J. Service Science & Management, 2010, 3: 117-129
doi:10.4236/jssm.2010.31015 Published Online March 2010 (http://www.SciRP.org/journal/jssm)
117
A Review on Deteriorating Inventory Study
Ruxian Li1, Hongjie Lan1*, John R. Mawhinney2
1School of Economics and Management, Beijing Jiaotong University, Beijing, China; 2Business School, Duquesne University,
Pittsburgh, USA.
Received September 2nd, 2009; revised October 21st, 2009; accepted November 29th, 2009.
ABSTRACT
To provide a comprehensive introduction about the deteriorating items inventory management research status, this pa-
per reviews the recent studies in relevant fields. Compared with the extant reviews (Raafat 1991; S.K.Goyal 2001), this
paper reviews the recent studies from a different perspecti ve. First, this paper proposes some key facto rs which should
be considered in the deteriorating inventory studies; then, from the perspective of study scope, the current literatures
are distinguished into two categories: the studies based on an enterprise and those based on supply chain. Literatures
in each category are reviewed according to the key factors mentioned above. The literature review framework in this
paper provides a clear overview of the deteriorating inventory study field, which can be used as a starting point for
further study.
Keywords: Deteriorating Items, Inventory Model, Literature Review
1. Introduction
Deteriorating items are common in our daily life; how-
ever, academia has not reached a consensus on the defini-
tion of the deteriorating items. According to the study of
Wee HM in 1993 [1], deteriorating items refers to the
items that become decayed, damaged, evaporative, ex-
pired, invalid, devaluation and so on through time. Ac-
cording to the definition, deteriorating items can be clas-
sified into two categories. The first category refers to the
items that become decayed, damaged, evaporative, or
expired through time, like meat, vegetables, fruit, medi-
cine, flowers, film and so on; the other category refers to
the items that lose part or total value through time be-
cause of new technology or the introduction of alterna-
tives, like computer chips, mobile phones, fashion and
seasonal goods, and so on. Both of the two categories
have the characteristic of short life cycle. For the first
category, the items have a short natural life cycle. After a
specific period (such as durability), the natural attributes
of the items will change and then lose useable value and
economic value; for the second category, the items have a
short market life cycle. After a period of popu larity in the
market, the items lose the original economic value due to
the changes in consumer preference, product upgrading
and other reasons.
The inventory problem of deteriorating items was first
studied by Whitin [2], he studied fashion items deterio-
rating at the end of the storage period. Then Ghare and
Schrader [3]concluded in their study that the consump-
tion of the deteriorating items was closely relative to a
negative exponential function of time. They proposed the
deteriorating items inventory model as stated below:
() () () dI t
I
tft
dt
 
In the function,

stands for the deteriorating rate of
the item, ()
I
t refers to the inventory level at time t and
then ()
f
t is the demand rate at time t. This inventory
model laid found ations for the follow-up study. Raafat [4]
and Goyal and Giri [5] made comprehensive literature
reviews on deteriorating inventory items in 1991 and
2001 respectively.
From a different perspective, this paper reviews the
recent trends in deteriorating inventory studies. Figure 1
presents the factors used to analyze and organize this
review. From the perspective of scope, we make a dis-
tinction between the studies which focus on the deterio-
rating items inventory study in a single enterprise from
those studies whose focus is on studying the deteriorating
items inventory problems across a supply chain. The
former is the focus of the early stage in the deteriorating
items study and the latter now is attracting more and
more attention from the researchers. From the perspective
of the factors which should be taken into consideration in
deteriorating items inventory study, we involve the im-
portant factors such as demand, deteriorating rate and

A Review on Deteriorating Inventory Study 118
Figure 1. Deteriorating items inventory literature and its
relation i n the re view
other factors such as price discount, allow shortage or not,
inflation, time-value of money and so on in our study. By
way of integrating different factors in different scope (in
a single enterprise or across a supply chain), different
model s can b e e s tablished.
This paper is organized as follow. Section two dis-
cusses key factors which should be taken into considera-
tion in the deteriorating inventory study. Section three
presents an overview on deteriorating items inventory
study in a single enterprise. For the purpose of classifica-
tion, three lines of research can be distinguished: the first
and the second lines study the inventory problems in the
enterprises which sell deteriorating items. The main dif-
ference between the two lines is that the first line in-
volves a warehouse while the second one involves two
warehouses. The third line reviews the produce and in-
ventory problems in the deteriorating items manufactur-
ers. The studies on the deteriorating items inventory
problems from the perspective of the supply chain are
sketched in section four. Then this paper discusses the
directions of the future research.
2. Key Factors in the Deteriorating
Inventory Study
Factors such as demand, deteriorating rate, and so on
should be taken into consideration in the deteriorating
inventory study. Among them, demand acts as driving
force of the entire inventory system and the deteriorating
rate stands for the characteristics of the deteriorating
items. Other factors like price discount, allow shortage or
not, inflation, and the time-value of money are also im-
portant in the study of deteriorating items inventory. By
making different combinations of these factors stated
above, we can get different inventory models.
