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\begin{document}
\title{Inventory models for managing deteriorating products: a literature
review}
\author[1]{Freddy Perez}%
\author[2]{Fidel Torres}%
\affil[1]{Universidad de la Costa}%
\affil[2]{Universidad de los Andes}%
\vspace{-1em}
\date{\today}
\begingroup
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\selectlanguage{english}
\begin{abstract}
The problem of determining the economic order quantity has long
attracted the attention of researchers, and several models have been
developed to meet requirements under different circumstances at minimum
cost. In the present paper, we conduct a structural content analysis of
317 selected peer-reviewed research articles that were published during
the period 2001-2018 and ranked with a quartile score of Q1 by either
ISI or Scopus database. By discussing the main topics of the inventory
modeling literature, we provide a comprehensive view of the past
research dealing with the management of deteriorating items. Here, we
focus on items undergoing physical modification during the planning
period, which encompasses a wide variety of products such as fresh
produce, processed food, pharmaceuticals and blood products. Therefore,
based on our holistic analysis, we identify new trends and we highlights
crucial research opportunities to develop more comprehensive and
practical models.%
\end{abstract}%
\sloppy
\emph{Informative title}
Inventory models for managing deteriorating products: a literature
review
Running title
Deteriorating inventory models: a review
Full names of authors
Freddy Andr\selectlanguage{ngerman}és Pérez Mantilla
José Fidel Torres Delgado
Author's institutional affiliations
Freddy Pérez (ORCID 0000-0002-2456-3268), PhD., Associated Professor,
Universidad de la Costa, Barranquilla, Colombia (fperez23@cuc.edu.co).
Fidel Torres (0000-0001-7379-6212), PhD., Associated Professor,
Universidad de los Andes, Bogotá, Colombia
(\emph{ftorres@uniandes.edu.co}).
Acknowledgments
We thank Caitlin Simpson May and Diego Castiblanco, from Universidad de
los Andes, for editing a draft of this manuscript.
Abstract and keywords
The problem of determining the economic order quantity has long
attracted the attention of researchers, and several models have been
developed to meet requirements under different circumstances at minimum
cost. In the present paper, we conduct a structural content analysis of
317 selected peer-reviewed research articles that were published during
the period 2001-2018 and ranked with a quartile score of Q1 by either
ISI or Scopus database. By discussing the main topics of the inventory
modeling literature, we provide a comprehensive view of the past
research dealing with the management of deteriorating items. Here, we
focus on items undergoing physical modification during the planning
period, which encompasses a wide variety of products such as fresh
produce, processed food, pharmaceuticals and blood products. Therefore,
based on our holistic analysis, we identify new trends and we highlights
crucial research opportunities to develop more comprehensive and
practical models.
\textbf{Keywords:} inventory models, deteriorating inventory,
perishability, shelf life, review.
Main text
\section*{Introduction}
{\label{introduction}}
Inventory management is one of the most fundamental and challenging
activities for any company dealing with raw materials, work-in-process
and/or finished goods. Since organizations usually make a significant
investment in inventories, the correct management of this tied-up
capital provides a very important opportunity for business improvements.
Under these circumstances, scientific methods for inventory decisions
can be decisive to achieve a significant competitive advantage in
today's business world. On the one hand, there is a wish to make large
replenishment orders to get trade credit benefits and volume discounts,
to reduce production costs by means of long production runs and
economies of scale, and to increase sales by providing a high customer
service level. On the other hand, there is a wish to keep stock levels
down to avoid the risk of suffering financial difficulty as a result of
low or tight liquidity and to avoid excessive costs incurred for keeping
and managing large inventories. In order to balance these and other
conflicting goals, novel approaches are required to provide an answer to
at least the following three questions:
\begin{enumerate}
\tightlist
\item
How often should the inventory status be reviewed and determined?
\item
When should an order be placed?
\item
How large should the order quantity be?
\end{enumerate}
One way to tackle these issues is to use an inventory model as a
decision-support tool. Generally speaking, inventory models are
approximations or simplifications of real inventory management systems.
It does not reflect every aspect of reality in a particular context.
However, they can be useful for decision-making processes. Despite all
the effort invested in research, there is still a lack of research on
inventory management where useful endeavors may result not only in a
significant improvement to companies but to society in general. The
evidence supporting this is overwhelming. In the agricultural industry,
for example, post-harvest losses are significant and unavoidable
{[}1{]}. Roughly, one-third of food produced for human consumption is
globally lost or wasted throughout the food supply chain, which is about
1.3 billion tons per year and has a negative impact on economic
development and on the environment {[}2{]}. In the grocery retailing
industry, perishable products within the grocery-food category account
for approximately 50\% of total supermarkets sales {[}3-5{]}, and the
losses of these kinds of products due to inventory spoilage at the
retail level are susceptible to range from 5\% to 22.8\% {[}2, 6{]}.
While reducing perishable inventory waste 20\% can increase total store
profit by 33\% {[}7{]}, mismanagement of perishable products can
represent a major threat to the profitability of companies in the
grocery retailing industry {[}8{]}. Therefore, finding suitable
inventory management policies has always been of great importance to
both researchers and practitioners.
The mathematical modeling of inventory systems has its roots in the
Economic Order Quantity Model (EOQ) proposed by Harris {[}9{]} in 1913,
which assists in determining the optimal number of units to order with a
view to minimizing the total cost associated with the purchase, delivery
and storage of the product. However, in the EOQ model, many of its basic
assumptions are far removed from practice. When, for example,
deterioration has a significant economic impact within inventory
systems, the common assumption of unlimited shelf life for lot-size
determination becomes very inaccurate. A challenging task for this class
of items is to maintain product availability while avoiding excessive
product loss so that effective inventory management is possible. Because
achieving this effectiveness represents a formidable challenge to both
academics and practitioners, the study of inventory systems dealing with
deteriorating products is still one of the most important research areas
that emerged from the first EOQ model.
