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1. AIP Conference Proceedings 2481, 040023 (2022); https://doi.org/10.1063/5.0103757 2481, 040023
© 2022 Author(s).
A review on fuzzy economic order quantity
model under shortage
Cite as: AIP Conference Proceedings 2481, 040023 (2022); https://doi.org/10.1063/5.0103757
Published Online: 08 November 2022
Preety Poswal, Anand Chauhan, Rahul Boadh, et al.

2. A Review on Fuzzy Economic Order Quantity Model
under Shortage
Preety Poswal1,a)
, Anand Chauhan2,b)
, Rahul Boadh1,c)
, and Yogendra Kumar
Rajoria1,d)
1
Department of Mathematics, K.R. Mangalam University, Gurgaon, 122103, Haryana, India.
2
Department of Mathematics, Graphic Era Deemed University, Dehradun, U.K, India
a)
preety2128@gmail.com,
b) dranandchauhan@geu.ac.in,
c) rahul.boadh@krmangalam.edu.in
d)
Corresponding author: yogendrakr.rajoria@krmangalam.edu.in
Abstract. Fuzzy set theory has a remarkable progress in the field of research. It initiates many areas in both practical &
theoretical studies. It is really useful for several people engaged in research and development including medical
researchers, mathematician, businessman, social scientists, natural researchers etc. This field of mathematics has
introduced new life into technical and scientific fields that have been undeveloped for a long period. Thousands of
scholars are operating and working with fuzzy set theory and presented a lot of research papers. In this paper latest
review of existing literature & numerous types of fuzzy EOQ inventory models under shortage situation. It assists to
categorize how the concept of fuzzy sets theory has been applied in inventory models which motivate researchers to
concentrating on new technique in study of inventory control models in fuzzy environment. In this review, we study
desirable constraints of existing models in fuzzy environment under shortage of supplies. A lot of effort is attempted to
deliver the latest review of existing literature of inventory and fuzzy models. The purpose of the work is to obtain a
continuous and comprehensive assessment of existing literature and recognize upcoming research guidelines. This
review helps other researchers to draft an outcome during the situation of short supplies. This will also help to manage
inventory according to the situation & reduce the loss during shortage of supplies, this review also helps to reduce loss
with proper inventory management in real-life applications and marketable products.
Keywords: Optimization, Inventory parameters, EOQ model, Partially backlogging, Fuzzy inventory
model, Defuzzification.
INTRODUCTION
Today’s free-trade area atmosphere needs production to design, development, test, manufacturing, and
organize most reliable goods in fewer time at low budget. For accomplishing this, billions of dollars are
being expended annually all over the world to develop consistent and well-organized products. For this,
problems related to inventory are very common in maintenance service, manufacturing, and corporate
dealings generally. Different models and methods have been planned to deal with different kind of
inventory issues. Usually, EOQ (Economic Order Quantity) model deals with constant assessment of
inventory complications. However, various costs are consumed by an organization to keep a necessary
level of inventory are called relevant costs. An effective organization of inventory helps in reducing costs
and to assurance a smooth process of any organization.
Perishable goods are very normal in our day-to-day lives. From study reports, the goods that turn into rotten,
depreciated, expired, etc. over the time period time are called deteriorating goods. These items are divided into
two classes. First type contains those substances that decay, defective or elapsed over the time. For e.g., as
milk products, meat products, fruit& vegetables, medicines etc. Second type refers those products that losing
partial or full value over the period due to change in technology, fashion, trend, occasions& festival. For e.g.,
seasonal goods, software, smart phones, fashion and so on.
EOQ (Economic Order Quantity) model is used to calculate the minimum order quantity of an inventory
from purchaser/vendor’s perspective. The purpose of this model is to ensure that the right amount of stock
quantity is ordered and solves the problems related to blockage of funds and extra storing cost. From this a
International Conference on Advancements in Engineering and Sciences (ICAES2021)
AIP Conf. Proc. 2481, 040023-1–040023-13; https://doi.org/10.1063/5.0103757
Published by AIP Publishing. 978-0-7354-4218-4/$30.00
040023-1

3. company minimizes its inventory cost like holding cost, order cost, shortage cost. Firstly, the EOQ model
and formula developed originally by Ford W. Harris in 1913. It is one of the oldest classical models. Later,
which was improved and extended by numerous researchers, by varying the assumptions, and objectives to
make it more realistic? By using EOQ model we are able to make better decisions on how much product to
order in a given period of time. In short, The EOQ formula proves useful in optimizing resources and thus,
minimizing associated cost. In the classical EOQ model, we considered that the parameters as crisp
numbers. This model is basically derived under some assumptions and limitations which are as follows:
x Assumes that the rate of demand in any inventory is already known with certainty and is
deterministic over time.
x Shortages of goods are not acceptable.
x The lead time (It is the amount of time that passes from the start of process until its conclusion) of
orders do not vary.
x The order quantity is received in a single delivery.
x The cost of holding a unit of stock does not depend on the quantity in stock.