2.1 Demand
Acting as the driving force of the whole inventory system,
demand is a key factor that should be taken into consid-
eration in an inventory study. There are mainly two cate-
gories demands in the present studies, one is determinis-
tic demand and the other is stochastic demand. Constant
demand [6,7], time-dependent demand [8,9] inventory
level-dependent demand [10–13] and price-dependent
demand [14] are all deterministic demand. Among them,
ramp type demand is a special type of time-dependent
demand. Hill [15] was the first to introduce the ramp type
demand to the inventory study. Th en Mand al and Pal [16]
introduced the ramp type demand to the inventory study
of the deteriorating items. After that, many researchers
have extensively studied this type of demand [17,18].
Stochastic demand includes two types of demands: the
first type characterized by a known demand distribution
and on the contrary the second type characterized by ar-
bitrary dem and dist ri b ut i o n.
2.2 Deteriorating Rate
Deteriorating rate is another key factor in the study of
deteriorating items inventory, which describes the dete-
rioration nature of the items. When it comes to the study
of deteriorating rate, there are several situations. In the
early stage of the study, most of the deteriorating rates in
the models are constant, such as Ghare and Schrader [3],
Shah and Jaiswal [19], Aggarwal [20], Padmanabhana
and Vratb [21], and Bhunia and Maiti [22]. In recent re-
search, more and more studies have begun to consider the
relationship between time and deteriorating rate. In this
situation there are several scenarios; including deterio-
rating rate is a linear increasing function of time [23,24],
deteriorating rate is two-parameter Weibull distributed
[25,26], deteriorating rate is three-parameter Weibull dis-
tributed [27], and deteriorating rate is other function of
time [28].
2.3 Other Factors
Beside demand and deteriorating rate, other factors like
price discount, allow shortage or not, inflation, and
time-value of money are also important factors. Price
discount is an important strategy which the seller always
uses to encourage the buyer to purchase in large quanti-
ties; many researchershave taken this factor into consid-
eration in deteriorating items inventory modeling. Al-
lowing shortage or not is another factor which research-
ers always focus on. Among them, some studies sup-
posed that shortage is not allowed [29], the rest supposed
that shortage is allowed and then corresponding inventory
strategy can be made according to the two assumptions
respectively. In fact, shortages usu ally happen in our dai-
ly life and what’s more, in the circumstance of high dete-
riorating rate, the demand may need to be backlogged to
reduce cost due to deterioration, so there are more studies
that concentrate on the assumption that shortage is al-
lowed. There are two cases when dealing with the short-
age, one case supposes that the shortage items are totally
backlogged [30,31] and the other case supposes that the
shortage items are partly backlogged, that is to say, the
customers are only willing to accept part of the items that
are out of stock this period and can only be supplied by

A Review on Deteriorating Inventory Study 119
the seller in the next period. There are some different
considerations in dealing with the backlogging function.
In some of the studies, the backlogging function was as-
sumed to be closely relative to the amount of demand
backlogged [32], therefore the more the amount of de-
mand backlogged, the smaller the demand to accept
backlogging would be. While in our real life, for deterio-
rating items with short life time, the waiting time for the
next replenishment is the main factor for deciding
whether the backlog will be accept or not and when we
are willing to accept the backlog, what is the accepting
proportion? It is easy for us to know that the willingness
of a customer to wait for backlogged items during a
shortage period declines with the length of the waiting
time. In order to reflect this phenomenon, some studies
[33–35] developed inventory models in which the back-
logging rate is a function of waiting time.
Trade credit is another factor. Trade credit (permissible
delay in payment) is a widely used business strategy. To
suppliers, trade credit helps to expand sale but it also
adds to the risk of bad debts at the same time. To buyers,
trade credit provides a very big advantage, due to the fact
that they do not have to pay the seller immediately after
receiving the items, but instead can delay their payment
until the end of the allowed period. The buyer pays no
interest during the agreed time for payment, but it in-
creases the risk of deterioration cost when large amounts
are purchased. However, both sellers and buyers need to
reach trade-off between the advantages and disadvan-
tages they can get from this inv entory policy. At present,
there are three kinds of trade credit, in the first case, the
length of th e credit period is fixed [36–38 ], in the second
case, the length of the credit period is closely relative to
the ordering quantity [39–41]. In the third case, the sell-
ers provide trade credit to part of the ordering quantity
and the purchase cost of the rest quantity should be paid
immediately after receiving the items.
Inflation and time value of money have also attracted
attention of researchers. Taking the two factors into con-
sideration is of vital importance. With the integration of
the global economy, the economic relationships among
countries are closer and the mutual influences are deep.
Currency’s purchasing power will change from time to
time and inflation should not be neglected. Many re-
searchers [14,44–47] have made complementally com-
prehensive consideration in deteriorating items modeling.