This research aims to gain a more in-depth understanding of the state of
the inventory modeling literature stream for deteriorating items. This
responds to the need for evaluating what the lot-sizing theory applied
for perishable products has collectively accomplished and what
directions might be fruitful for future research. In general, the
identification, evaluation, and interpretation of existing knowledge in
literature reviews is an essential part of all kinds of research
processes. This is frequently pointed out by textbooks on research
methodologies {[}10-12{]}, as well as methodological articles in high
impact journals {[}13, 14{]}. In particular, since the last literature
reviews on deteriorating inventory modeling {[}15-17{]}, more than 300
papers have been recorded over the last years. This not only raises
concerns about the state of art of this research stream but justifies
the need to provide a starting point for research by identifying
patterns, themes, and issues from the existing body of recorded
documents.
Since the earliest works on deteriorating inventory modeling during the
decades of the 50's and 60's {[}18-20{]}, many studies have been
published every year. The first review on this research area was
developed by Nahmias {[}21{]} in 1982. This review discussed the
relevant literature dealing with the inventory problem of finding
suitable ordering policies for either fixed or random lifetime items.
Next, nine years later, and following the classification scheme of
Silver {[}22{]}, Raafat {[}23{]} reviewed the advances of deteriorating
inventory literature but limited to those studies that investigated the
effect of deterioration as a function of the on-hand inventory level .
Then, in the year 2001, Goyal and Giri {[}24{]} extended Raafat {[}23{]}
but included inventory models subject to fixed lifetime items. After
that, in the year 2012, Bakker, Riezebos {[}15{]} updated Goyal and Giri
{[}24{]} by providing an overview and classification very close to that
of Goyal and Giri's to facilitate comparisons between them. Finally,
Janssen, Claus {[}17{]} updated Bakker, Riezebos {[}15{]} by analyzing
relevant papers from 2012 to 2015 and by discussing new topics. Unlike
all the previous reviews mentioned, they included newsvendor and
transport models. Apart from this stream of surveys, there are other
works that have been reported in the literature. However, some of them
only focused on specific topics of deteriorating inventory modeling
{[}25-30{]}, and others followed a different classification and/or
analysis approach {[}16, 25{]}.
In this paper, we complement and update the surveys in {[}15, 17, 24{]}.
As in these works, we discuss the basic features, extensions and
generalization of new reported published literature in the mathematical
deteriorating inventory modeling (in our case, 167 papers from 2016 to
2018). However, our scope is broader and does not only include a
classification of the new reported literature since the work of Janssen,
Claus {[}17{]}, but also includes a relevant sample of published papers
from 2001 to 2015. To keep the scope of this research treatable, we
limited ourselves to inventory models dealing with products which
naturally undergo physical degradation. This includes most inventory
models with fixed and random lifetime products, and even some models for
seasonal products.
The remainder of this paper is organized as follows. In Section 2, we
first describe the research materials and methodology utilized to search
and select the papers included in our review. Next, in Section 3, we
provide the different categories upon which our thorough evaluation of
the selected literature is conducted. Discussions are then presented in
Section 4, while conclusions and future research opportunities are given
in Section 5.
\section*{Methodology and descriptive
analysis}
{\label{methodology-and-descriptive-analysis}}
With a view to review the body of research on deteriorating inventory
models, the methodology applied in our study is based on the work of
Seuring, Müller {[}31{]}, which discusses the basics on how to conduct a
literature review through a structural content analysis. By doing so,
the following subsection contains a description of the material
collection process as well as of the background of the collected
material.
To ensure the quality of selected studies, we confined our searching to
only peer-reviewed research papers that were written in English and
ranked with a quartile score of Q1 by either ISI or Scopus database. By
means of this quality selection rule, material for our review was
collected in two phases. In the first phase, we extracted a sample of
150 articles published between 2001 and 2015 from previous related
reviews {[}15, 17{]}. We felt that 10 papers per year, meeting our above
quality selection rule, was good enough to draw additional inferences.
In the second phase, we used the Web of Science (WOS) Core Collection
database for searching research papers from 2016 to 2018. The search in
the WOS platform was conducted in 05/14/2019 by using the
expression\emph{TS=((deteriorat* OR perish* OR decay* OR ''shelf life''
OR spoil* OR outdate* OR ``waste'' OR lifetime*) AND ``Inventory'')} ,
and then filtering matched records by selecting all the Research Areas
of the WOS that generated at least one relevant paper for this review.
All selected Research Areas from the WOS database are displayed in
Figure 1.
As can been seen in Figure 1, the above searching process yielded 1204
records. In our sample we do not include book chapters and data papers,
so we first discarded them. Next, we scanned the title, keywords, and in
some cases, the abstract to select all potential articles which develop
an inventory model. Here, we were as flexible as possible, and we
obtained a preliminary sample of 488 articles. Then, as was done for
papers published between 2001 and 2015, we chose all papers ranked with
a quartile score of Q1 by either ISI or Scopus database. This resulted
in a sample of 325 papers. Last of all, we proceed to classify all the
inventory models with the potential of being applied to products which
naturally undergo physical degradation. As a result, we obtained a
relevant sample of 167 papers published between 2016 and 2018.
Note that when executing a Topic Search in WOS (TS =\ldots{}), the
search engine looks for further matches of the words entered by using an
extended keywords (``keywords plus''), which is harvested from the title
of indexed articles by WOS. We realized that this ``keywords plus'',
which is unique to WOS, can change over time. Thus, users may be aware
of this when executing the same topic search at different times. It is
also worth noticing that we employed all the search terms in {[}15,
17{]}. However, as is evident from Figure 1 and Table 1, there is a
significant amount of unrelated studies even after limiting the search
to specific research areas. In this regard, we had to discard 699 papers
by scanning the reference manager files (RIS) exported from WOS
database, and most of them coming from records found using the word
``waste'' (see Table 1).
