bðQ
s
Q
b
Fx
ðÞ
dx
!
(A.16)
Owing to Q
sQ
bþh1u
ðÞ
@e
@oQ
sQ
b
ðÐQ
s
Q
b
Fx
ðÞ
dxÞ<0, we have @C
bd
@o<0. Hence when o¼0, the gov-
ernment’s expected cost is maximal, as follows:
C
bd ¼xþhv
ðÞ
Q
bQ
d
ðÞ
þoq
0þq
1
ðÞ
þhvQ
bQ
dðQ
b
Q
d
Fx
ðÞ
dx
!
þkp þ1k
ðÞ
M
ðQ
sþS
Q
dþS
Fx
ðÞ
dx Q
sþQ
d
!
þc0þUmo
ðÞ
kp þ1k
ðÞ
Mv
Ucþhv
ðÞ
þhkp þ1k
ðÞ
M
S
!
q
0þq
1ðQ
s
Q
b
FðxÞdx
!
(A.17)
By substituting c0¼Ukpþ1k
ðÞ
Mv
ðÞ
cþhvm
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
ðÞ
Sþvinto
Eq. (A. 17), we get
C
bdmax ¼xþhv
ðÞ
Q
bQ
d
ðÞ
þoq
0þq
1
ðÞ
þhvQ
bQ
dðQ
b
Q
d
Fx
ðÞ
dx
!
þkp þ1k
ðÞ
M
ÐQ
sþS
Q
dþSFx
ðÞ
dx Q
sþQ
d
þUkpþ1k
ðÞ
Mv
cþhv
ðÞ
Ucþhv
ðÞ
þhkpþ1k
ðÞ
M
Sþv
!
q
0þq
1ðQ
s
Q
b
Fx
ðÞ
dx
!
(A.18)
Owing to @C
bdmax
@S<0, S¼0 maximizes C
bdmax.Sowe
get: C
bdmax <xþhv
ðÞ
Q
bQ
d
ðÞ
þhvkp
ð
1k
ðÞ
MÞQ
bQ
dþÐQ
d
Q
bFx
ðÞ
dx
¼0, i.e., C
bdmax <0.
To sum up, when 0 <o<oband 0 <S<Sb, the enter-
prise and the government will be better off.
JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1473