3. Deteriorating Items Inventory Study in a
Single Enterprise
Inventory cost is an important part of the enterprise op-
eration cost. For deteriorating items, especially those with
high deteriorating rate, deterioration is a key characteris-
tic and its impact on modeling of inventory systems can-
not be neglected. So the deterioration rate should be tak-
en into consideration in the development of inventory
strategy. For different kinds of enterprises, the emphasis
on the deteriorating items inventory study is different.
For the seller of deteriorating items, the current studies
can be divided into two types; the first type emphasizes
the inventory strategies for the retailer of the deteriorat-
ing items, the second type focuses on the inventory pol-
icy under a two-warehouse system. For the manufactures
of deteriorating items, the current emphasis is on devel-
oping an optimal production-inventory strategy. So, this
paper has divided the present studies on deteriorating
items inventory for a single enterprise into three catego-
ries as stated above.
3.1 Deteriorating Inventory Strategy
The sellers of deteriorating items such as retailers must
frequently assess the replenishment strategy for deterio-
rating items. Deteriorating items have a shorter life time
compare with other items. In order to avoid the loss due
to damage or expiration, it is of vital importance to de-
velop a proper inventory strategy. Many researchers have
conducted extensive studies on the deteriorating inven-
tory strategy. Compared with normal inventory models,
besides demand, deteriorating rate is another key factor
that impacts significantly on an inventory management
system. So when discussing models below, this paper
will mainly concentrate on demand and de terior ating rate,
and other factors, such as price discount, inflation, and so
on, will also be taken into consideration.
3.1.1 Deterministic Demand
Sarker, Jamal and Wang [48], Chang [49], Chung and
Liao [50], Huang and Liao [51]), ZHANG, DAI and
HAN [42], ZHANG, DAI, HAN and LI [52] all devel-
oped inventory models in which both the demand and
deteriorating rate are constant. In their research, the buyer
is allowed a delay period to pay for the items purchased.
The purpose of their studies is to help the buyers to make
economic inventory strategy decisions under the influ-
ence of trade credit. Although the constant demand as-
sumption helps to simplify the prob lem, it is far from the
actual situation where demand is always in change. In
order to make research more practical, many researchers
have studied other forms of demand. Among them,
time-dependant demand has attracted considerable atten-
tion. WANG Sheng-dong and WANG jun-ping [53] de-
veloped a model to determine optimal ordering policy for
deteriorating items under inflation, partial backlogging,
and time-dependant demand. The effect of the time value
of money was also considered in the paper. Panda, et al.
[54] discussed an inventory model for a seasonal product.
In the model, the demand rate is represented by a
ramp-type time dependent function and the deteriorating
rate is a constant. With the assumption that shortages are
not allowed, the paper aimed to develop an optimal re-
plenishment policy for retailer. In addition, Papachristos
and Skouri [55], Chu and Chen [56], Khanra and Chaud-

A Review on Deteriorating Inventory Study 120
huri [57], Yang [58], Dye et al. [59] all conducted re-
search on deteriorating items inventory under the premise
that the demand is time-dependent.
Besides time, the inventory level is another factor that
has a close relationship with demand. For some item
types, the demand rate may fluctuate with the on-hand
stock level. It is a common phenomenon that a large
amount of goods displayed in the supermarket will lead
the customers to buy more, so the demand is closely de-
pendant on the inventory level. Gupta and Vrat [60] first
proposed an inventory model in 1986 in which demand
rate is stock-dependent. Then Mandal and Phaujdar [61]
developed a production-inventory model for deteriorating
items with the assumption that demand is a linear func-
tion of inventory level. Padmanabhan and Vrat [62] de-
veloped an EOQ model for deteriorating items with
stock-dependant demand, they proposed three models:
completely backlogging, partial backlogging and without
backlogging, for the purpose of maximizing profit. In this
paper, the backlogging function was assumed to be
closely relative to the amount of demand backlogged. As
we discussed before, in real life it is more common that
the willingness of a customer to wait for a backlog during
a shortage period declines with the length of the waiting
time, that is to say, the longer the waiting time is, th e less
likely the customers willing to wait. Then in 2005, Dye
and Ouyang [63] extend Padmanabhan’s model (Padma-
nabhan and Vrat, 1995) by proposing a time-proportional
backlogging rate. While the opportunity cost due to lost
sales was not taken into consideration in Padmanabhan’s
model, Chung-Yuan Dye took consid erable consideration
in his improved model. Hou [64] established an inventory
model for deteriorating items with stock-dependent de-
mand rate and shortages under inflation and time dis-
counting. There are also some who studied other types of
demand, like price-dependent demand [65–67], infla-
tion-dependent demand [68] and so on.