To overcome this shortcoming in future literature reviews, along with
the selected Research Areas from WOS (see Figure 1), we suggest a far
better search expression that includes both all the papers selected in
our final sample and all the inventory models for deteriorating items
that we identified as relevant in a further examination. This search
expression generates 493 records instead of 1804, and is provided as
follows:
\emph{TS=((deteriorat* OR perish* OR decay* OR ''shelf life'' OR
lifetime\$ OR ``expiration'' OR ``evaporating'') NEAR/14 (product\$ OR
item\$ OR produce\$ OR ``inventory'' OR ``goods'' OR food\$ OR ``cost''
OR ``weibull'' OR ``storage'') AND ``inventory'' NOT ``life cycle
assessment'')}
In the above expression, the wildcards (*, \$) represent unknown
characters and are used to find inflected forms of the corresponding
words. The asterisk (*) represents any group of characters while the
dollar sign (\$) represent zero or one character. The quotation mark (``
'') is used to find exact phrases. And the proximity operator NEAR/14 is
used to find only records where the terms joined by the operator are
within 14 words of each other. Readers are referred to the WOS core
collection help for more details and tips.
Note that we did not recommend to use in the query the keywords spoil*,
``waste'' and outdate*. For the case of spoil* and outdate*, we found
that all the relevant papers provided by these keywords search are also
provided by the keywords perish* and ``shelf life''. And for the case of
the ``waste'' keyword we found on closer examination that the records
solely belonging to this query (15 of the 39 in Table 1) are not
actually relevant for a review of deteriorating inventory models.
Table 2 shows the list of the Journals to which most of the selected
articles belong, and ranked in descending order of papers published. It
can be verified that these journals published 121 (72.5\%) of the total
number of papers included in our review in the research period
(2016-2018). The rest of journals with only one or two paper each are
provided in Table 3.
Among all the journals, the International Journal of Production
Economics alone accounts for 25 papers (17.5\% of all publications).
Second and third are the Computers \& Industrial Engineering and the
European Journal of Industrial Engineering with fifteen and fourteen
papers each. There is a dominance of traditional Operations Research \&
Management Science journals, but in recent years, environmental
management-related journals have increasingly been used as a channel for
publication.
\section*{Main characteristics of inventory
systems}
{\label{main-characteristics-of-inventory-systems}}
In general, inventory models can be broadly classify according to the
demand and the type deterioration. Depending on the type of demand,
there are deterministic inventory models or stochastic one. If the
demand is deterministic, the variation of inventories over time on each
inventory cycle may be affected by a prediction of a constant
demand\(\ \left(\frac{\text{dI}\left(t\right)}{\text{dt}}=-D_{1}\right)\), or by the combined effects of a constant
demand \(D_{1}\) and a fixed fraction \(D_{2}\) of the
instantaneous stock level\(\left(\frac{\text{dI}\left(t\right)}{\text{dt}}={-D}_{1}-D_{2}I(t)\right)\). In more elaborate inventory
models, the depletion of the inventory can also occur due to a known
function of demand depending on time or further depending on the selling
price and/or one of various marketing parameters (e.g., selling price,
frequency of advertisement, credit, and freshness of products). In turn,
when the demand is uncertain, it may follow a known probability
distribution, or it may be represented through an additive or
multiplicative functional-form with random components. When the accuracy
of the stochastic demand distribution/function is unknown, modeling a
fuzzy/hybrid demand {[}32-35{]} can be useful to address this type of
uncertainty.
According to Raafat {[}23{]}, any stocked items restrained by any
process from being used for its original intended use is known as
inventory subject to deterioration or decay. This definition encompasses
many different types of products. However, they have been traditionally
classified into three main categories: items with fixed lifetime, items
with random lifetime, and items subject to obsolescence.
Fixed lifetime refers to the best-before date (BBD) of most packaged
products. Although these types of products are not usually spoiled at
the end of its BBD, sellers discard them in order to follow regulations.
Random lifetime refers to the uncertainty in the spoiled time of items
like fresh produce. Here, the time to spoilage may be uncertain for each
individual stocked item, but, in practice, or from a modeling
perspective, the total amount of spoiled items within any specific
interval of time may follows either a deterministic or probabilistic
function. Finally, obsolescence refers to the rapid loss in value of
unsold items due to the introduction of a new product or the end of a
shopping season. Unlike fixed lifetime items, this type of products does
not suffer physical degradation due to its own nature, and thus, they do
not necessarily need to be removed from the inventory throughout their
selling season. Typical examples are found in the fashion and technology
industry, and almost all the industry applications of studies further
investigating or extending the classic newsvendor problem.
Table 4 shows the classification of deteriorating inventory items
adopted for most authors in the literature. Note that some of the most
common terms used in the literature of inventory models such as
``perishable items'' and ``random lifetime products'' may be used in
different context. For example, the term ``perishable products'' may be
used for either items with fixed life time or items subject to
obsolescence, and the ``random lifetime'' term may be utilized for
models that do not necessarily consider deterioration as a stochastic or
random process. As a result, for the sake of transparency, in the
present review we use three categories that represent the way in which
spoiled items are model or represented into the mathematical models.
These categories are fixed lifetime items, constant deterioration rate,
and time-varying deterioration rate.
The first category, inventory models with a \emph{known fixed
lifetime,}is used for products discarded in a particular point of time,
i.e., when their expiration date has lapsed (e.g., 2 days, 1 week,
etc.). The second category, inventory models with a \emph{constant
deterioration rate,} is used to those models where the variation of
spoiled items at each period or instant of time \(t\) is
represented by a constant fraction \(\theta\) of the
instantaneous stock level\(\left(e.g.,\ \frac{\text{dI}\left(t\right)}{\text{dt}}={-D}_{1}-\selectlanguage{greek}\text{θI}\selectlanguage{english}\left(t\right)\right)\). The third category,
inventory models with a \emph{varying deterioration rate,} is used to
discuss those models where the variation of spoiled items at each period
or instant of time \(t\) is represented by a non-uniform
fraction over time of the instantaneous stock level\(\left(e.g.,\ \frac{\text{dI}\left(t\right)}{\text{dt}}=\ {-D}_{1}-\theta\left(t\right)I\left(t\right)\right)\).
Note that papers such as {[}36, 37{]} that claim the inclusion of a
deterioration rate following a probability distribution are classified
in the second category when, in the mathematical model, the mean of a
probability density function is used as a constant rate, and thus, the
amount of spoiled items is the same over time. Contributions such as
{[}38-50{]}, in which the rate of deterioration can be reduced by
investing \(\xi\) monetary units in a preservation
technology, are classified in the second or third category depending on
the variability over time (or not) of spoiled items in the inventory
model\(\left(e.g.,\ \frac{\text{dI}\left(t\right)}{\text{dt}}={-D}_{1}-\left[\theta-m\left(\xi\right)\right]I\left(t\right)\rightarrow category\ 2;\ \frac{\text{dI}\left(t\right)}{\text{dt}}={-D}_{1}-\left[1-m\left(\xi\right)\right]\theta\left(t\right)I\left(t\right)\rightarrow category\ 3\right)\). Inventory models in which an item can randomly
expire before or up until their maximum lifetime (uncertain lifetime)
are classified in the third category due to the amount of the same item
perishing at each period or instant of time \(t\)
explicitly varies with respect to time. Meanwhile, studies such as
{[}51-74{]}, in which the inventory loses value but it is not physically
destroyed are classified in the first category.
Among other interactions, the assumptions or restrictions to be
considered for the correct application of inventory models for
deteriorating items include the lead time (negligible, constant, or with
a known or unknown distribution), the inventory review policy (periodic
or continuous), the existence of shortages (lost sales or backorders),
the inclusion of multiple products (with complementary and/or substitute
items), the production rate (finite, infinite or uncertain), the selling
price (fixed, variable or uncertain), the time value of money, the
number of echelons within the supply chain (closed loop supply chain,
VMI, non-cooperative, etc.), a permissible delay in payments (fixed or
conditioned), and the set of uncertain parameters or variables.
\section*{Classification of inventory
models}
{\label{classification-of-inventory-models}}
In this section, 317 articles that developed an inventory model for
deteriorating items are classified and analyzed. In Section 4.1, the
studies published between 2001 and 2018 are classified according to the
type of demand and deterioration assumed. Similarly, in Section 4.2, the
different papers referenced are also classified according to the
inclusion of the following characteristics: a determination of an
optimal price policy, considerations of a stock-out period, the
inclusion of multiple products, consideration of two warehouses, a study
of two or more echelon within the supply chain, considerations of delay
in payments, the inclusion of the time value of money, and the inclusion
of uncertain parameters or variables.
\subsection*{Classification according to the demand and
deterioration}
{\label{classification-according-to-the-demand-and-deterioration}}
Table 5 shows the type of demand and deterioration considered in the
models of our sample according to the characteristics discussed in
Section 3. Here, 230/317 of the papers reviewed have included a
deterministic demand, and 92/317 included an uncertain demand. In
addition, 153/317 considered a constant deterioration rate, 77/317
considered a variable deterioration rate, and 94/317 studied
deterioration with a fixed lifetime. Note that of all of the authors who
considered an inventory model for deteriorating items with a fixed
lifetime, 63/94 have taken an uncertain demand into account. Also note
that from all authors who considered a deterministic demand, 148/230 did
not include any of the marketing strategies that companies commonly use
to influence demand, such as pricing, advertising, markdowns, post sales
services, among others.
When marketing strategies are used to stimulate customers' consumption,
the integration between marketing and inventory management decisions
becomes important to maximize mutual benefits and avoid potential
conflicts. Of the 230 deterministic inventory models, 82 incorporated an
optimal decision for marketing strategies (pricing and others marketing
dependent demand factors) in combination with an optimal inventory
replenishment policy. Of these 82 models, 32 considered that the demand
for an item depends on characteristics further than selling price, thus
affecting marketing decisions: in {[}75-79{]}, the demand rate is a
function of the trade credit period offered by the retailer to their
customers; in {[}53, 80-86{]}, the demand rate is influenced by the
frequency of advertisement or the promotional expenditure; in the work
of {[}50, 86-88{]}, the demand rate is affected by the customer service
level such as the warranty period and the aftersales service
expenditure; in {[}52-54, 56, 59, 61-64, 71, 89-94{]}, the consumption
rate depends on the product's freshness or quality of an item during the
planning period. Finally, in {[}47, 80{]}, customer's environmental
concern affecting the demand are considered.
If environmental factors such as economic and marketing conditions
change during the product life and have a significant effect on the
demand, then the assumption that the demand in each period is a random
variable and is independent of environmental factors apart from time
will be incorrect. In such real life situations, the Markov chain
approach provides a flexible alternative for modeling the demand process
{[}55, 276, 321, 335, 336, 339-341, 343-346{]}; not only does it
significantly generalize the Poisson process {[}55, 272, 278, 280, 281,
287, 290, 297, 303, 304, 307, 308, 316, 333, 338, 341{]}, but it is also
a convenient tool for modeling both the renewal and non-renewal demand
arrivals. However, despite this fact, the Markovian assumption holds for
demand processes with relatively low variation coefficients, i.e., in
cases where high demand variances are observed, the non-stationary
assumption does not hold in the Markovian environment because standard
periods of constant length may introduce memory and generate correlated
demand distributions within periods. For these cases, the models in
{[}8, 285, 286, 292, 310, 319{]} provide a reasonable alternative. Of
the inventory models with stochastic demand explicitly guiding marketing
strategies, the models in {[}66, 67, 69, 283, 289, 293, 300, 314, 325,
326, 328, 330, 334, 342{]} are developed to define both a price decision
and inventory control policies. In turn, the inventory models in {[}51,
66, 67, 69, 283, 289, 300, 302, 307{]} take into account that the market
demand is random and sensitive to the freshness of the product, which is
more assertive for deteriorating items.