In fact, the real situation is complex and the demand is
always affected by several factors such as time, inventory
level, price, and so on. Balkhi and Benkherouf [69], Pal
et al [70], Hsu et al [71] combined several of the factors
together and considered the impact of the combination on
the demand, in this premise the optimal inventory strat-
egy was discussed. WEN et al [72] developed an inven-
tory model in which the demand is affected by time and
inflation. The aim of the paper is to determine the optimal
replenishment strategy including the replenishment time
and order quantities which minimize the present value of
the total cost.
All the models discussed above have the characteristics
of a constant deteriorating rate. While the constant rate
simplifies the problems, it cannot reflect the real situatio n
of the deterioration; in fact the deteriorating rate varies
with time. Wee and Law [73] proposed a deteriorating
inventory model which took into consideration the
time-value of money and price-dependent demand. Pa-
pachristos and Skouri [74] considered a model where the
demand rate is a function of the selling price and the
backlogging rate is a time-dependent function. Both of
them assumed that the deteriorating rate is two-parameter
Weibull distributed. Dye et al [75] developed an inven-
tory model for deteriorating items with price-dependent
demand. In the model the deterioration rate is a function
of time and shortages are allowed. The unsatisfied de-
mand is partially backlogged and the backlogging rate is
a negative exponential rate of the waiting time. The aim
of the model was to maximize the profit, and according to
the model, the inventory strategy, including price, eco-
nomic ordering quantity and so on, can be determined.
Then Dye et al [76] took the time-value of money into
consideration and extended their earlier research. Chang,
et al [77] established an EOQ model for deteriorating
items. In the model the supplier provided trade credit to
the purchaser on the cond ition that the order qu antity was
greater than or equal to the predetermined quantity. The
paper proposed Taylor’s series approximation to obtain
the optimal order quantity and replenishment time. Liao
[78] then amended the solution for the model proposed
by Chun-Tao Chang, etc and developed an alternative
approach to determine the optimal ordering policy.
HUANG Wei-lai and HUANG song [79]studied the eco-
nomic order quantity problem with deteriorating items
taking time-value of money into consideration. In the
problem, the deteriorating rate was two-parameter Wei-
bull distributed and the demand rate is linear function of
the inventory level. An optimal order quantity mod e l with
deteriorating items based on the minimization of total
cost was established, taking account of time value of
money.
3.1.2 Stochastic Demand
In fact, with the growing uncertainty in the modern busi-
ness environment, the assumption of deterministic de-
mand is far from truth, so stochastic demand has attracted
more and more attentions.
It is common that when dealing with the stochastic
demand, a large number of researchers make the assump-
tion that the demand is Poisson distributed. S. KAL-
PAKAM and S.SHANTHI [80,81] analyzed inventory
systems with Poisson demands for deteriorating items.
With the assumption of constant demand, in solving the
model, Zhu [82] proposed a data processing method in
which the demand was accumulated, making the model
easily solved by computer and then enhancing the practi-
cality of the model. Li et al. [83] introduced the
three-parameter weibull function to describe the charac-
teristics of deterioration. In this paper, demand is sup-
posed to be closely relative to d eterioration, that is to say,
the more items deteriorating, the less the demand will be.
Beside this, shortages were allowed and totally back-
logged. According to the model, the optimal inventory

A Review on Deteriorating Inventory Study 121
policy including the replenishment time, order quantity
can be determined.
With the increasing complexity in the business envi-
ronment, it is of vital importance for enterprises to take
factors such as demand, price discounting, inflation,
time-value of money, and so on into consideration in in-
ventory decision making. The current studies have given
considerable con sideration to the factors stated above and
so they play an important role in the inventory deci-
sion-making.
3.2 Two Warehouse Deteri orating Inventor y Study
Two-warehouse inventory problem for deteriorating
items is a main category in the current inventory study
for deteriorating items. The storage capacity of an en-
terprise is limited and in some real life situations,
when the supplier provides a price discount for bulk
purchases, or when the item is a seasonal product, or
when the order cost is high, or the demand goes up
quickly, or the cost of out-stock is high, the buyer may
purchase a large amount of an item [84]. When the
amount of the purchased items exceeds the storage
capacity of the enterprise’s own warehouse (OW), the
excess quantities have to be stored in a rented ware-
house (RW). This is the real background of the two-
warehouse inventory problem. There are some com-
mon assumptions in the two-warehouse inventory
problem. It is often believed that the storage capability
of the OW is limited but the storage capability of the
rented warehouse (RW) is unlimited, and that the RW
often provides a better preserving facility than the OW.
So compared with the OW, the inventory cost in RW is
higher but the deterioration rate is lower. According to
these assumptions, it is easy for us to understand that
in order to reduce the inventory cost; it will be a good
choice to consume the items in the RW first and store
items in OW before RW (last in first out policy, LIFO).
A large number of researchers have studied in this area
according to the assumptions stated above, such as
[85], yang, S.L. and Zhou, Y.W. [86], Benkherouf [87]
and so on.