From studies with varying deterioration rates and deterministic demand,
in the inventory models {[}38, 89, 108, 199-202, 206, 207, 211, 212,
215, 219, 244, 263-266{]}, the deterioration rate is assumed to be a
general or arbitrary function of time \(\theta(t)\); in the models
described in{[}268{]}, the deterioration rate is studied as time
proportional\(\theta\left(t\right)=\theta t\); in the papers {[}94, 155, 203, 204, 209,
262{]} a three-parameter Weibull distribution\(\theta\left(t\right)=\alpha\beta\left(t-\gamma\right)^{\beta-1}\)is
included; in the models described in {[}87, 93, 157, 161, 205, 210, 213,
214, 217, 221, 222, 245, 269, 271{]}, the deterioration rate follows a
two Weibull distribution\(\theta\left(t\right)=\alpha\beta\left(t\right)^{\beta-1}\); and in the models developed
in {[}76-79, 154, 158-160, 162, 218, 220, 246, 267, 270{]}, the products
deteriorate at a rate \(\theta\left(t\right)=\frac{1}{1+m-t}\), where \(m\) is
the maximum lifetime at which the total on-hand inventory deteriorates.
In other particular works: {[}245{]} present the deterioration rate as a
function of both quality level and time, where the time-related function
follows a two-parameter Weibull distribution; in {[}208{]} the time to
deterioration is assumed normally distributed over time; and {[}255{]}
introduce a deterioration rate depending on the time and storage
temperature to which food products are exposed.
With respect to inventory models with an uncertain demand, it is
interesting to note that most of these studies assumed a fixed life
time, but few of them {[}34, 51, 55, 66-70, 73, 74, 90, 289{]} consider
the influence of product freshness over demand. From inventory models
with uncertain demand and constant deterioration, the deterioration of
items in {[}32, 33, 35, 112, 323-333{]} is assumed to be a constant
fraction \(\theta\) between {[}0,1{]} from the on-hand
inventory. Meanwhile, from inventory models with uncertain demand and
time-varying deterioration, the product lifetime in {[}282, 337, 339,
340, 343{]} is random and can be described by a discrete distribution,
while the lifetime of each item in {[}297, 334-336, 338, 341, 344-346{]}
has negative exponential distribution with parameter \(\gamma\ (>0)\),
which may be suitable for inventory systems where item lifetimes are
typically small but occasionally have long lifetime.
In nearly all of the inventory models considering deteriorating items,
the demanded items are immediately delivered to the customers from the
goods in stock. However, when the items from the on-hand inventory are
not delivered at the time of demand, but after some positive service
time that usually is random, the inventory managers need to consider the
replenishment policy as well as the effect of the formation of queues in
the inventory system, in order to implement suitable control policies.
Amirthakodi, Radhamani {[}335{]}, Manuel, Sivakumar {[}344{]}, and
Yadavalli, Sivakumar {[}346{]} analyzed an inventory system with a
service facility assuming that the customers arrive according to a
Markovian arrival process with a product lifetime following an
exponential distribution, but {[}335, 344{]} consider that the service
facility has a single server, and {[}346{]} consider a multi-server
perishable inventory system.
\subsection*{Classification according to shared characteristics with
other research
streams}
{\label{classification-according-to-shared-characteristics-with-other-research-streams}}
As can be observed in Table 6, 167/317 of the models allowed a stock-out
period in their models (category 1), 65/317 took into account an
inventory system with a permissible delay in payment (category 2),
82/317 considered the determination of an optimal pricing policy
(category 3), 77/317 studied a multi-echelon deteriorating inventory
model (category 4), 44/317 developed an inventory model considering the
time value of money (category 5), 30/317 studied a two-warehouse
inventory model (category 6), 32/317 took into account the existence of
multiple products (category 7), and 12/317 studied an inventory model in
a fuzzy environment (category 8).
Out of the 167 investigations allowing shortages, a total of 18, 31, 29,
31, 18, 18 and 4 models take into account the category C2, C3, C4, C5,
C6, C7 and C8, respectively. Of the 64 investigations considering a
delay in payment or prepayments, 11, 7, 15, 11, 0 and 5 models
considered categories C3, C4, C5, C6, C7, and C8, respectively. Of the
82 studies determining a pricing policy, 20, 9, 4, 2 and 2 considered
categories C4, C5, C6, C7 and C8, respectively. Of the 77 multi-echelon
inventory models, 5, 5, 10, and 1 models included categories C5, C6, C7
and C8, respectively. Of the 44 models considering the effect of the
time value of money, 8, 2, and 2 considered categories C6, C7, and C8,
respectively. Of the 30 investigations considering two or more
warehouses, 2 and 2 considered the categories C7 and C8 in the model,
respectively. Finally, of 32 inventory models with multiple items, 12
papers considered category C8.
\subsection*{Inventory models including a stock-out
period}
{\label{inventory-models-including-a-stock-out-period}}
When demand is higher than the previous forecast and cannot be fulfilled
immediately with the inventory on hand, then there is an excess of
demand, or shortages. Depending on the customer-company relationship,
the excess of demand (shortage) can be lost {[}89, 90, 98, 99, 101, 155,
258{]}, or the excess demand can be accumulated in different ways with
an associated shortage cost. Therefore, the behavior of the inventory
systems, when a stock-out period is allowed, differs from one model to
another.
Of the deterministic inventory models developed in {[}35, 45, 83, 102,
105, 112, 123, 124, 128, 139-144, 151, 161, 167, 169, 171, 181, 186,
203, 207-210, 214, 216, 217, 227, 232, 242, 251, 257, 266, 271, 274{]},
all of the excess of demand is willing to wait for the next
replenishment (shortages are fully backlogged). In the papers {[}87,
119, 122, 144, 157, 160, 172, 201, 211, 213, 226, 227, 229{]}, shortages
are allowed, but only a fixed fraction is backordered, and the rest is
lost. Finally, the deterministic inventory models found in {[}40, 41,
43-46, 82, 84, 85, 138, 146, 152, 166, 173, 176, 177, 185, 187-189, 192,
194-196, 205, 218-222, 225, 228, 234, 241, 245, 246, 250, 263-265, 269,
270{]} consider that shortages are allowed, but the unsatisfied demand
is partially backlogged depending on the waiting time until the arrival
of the fresh lot.
With respect to the inventory models with uncertain demand and
shortages, nearly all of the authors assumed that all of the excess
demand during the stock-out period become either lost sales {[}8, 51,
66, 68, 70, 73, 90, 98, 272, 273, 275, 276, 278, 281, 282, 288-290, 294,
296-300, 302, 303, 306, 309-311, 313, 316, 320-322, 326, 332, 335,
337-340{]}, or shortages that are fully backlogged {[}8, 35, 112, 278,
284, 286, 287, 295, 297, 308, 315, 319, 321, 323, 325, 329, 341, 343,
347{]}. However, some models with uncertain demand consider a partial
backlogging rate: {[}305, 330, 331, 333{]} assume that a fixed fraction
of shortages are backlogged while {[}328, 342{]} assume a waiting time
function for unmet demand. Finally, inventory models considering a
maximum allowable shortages constraint can be found in {[}280, 336{]},
and inventory models in which unmet demand during the stock-out period
temporarily leave the service area and repeat or retry their request
after some random time (until they find a positive stock level) is
studied in {[}345, 346{]}. In the latter case, it is important to note
that, as opposed to the lost-sales and backlog cases, the company does
not incur any expenditure toward lost sales or for holding unsatisfied
demands.
Wu, Ouyang {[}194{]} proposed a model for non-instantaneous
deterioration of items with stock-dependent demand and in which the
backlogging rate is variable and dependent on the waiting time for the
next replenishment. Olsson and Tydesj\selectlanguage{ngerman}ö {[}308{]} described a model where
demand is generated by a stationary Poisson process, but it is assumed
that unmet demand is immediately backordered. Meanwhile, Dehghani and
Abbasi {[}278{]} proposed a new age-based lateral-transshipment policy,
which may be useful for reducing stockouts and improve performance in
supply chains.
\subsection*{Permissible delay in payment and/or
prepayments}
{\label{permissible-delay-in-payment-andor-prepayments}}
In the traditional inventory economic order quantity model, it was
tacitly assumed that the supplier is paid for the items as soon as the
items are received. However, in business transactions it is observed
that the supplier provides a grace period whereby purchasers can repay
their debts without having to pay any interest (trade credit period) or
may delay the payment beyond the permitted time, in which case interest
is charged.
Inventory models in which the supplier provides a permissible delay to
the buyers if and only if the order quantity is greater than or equal to
a predetermined quantity \(W\) can be found in {[}107, 108,
116, 168{]}. Authors who assume that there already exists a regular
credit policy between the retailer and the vendor (fixed delay in
payment) can be consulted in {[}48, 49, 111, 119, 125, 134, 138, 146,
152, 154, 165, 180, 182, 190, 191, 202, 204-207, 243, 250, 257, 259,
333{]}, whereas authors who consider a permissible delay of payments,
but as a decision variable, can be found in {[}33, 75, 76, 78, 79, 118,
164{]}.
The inventory models submitted in {[}32, 75, 76, 79, 110, 114, 115, 145,
158, 162, 164, 220, 237, 264, 324, 328{]} take into consideration
affairs in which the supplier not only offers a fixed credit period to
the retailer but the retailer also adopts the trade credit policy with
his/her customer. A two-level trade credit is also considered in {[}33,
77, 126, 159, 227, 267, 270{]} but with a partial trade credit in one or
both echelon of the supply chain.
Alternatively, in the models presented in {[}117, 120, 129, 267{]}, the
supplier provides not only a permissible delay in payments to the
customer but also a cash discount, to wit a cash discount is offered by
the supplier if full payment is paid within time \(M1\)
(period of cash discount); otherwise, the full payment is paid within
time \(M2\) (with \(M2\ >M1\)).
Ouyang, Teng {[}131{]} and Guchhait, Maiti {[}230{]} consider situations
in which a supplier also offers a partial permissible delay in payments
even if the order quantity is less than a predetermined amount
\(W\) of units of an item. This means that if the order
quantity\(Q\) is less than \(W\), then the
retailer must pay a fraction\(\ 0\leq\beta\leq 1\) of the total purchase
costcQ when the order is filled and pay the
rest,\(cQ(1-\beta)\), at the end of the trade credit
\(M\).
Chang, Ouyang {[}109{]} consider a situation in which the supplier
offers his/her customers a permissible delay in payments,
\(M1\), and a price discount if the order quantity is
greater than or equal to a predetermined quantity \(Y\), a
permissible delay in payments, \(M2\), if the order
quantity is greater than or equal to a predetermined quantity
\(X\) (with \(M2>M1\)), and a trade credit
period, \(M1\), if the order quantity is less than
\(X\) (or\(Y\)). In turn, {[}122, 142, 160,
227{]} develop an EOQ model with multiple prepayments
\subsection*{Inventory models determining an optimal price
decision}
{\label{inventory-models-determining-an-optimal-price-decision}}
In competitive environments, it is common to change the price of items
to stimulate demand and decrease the rate of deterioration of stored
commodities: in this context, it is important to define a price strategy
during the horizon planning, and thus, it is important to include price
decisions in the deteriorating inventory models.