Sarma [88] first studied the two-warehouse inventory
problem for deteriorating items. In his study, the deterio-
rating item was first stored in the OW and the excess
quantity was stored in the RW. Then in the inventory
model an infinite replenishment rate was considered and
shortages were allowed. Then Pakkala and Achary [89]
extended Sarma’s study to the situation of an infinite
replenishment rate. Yang [90] considered a two-ware-
house inventory problem for deteriorating items with
constant demand rate and shortages under the circum-
stances of inflation. Then Yang [91] extended the former
study and considered partial backlogging in the two–
warehouse inventory model, in which the backlogging
rate is closely relative to the waiting time. Then Hui-Ling
Yang compar ed th e two two-w ar ehou s e inven tory mod e ls
based on the minimum cost approach and the result
showed that the model 2 was less expensive than model 1
if partial backlogging and inflation are considered. QIU
and LIANG [92]) developed a two-warehouse inventory
model on the basis of minimum cost. In the model the
factors such as a constant deteriorating rate and demand
trade credit were taken into cons ideration. The numerical
results can aid the manager to decide the order quantity,
order cycle, and so on.
All the two-warehouse inventory models discussed
above have the characteristic of constant demand; in fact
many other researchers have studied other types of de-
mand, like time-dependent demand, etc. XU and LI [93]
proposed an optimal inventory policy for a two-ware-
house inventory model with time-dependent demand and
constant deterioration rate. In the model, shortages are
not allowed and according to the model the total order
quantity and the quantity stored in the rented warehouse
in a cycle time can be determined. Dey et al. [94] studied
the two-warehouse inventory problem from the perspec-
tive of retailers. In their study, the storage capability of
OW which is always located at a busy market place is
limited. So the excess items should be stored in a RW
which may be located away from the market place. The
paper assumed that the inventory cost in the RW is
greater than OW and so the items are first stored in OW
and only excess items are stored in RW. This is the same
as we discussed before. In order to reduce the inventory
cost, the RW was emptied first by transporting the stock
from RW to OW in a continuou s release patt ern. That is to
say, the demand of items was met at OW only. Then the
paper developed two-warehouse inventory model with the
consideration of inflation, time-dependent demand, par-
tially backlogging, and time-value of money. Rong et al.
[95] studied the sim ilar probl em as Dey et al. [94] did.
As we stated at the beginning of 3.2, in order to reduce
inventory cost, a considerable number of studies on two-
warehouse inventory problems for deteriorating items
follow the rule of LIFO (last in first out). But Lee [96]
held a different opinion in this respect; he believed th at in
the RW, especially in the public warehouse, the vendor of
the warehouse carried a lower operating cost because of
well equipment set ups, learning effect of trained workers,
and economies of scale. What’s more, in the buyer’s
market more and more warehouses have to offer valued
added service with completive lower prices to attract
customers. So the assumption that the holding costs are
higher in the rented warehouses is not so reasonable. Be-
sides all stated above, the researcher still believ ed that for
deteriorating items, under the rule of LIFO, the cost of
item deterioration and related opportunity cost may far
exceed the cost saving benefit derived from the rent (on
the condition that the holding costs in the rented ware-
houses are really higher than those in the OW). So it may

A Review on Deteriorating Inventory Study 122
be more reasonable to operate under the rule of FIFO. In
his study, Chun Chen Lee reconsidered the LIFO model
proposed by Pakkala and Achary [97] and then developed
a FIFO model. Comparison of the two models showed
that the FIFO model was less expensive than the LIFO
model, on the condition that the mixed effects of deterio-
ration and holding cost in RW are less than that of the
OW. Niu and Xie [98]then amended the model proposed
by Chun Chen Lee and so the modified LIFO model al-
ways had a lower cost than the FIFO model proposed by
Pakkala and Achary.
It is a common phenomenon that seasonal or cyclical
items are often purchased in large quantities and so the
study of two-warehouse inventory is of important practi-
cal significance for these items.
3.3 Production-Inventory Study for
Deteriorating Items
For deteriorating items manufacturing, decision makers
should take inventory into consideration in production
policy making. Proposing an efficient production-inventory
decision is conducive to enhancing the competitiveness
of enterprises [43]. Many researchers have extensively
studied in this area, such as Goyal and Gunasekaran [99],
Jiang and Du [100], Chen et al. [101], Gong and WANG
[102], Maity et al. [103] and so on. With the assumption
that the demand rate, production rate, and deteriorating
rate are all constant, Jui-Jung Liao [104,105] established
a production-inventory model for deteriorating items un-
der the condition of the supplier providing the retailer
with trade credit. Teng and Chang [106] established an
economic production quantity (EPQ) model for deterio-
rating items when the demand rate was dependant on
both the stock level and the selling price per unit, and in
which the deteriorating rate was constant. The paper also
provided the necessary conditions to decide the optimal
solution which maximized profit of the EPQ model. From
the model, decision makers could get aid in developing
the optimal price and in production run time decisions.