The selling price as a decision variable was first considered for
perishable products in 1996 by Eilon and Mallya {[}348{]}. Later, Kang
and Kim {[}349{]} and Aggarwal and Jaggi {[}350{]} reformulated and
extended this model to make way for further investigations in which both
the selling price {[}42, 45, 47, 50, 61, 64, 67, 69, 80-83, 86-89, 93,
128, 164, 167, 179, 218, 223, 230, 232, 240, 250-257, 259, 261, 263-271,
275, 325, 328{]} and the discounted selling price {[}68, 89, 145, 170,
175, 249, 298, 314{]} are taken into account as decision variables.
Traditionally, a common practice followed by companies is to maintain a
constant price on the goods offered and apply discounts when items are
close to expiry or when the demand decreases. However, applying an
aged-dependent selling price strategy may be more beneficial {[}53, 62,
63, 289{]}. Inventory models considering a dynamic pricing policy are
described in {[}38, 53, 55, 62, 63, 65, 71, 91, 92, 94, 174, 178, 191,
233, 235, 247, 260, 283, 289, 293, 300, 325, 326, 330, 342{]}.
In relation to the particular works in this category, new approaches for
determining a pricing policy when reference price play a critical role
in customer purchase decisions are proposed in {[}38, 47, 258{]}, a
clearance sales as a strategy to sell items approaching their expiration
dates at a reduced price is studied by Li, Yu {[}298{]}, and an optimal
replenishment and pricing policy for deteriorating items with
heterogeneous consumer sensitivities is investigated by Herbon {[}64{]}.
\subsection*{Multi-echelon inventory
models}
{\label{multi-echelon-inventory-models}}
In the present literature reviewed, 39/77 of the inventory models that
consider more than one echelon in the supply chain are restricted to
interactions that occur between a supplier/producer and a buyer {[}37,
39, 58, 63, 67, 73, 80, 81, 87, 88, 97, 101, 103, 104, 106, 112, 114,
118, 143, 145, 147, 149, 170, 176, 198, 229, 231, 237, 242, 248, 251,
252, 256, 275, 293, 305, 321, 331, 332{]}. However, a supply chain
involving a single-vendor and multiple buyers is studied in {[}34, 36,
60, 72, 82, 100, 140, 153, 197, 210, 253, 261, 277, 281, 290, 296, 326,
334{]}, while a two-level supply chain with multiple suppliers and
multiple buyers is considered in {[}48, 95, 315, 327{]}.
Inventory models for a three-level supply chain can be found in {[}42,
49, 86, 90, 96, 98, 133, 135, 136, 148, 155, 156, 312, 318, 333{]}, and
inventory models for a closed-loop supply chain (recovery system) are
presented in {[}34, 73, 149, 163, 261{]}. Wee, Lee {[}149{]} incorporate
VMI strategies for a green electronic product in a two-level supply
chain. Yang, Chung {[}261{]} also consider a closed-loop supply system,
but involve a single producer and multiple buyers. Meanwhile, Alamri
{[}163{]} consider a recovery system in a production environment
consisting of three shops: one of manufacturing new items, and the
others for collecting and remanufacturing returned items.
Finally, it is worth noticing that most of the above studies addressed a
coordinated policy for integrating a supply chain. However, few studies
have proposed new mechanism for achieving a cooperation in a
non-cooperative environment {[}67, 73, 88, 133, 256, 281{]}. Other few
studies have also addressed the problem of a non-coordinated supply
chain {[}63, 248, 275{]} but in a competitive environment.
\subsection*{Inventory models under the effect of time value of
money}
{\label{inventory-models-under-the-effect-of-time-value-of-money}}
The effect of the time value of money (TVM) and inflation is another
important extension that makes inventory models applicable to real-life
situations. It plays an important role in business, especially in
countries with double-digit Gross National Product rate {[}351{]}. Table
7 shows the papers incorporating the time value of money effect into an
inventory models. Note that few studies have been developed for
determining an inventory policy with multiples items or in a fuzzy
environment under time value of money.
Hou and Lin {[}232{]} discuss an inventory model for deteriorating items
with price and stock dependent selling price in which shortages are
completely backordered. Ghiami and Beullens {[}157{]} provide a Net
Present Value analysis for a production-inventory system without the
inventory cost parameters commonly used in this context. Meanwhile, Tat,
Taleizadeh {[}143{]} study the optimal ordering policy in an inventory
system considering backorders and delay in payments.
In the previous models, the inflation rate has been considered as a
constant and known value. However, Mirzazadeh, Seyyed Esfahani {[}329{]}
consider that the inflation changes over the time horizon, specifically,
the inflation rate was assumed to be stochastic with known pdfs over the
time horizon. Here, the demand rate is a linear function of the internal
and external inflation rates, shortages are allowed and fully
backlogged, and a constant fraction of the on-hand inventory
deteriorates per unit time.
\subsection*{Two or more warehouse for deteriorating inventory
models}
{\label{two-or-more-warehouse-for-deteriorating-inventory-models}}
In most of the existing inventory models for deteriorating items, it is
assumed that all of the items are stored in a single warehouse (owned
warehouse, OW). However, in some cases, organizations may require
another storage facility (rented warehouse, RW) with better preserving
facilities and ample capacity in which deterioration, costs, demand, and
other parameters are different. These facilities are usually rented
either to reduce the losses due to deterioration or to store excessive
goods obtained at a discount price or simply to avoid inflation rate.