In all the models stated above, shortages are not al-
lowed. In conditions of shortages, each production- in-
ventory cycle in the planning horizon can be described in
more detail. Generally speaking, each production- inven-
tory cycle can be divided into four stage, there are pro-
duction starts——con sumption stage; production stops—
—consumption stage; production stops——shortage
stage, and production starts——shortage stage. The order
of the four stages depends mainly on the shortage occur-
ring at the beginning or the end of the cycle. Lin Hsinyi
et al. [107] established a production——inventory model
with constant production rate, demand rate and deterio-
rating rate. Shortages are allowed and occur at the start of
each cycle. So the production——inventory cycle can be
divided into production stops——shortage stage; produc-
tio n starts——shortage stage; product ion —— con su mpt io n
stage; production stops——consumption stage. From the
model, we can get the number of production-inventory cy-
cles and the starting point of the four stages in every cy-
cle. Incorporating these parameters with the known con-
stant production rate, we can get the optimal produc-
tion-inventory policy. Zhou et al. [108] also considered
production-inventory problem in which each cycle of a
production-inventory schedule starts with replenishment
and ends with a shortage. In this model, time-dependent
demand was taken into consideration. In the research of
Sana et al. [109] and Zhou and Gu [110], shortages are
allowed and occur at the end of a cycle. So the cycle con-
sists of four stages as below: production starts——con-
sumption stage; production stops——consumption stage;
production stops——shortage stage and production st a rt s —
—shortage stage. With the consideration of time variable
demand and constant deteriorating rate, the optimal pro-
duction-in ventory policy was studi e d.
All the deteriorating rates in the models stated above
are constant and some researchers have studied time-
dependent deteriorating rates. With the time-varying de-
mand and deteriorating rate, both Skouri and Papachris-
tos [111] and Chen et al. [112] developed a produc-
tion-inventory model in which the storages are allowed at
the beginning of the cycle. In contrast, in Manna and
Chaudhuri [114] and Balkhi’s research [113], shortages
are also allowed but occur at the end of each cycle. Abad
[116] studied the pricing an d lot-sizing problem for dete-
riorating items under the conditions of finite production,
partial backlogging, and lost sale. Teng et al [115]extend
P.L. Abad’s model by adding the backlogging cost and
the cost of lost goodwill. Then, the paper made a com-
parison between Abad’s model [116] and Goyal’s model
[117] in which shortages occur at the beginning of the
cycle. The numerical results show that there is no domi-
nant one between these two models. The paper also pro-
vides certain conditions und er which one model had more
net profit per unit time than the other.
4. Deteriorating Items Inventory in the
Supply Chain
With the integration of the global economy, there is more
and more diversity and uncertainty in the market. In ord er
to deal with the uncertainties in the market and respond
quickly to the diverse and personal demand of customer,
enterprises need to cooperate with each other in the form
of an integrated supply chain. The traditional inventory
theory can not adapt to the current situ ation any more, the
inventory problem should be considered in the supply
chain. The deterioration of the deteriorating items makes it
is important for the relative firms in the supply chain to
make the optimal inventory policy together to minimize
the total inventory cost across the supply chain. In the fol-
lowing part of the paper, we will discuss the current studies
on the deteriorati ng item s inventory in t he supply chain.

A Review on Deteriorating Inventory Study 123
4.1 Deterministic Deteriorating Items Inventory
in the Supply Chain
Yang and Wee [118,119] have conducted research on the
inventory policy for deteriorating item in the supply
chain including a single-vendor and multi-buyers. Yang
and Wee [118] developed a multi-lot-size production and
inventory model for deteriorating items with constant
production and demand rates. In this paper, a supply
chain including a single-vendor and multi-buyers was
established and the supplier was dominant in the supply
chain. The supplier delivered the goods in the form of JIT
and the aim of the paper was to find the optimal produc-
tion-inventory policy on the basis of minimizing the in-
ventory cost in the supply chain. According to the nu-
merical results, the optimal policy using the integrated
approach can reduce the total cost for the producer and
the buyer. However, the producer’s cost is higher in the
form of supply chain integration than the independent
decision without considering the buyer’s perspective. So
in order to maintain the stability of the supply chain, the
buyers have to supply some kinds of incentives to the
supplier. Then Yang and Wee [119] established a col-
laborative inventory system of single vendor and single
buyer with the price-dependent demand, shortages are not
allowed in the system. The paper developed different
inventory models under the cases of vendor and buyer
making decisions respectively; vendor and buyer making
decisions together but no incentives are incorporated in
the decisions; the two parties making decisions together
and the vendor providing trade credit to the buyer. The
numerical results showed that among the three cases, the
profit in the first case was the smallest. The profit in the
second case was the same with the one in the third case
and the profit in the secon d and th ird case was larger than
the one in the first case. As is well known to us all, the
optimal solution for the whole system is not always bene-
ficial to all the players in the system and in fact in the
second case, the buyer’s profit was seriously damaged
and so the buyer will refuse to cooperate with the sup-
plier. While in the third case, the trade credit was incor-
porated in the system and it helped to share profit be-
tween the two parties. Compared to the first case, both
vendor and buyer can get the extra profit and the supply
chain was stable in this case.
Compared to the deteriorating items inventory study in
a single enterprise, the inventory studies in the supply
chain have to emphasize how to maintain the stability of
the whole system while achieving the minimum of in-
ventory costs in the supply chain. Therefore, researchers
focused on supply chain coordination mechanism in the
deteriorating items inventory problem study. LI and
HUANG [120] established a three-level supply chain
including vendor, manufacture, and retailer in which the
manufacture is dominant. The paper developed two dif-
ferent models; in first case all the firms in the supply
chain make decisions independently, and in the second
the firms make decisions together. In the case of
non-integrated decision-making, the manufacturer was
the dominant member in the supply chain; the ordering
policies of vendor and supplier were subject to the policy
made by manufacturer, so the aim of the model is to mi-
nimize the costs of the manufacturer. While in the case o f
integration, the aim of the optimal production and inven-
tory policy making is to minimize the costs of the whole
supply chain. According to the numerical results, the
costs in the case of integration is lower than that in the
case of non-integration, but the optimal solution for the
whole system is not beneficial to all the members in the
supply chain. In order to realize integration in the supply
chain, the paper introduced the cost sharing mechanism
in the supply chain to guarantee the implementation of
integration and win-win among members. Both An and
Luo [121], Qin and Guo [122] stud ied the effect of quan-
tity discount across the supply chain and their studies
showed that quantity discount is a useful coordination
method among supply chain members. In Qin and Guo’s
study [122], quantity discoun t was used as a coordination
mechanism to achieve the stability of supply chain. As
the leader of the supply chain, the supplier develops an
optimal discount price policy and shares the policy with
the customer. Then the customer determines his unit
selling price and sales volume. The numerical example
showed that when the supplier determined the quantity
discount, bot h t he s uppl i e r an d c ustomer’s profit increase.
4.2 Stochastic Deteriorating Items Inventory in
the Supply Chain
The studies on stochastic deteriorating items inventory in
the supply chain at present are much le ss th an th e o n e s on
the deterministic deteriorating items inventory. Du et al.
[123] studied the deteriorating item stock replenishment
and shipment po licy for v endor- managed inven tory (V MI)
system with the assumption that the demand process fol-
lows a typical Poisson process. This research is important
to practical applications and we will discuss the paper in
detail.
In the circumstances of vendor-managed inventory, the
vendor acts as a supply chain manager and plans inventory
and shipment scheduling from the perspective of the whole
supply chain rather than obey orders passively and arrange
shipments respectively. The aim of VMI is to develop an
inventory and shipment policy in which the vendor can
meet the demand of the retailers and determine their own
inventory replenishment policy at the same time. The pol-
icy can realiz e the aim of minimizing the total costs in the
supply chain. This is the so called integrated shipment
which means that gathering the small amount of order
quantities from downstream enterprises into a large quan-
tity and then make integrated shipping scheduling. The

A Review on Deteriorating Inventory Study 124
integrated shipment policy can reduce relative costs in the
inventory and shipm ent delivery [124] .
Du et al. [123] made a comprehensive study on the in-
tegrated shipment policy under the circumstances of
vendor-managed inventory for the deteriorating items.
The paper established a supply chain system which in-
cludes a manufacturer, a vendor, and many retailers lo-
cated in different regions and the demand process follows
a typical Poisson process. In the mode of VMI, the ven-
dor is responsible for replenishing the retailers’ inventory
and arranging the corresponding shipment schedules. The
purpose of the study is to make a joint decision in those
two areas stated above. There are three parameters (Q, q,
T) in the integrated shipment policy, among them the Q
stands for the order-up to level, q stands for planned
shipments and T is the planned delivery period. The pol-
icy operates as below: vendor delivery order quantities
which accumulate during the predetermined replenish
cycle T is shipped to the retailer at the end of the cycle.
However, if the accumulated order quantity reaches the
amount of q, which is also a predetermined parameter,
before the end of the cycle, the vendo r will also, delivery
items to the retailers. Every time a delivery is planned,
the vendors have to decide whether the inventory for each
item needs to be replenished, and if it needs replenish-
ment, what is the replenishment amount. It must be
guaranteed that the inventory level of the vendor is equal
to or more than the order-up to level parameter Q after
delivery.
Integrated delivery is an importan t issu e which must be
considered when implementing vendor-managed inven-
tory. The studies on the integrated delivery problem for
deteriorating items can aid the vendors to make relevant
decisions about delivery and replenishment and so can
help reduce the costs in the supply chain.