Cases where a retailer owns two shops: one shop to sell fresh items
until a particular point in time and the other shop to offer non-fresh
items at a reduced price are included in this category. Two-warehouse
inventory models incorporating further characteristics related to
inventory systems are presented in Table 8. Notice that models
considering two or more storage facilities and the time value of money
are shown in Table 7.
Two-warehouse inventory models for products with imperfect quality, and
thus, subject to inspection are developed in {[}121, 200{]}. Alamri and
Syntetos {[}200{]} propose a new policy entitled
``Allocation-In-Fraction-Out (AIFO)''. Unlike Last-In-First-Out (LIFO)
or First-In-First-Out (FIFO) dispatching policy which are commonly
adopted in classical formulation of a two-warehouse inventory models,
AIFO implies simultaneous consumption fractions associated with RW and
OW. Jaggi, Tiwari {[}121{]} also study the effect of deterioration on
two-warehouse inventory model with imperfect quality, but Alamri and
Syntetos {[}200{]} assume that the percentage defective per lot reduces
according to a learning curve while Jaggi, Tiwari {[}121{]} assume that
each lot received contains a random proportion of defective item, an
thus, the screening times of OW and RW are included as decisions
variables.
Most of the studies uses a calculus-based approach for solving the
proposed inventory model. However, some formulations require the use of
metaheuristic due to the complexity of the mathematical model. For
example, Guchhait, Maiti {[}230{]} studied the features of PSO with a
genetic algorithm in a hybrid heuristic named PSGA (Particle
Swarm-Genetic Algorithm) to show that this performance is better
compared to the FGA (Fuzzy Genetic Algorithm) and the traditional PSGA
in an inventory system with stock and selling price demand being
dependent under crisp and fuzzy environments.
\subsection*{Multi-item inventory
systems}
{\label{multi-item-inventory-systems}}
Although in most inventory systems the assumption of a single item is
not real, very few studies have been found in the literature dealing
with multi-item inventory control compared to single-item models. Two or
more items in combination with shortages are taken into account in
{[}34, 35, 70, 99, 101, 102, 105, 123, 155, 245, 279, 284, 292, 296,
299, 311, 315, 341{]}. Inventory models considering an optimal price
decision and multiple items can be found in {[}62, 223{]}. Inventory
models taking into account multiple products in a fuzzy environment are
developed in {[}35, 137, 245{]}. And multi-item inventory models in a
supply chain system can be found in {[}34, 72, 100, 101, 103, 127, 155,
296, 312, 315{]}.
It is worth noticing that, although in recent years the volume of
inventory models with multiples items has increased (23 papers between
2016 and 2018 compared with 9 papers between 2001 and 2015), most of the
models only consider a different demand, perishing nature and/or cost
parameters for items. Thus, the effect of multiple product competing for
shared recourses (such as demand, space, etc.) is still a research area
where much more work is needed. Studies considering these issues can be
found in {[}57, 223, 236, 307{]}.
Maity and Maiti {[}236{]} propose an inventory control system wherein
multiple items are either complementary and/or substitutes, and the
deterioration rate is stock dependent. In this work, an optimal
inventory policy for complementary and substitute commodities is
considered, but shortages are not allowed, and demand is stock
dependent. Önal, Yenipazarli {[}223{]} develop a mathematical model with
price-and displayed-stock -dependent demand under shelf-space and
backroom storage capacity constraints. Claassen, Gerdessen {[}57{]}
study a production planning and scheduling problem in a food processing
industry where setups are usually sequence-dependent. Finally, Najafi,
Ahmadi {[}307{]} address a blood inventory management problem by
considering all types of blood and their substitution compatibilities
based on medical priorities.
\subsection*{Inventory models with uncertain
parameters}
{\label{inventory-models-with-uncertain-parameters}}
In most inventory models with stochastic demand, the lead time is
assumed to be either negligible {[}8, 69, 90, 112, 276, 277, 284, 286,
293, 300, 307, 319, 323, 324, 328-330, 334, 340, 343{]}, or a fixed
positive constant {[}272-274, 279, 282, 287, 288, 290-292, 296, 297,
300, 304, 306, 308, 312, 313, 316, 322, 331, 332, 337, 340, 341{]}.
However, there are inventory models that consider the lead time as an
unknown parameter following an arbitrary distribution {[}336, 338{]}, an
exponential distribution {[}344-346{]} or a phase-type distribution
{[}335{]}. Furthermore, although a non-negligible lead time is usually
associated with a stochastic demand, few studies consider an uncertain
lead time with deterministic demand {[}172, 176{]}.
In some mathematical models with stochastic demand, the inventory is
controlled explicitly with a periodic review system {[}8, 66, 273, 274,
276, 279, 280, 282, 284, 287, 289, 292, 293, 296, 300, 303, 320, 321,
324, 330, 339{]}, while in others, it is controlled with a continuous
review system {[}272, 278, 290, 297, 304, 306, 308, 309, 311, 312, 316,
323, 332, 335, 336, 338, 341, 343-346{]}. On one hand, among items
monitored continuously, {[}272, 306, 311, 312, 323, 332, 335, 336,
343-346{]} apply the traditional\((s,S)\) policy in which a
same amount of product \((Q=S-s)\) is ordered every time the
inventory level (on hand plus an order minus backorders) reaches the
level of \(s\). {[}278, 290, 297, 308, 309, 338{]} use a
\((S-1,S)\) ordering policy for stock replenishment in which in
which a reorder for an item is placed whenever the inventory level drops
by one unit either due to demand or because a perished unit. {[}304,
316{]} suggest a combined age-and-stock-based ordering policy,
\((Q,r,T)\) policy, in which a replenishment order of size
\(Q\) is placed either when the inventory drops to
\(r\) or when \(T\) units of time have
elapsed, whichever occurs first. Finally, {[}341{]} uses the
\((s_{i},c_{i},S_{i})\) policy suggested by {[}352-354{]}
with\(s_{i}