5. Conclusions
The inventory problems for deteriorating items have at-
tracted more and more attention and many researchers
have conducted extensive studies in this area. In this pa-
per, from a different perspective, we have tried to make a
review on deteriorating inventory literatures after Raafat
[4]’s and Goyal’s reviews [5]. According to the litera-
tures discussed in this paper, we can draw some useful
conclusions. Subsequent subsections present significant
findings, the gaps identified in the research and future
directions of the research in the relevant area.
5.1 Significant Findings
1) According to the research scope, the current studies
can be divided into two categories, the first category stu-
dies the deteriorating items inventory problems in a sin-
gle enterprise and the second category studies the dete-
riorating items inventory in supply chain. From the per-
spective of quantity, there are much less studies in the
second category compared with those belong to the first
category. The study on deteriorating items inventory
problems is a new research area compared with the in-
ventory research on common items and so the total re-
search amount is far less than the traditional one.
2) Also, studies in the single enterprise category are
less complex than the second category of the integrated
supply chain. Two situations are included: first, from the
perspective of the number of key factors involved in the
model, there are always fewer factors involved in the
supply chain deteriorating inventory models. For example,
in the first category inventory models, besides demand
and deteriorating rate, other factors such as price discount,
allow shortage or not, inflation, time-value of money and
so on are always combined in the model; then in the sec-
ond category, in order to simplify the situation, always
less factors considered.
3) Then when it comes to a single factor, the models in
supply chain circumstances always consider easier situa-
tion of the same factor. Take demand for example, Many
studies in the second category especially those under the
circumstances of multi-echelon supply chain involve
constant demand rate in the inventory model. Only few of
the studies in the second categ ory take stochastic demand
into consideration for it is really hard to deal with in these
kinds of models. For the first category, constant demand
is the focus of the early stage and now more and more
studies begin to research other kinds of demand, such as
time-dependent demand, inventory level- dependent de-
mand, price-dependent demand, and even stochastic de-
mand.
4) Deteriorating rate is another key factor in the study of
deteriorating items inventory and there are also several
kinds of deteriorating rate in the present study, such as
constant deteriorating rate, deteriorating rate is a linear
increasing function of time, deteriorating rate is two- pa-
rameter Weibull distributed, and deteriorating rate is
three-parameter Weibull distributed. Among them, con-
stant deteriorating rate is the easiest one and the
three-parameter Weibull distribution deteriorating rate is
more complex. Some studies which belong to the first
category have made extensive study in this factor and
some studies even involve the three-parameter Weibull
distribution deteriorating rate; the relatively complex one
in the model. For the second category, constant deteriorat-
ing rate is the most common one in model establishment.
5.2 Future Directions of the Research
1) As we all know, in order to respond quickly to the di-
versity and uncertainty in the market, all the enterprises
should be part of a supply chain and so it is more reason-
able to consider inventory problems in a supply chain. The
diversity and uncertainty that a supply chain will face are
much more complex than a single enterprise so more fac-
tors and more complex situation of a single factor should

A Review on Deteriorating Inventory Study 125
be taken into consideration of deteriorating inventory
problems in a supply chain. Only in this way, can the stud-
ies supply practical measurem ent s to m anagers.
2) In practical situation, the information about inven-
tory is not always precise, most of the time it is vague or
imprecise. So it is more reasonable to develop some
fuzzy, stochastic and dynamic research methods and this
is also the future trend of the deteriorating inventory
study.
3) Luckily, many researchers have studied inventory
under fuzzy circumstances. Katagiri and Ishii [125] es-
tablished inventory models for deteriorating items with
fuzzy shortage cost and due to the fuzziness of relative
parameters; the expected profit function in the study is
represented with a fuzzy set. Chen and Ouyang [126]
established a fuzzy inventory model for deteriorating
items with permissible delay in payment. The study in-
volved the fuzzy carrying cost rate, fuzzy interest paid
rate, and fuzzy interest earned rate simultaneously in the
model. With the assumption of constant demand and de-
teriorating rate, Arindam Roy, Samarjit Kar and Mano-
ranjan Maiti [127], Arindam Roy, Manas Kumar Maiti,
Samarjit Kar and Manoranjan Maiti [128] established an
EOQ model which involved imprecise parameters, such
as fuzzy inventory cost, fuzzy storage area, and so on.
The purpose of their research was to minimize the total
inventory cost.
It is recommended that more and more researchers be-
gin to study deteriorating items inventory problems in
supply chain with fuzzy, stochastic and dynamic research
methods, only in this way, can the researches be applied
to practice. It is hoped that this paper can provide an
overview of the deteriorating items inventory in the re-
cent years and so act as a cornerstone for future study in
this field.
6. Acknowledgements
This research was suppor ted by The Research on Process
Optimization of Cold Chain Logistics Based on Food
Safety of Foundation of Beijing Jiaotong University
(2007XM018).
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