Though the probability approach has been effectively useful to various practical engineering, reliability and
integrity problems, even now have some limitations to the probability way. Also, in actual real-world
problems sometimes we have inadequate information to handle correctly the statistics of parameters. In real
life circumstances, various decision-making problems are too complicated to be comprehended
quantitatively since the complication generally derived from uncertainty in the form of impreciseness. The
theory of probability has been an efficient tool to handle ambiguity but it can be implemented only to solve
the problems whose features are based on random processes.
Backlog happens due to deficiency. From time to time, some researchers considered partial backlog in
inventory model, while others assumed full backlog. In fact, if all buyers will wait till next order arrives,
this is known as complete backlog. However, in some cases, few customers can expect for next delivery, to
meet their requirement throughout the stock out time period, though some consumer can't delay their
orders, in that case customers fulfilled their demands from other sources. Consumers who experience stock-
out cannot order commodities again from respective providers, and they can be converted to some other
store to buy goods. As a result, a large portion of sales has been lost which leads to a small profit. So, the
partial backlogging factor is essential. We also discussed the backlogged in fuzzy environment. Here in this
review, we considered that the inventory demand throughout the shortage of supplies is either fully back-
logged or partially back logged.
FUZZY INVENTORY MODELS
Some research studies apply fuzzy set theory instead of classical set theory in the management of
inventory strategies. When the boundary of a piece of data is not clear, then fuzziness occurs. Much fuzzy
knowledge occurs in the real world: knowledge that is unspecified, uncertain, imprecise, inaccurate,
unclear, inexact, inexperienced or probable in nature. Human thinking and reasoning often involve vague
information, which naturally arises from inexperienced human concepts and can give satisfactory answers,
which are probably true. However, our complicated process is incapable to answer various questions. Most
systems are based on classical set theory that is unable to with handle unreliable and incomplete data. So, it
is understandable that, in current modest, productive and energetic corporate environment, it is not probable
to accessible entire essential data and information. Therefore, today’s inventory management structure is
not straightforward as supposed in classical inventory models. But fuzzy sets are capable of providing
solutions to many real-world imprecise data.
An inventory model having at least one fuzzy parameter is known as a fuzzy inventory model. In
fuzzy approach the constraints, parameters and goals are measured as fuzzy set with known membership
function or any kind of fuzzy number. Too much work has been done by numerous researchers in fuzzy
environment. The fuzzy inventory model provides a better and acceptable inference than the crisp inventory
model for its uncertain data. For this reason, many researchers have been attracted to work and contribute
more to its development.
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4. The foremost journal in fuzzy sets theory is given in 1965 [1]. In his publication, Zadeh presents the
concept of uncertainty. A scientific fuzzy model has formulated in the area of decision making
complications in management sciences and also in operation research sciences [2]. After this theory, people
initiate to acknowledge how uncertainty originating from human thinking can influence scientific problems.
After that, in 1965 Professor Zimmermann discussed about the concept and central idea of fuzzy set theory
and learns about its utilization. Professor Zimmermann contributes majorly to the literature on the fuzzy set
&decision analysis. Later some researchers also have proceeded to implement the fuzzy set theory in
inventory organization for maximize the profit and reduces the cost. There are several studies on fuzzy
modeling. Fuzzy logic has been effectively used in working with frequent practical applications. Fuzzy
inventory model with shortages has been considered in the literature. Here, we will be discussed about
either fully backlogged or partially backlogged in both fuzzy and non-fuzzy environment.
LITERATURE REVIEW OF INVENTORY MODELS UNDER SHORTAGES
Firstly, inventory model has been presented in 1996 [3]. The researchers have taken backorder in variable
lead time and lost sales. In his model, they have taken partial backlogging and assumed that shortages are
allowable. An inventory has established structure for perishable items with quantity discount, he also
includes price and partial backorder [4]. He minimizes the cost and maximizes the profit by assuming that
demand rate is decreases as amount for the product increases. An inventory model has been used for
perishable goods and permitting the partial backorder of unfulfilled demand [5]. It is more analytic to
accept that shortage rate in inventory is normally depend upon the interval of waiting period for the
upcoming replenishment or restock.
The inventory models have been improved with comprehensive ramp type of demand rate [6], [7], [8], [9].
The researcher assumed that the partial backorder rate is dependable on waiting period as well as on next
replenishment and instead used Weibull Distribution deterioration [6], [7], [9]. An inventory model was
developed and presented more accurate discussion related to the inventory problems over a finite planning
horizon for decaying items [10]. The author is taken the time-varying demands and shortages are allowable
in his model. Also, he defines a suitable partial backordered rate along with; he presented the opportunity
cost as considering to lost sales and an inventory control model also described in 2001 [11]. In this model,
lead time as well as examination period is measured as decision variable quantity, also taken a combination
of lost sale and backorders. To find the optimal solution they developed an algorithm.
One more addition on hypo the size the model [12] and analyzed a model with perishable products,
quantity discount model for the unit cost [13]. Partial shortage rate is considered as dependent function on
time. An inventory model is formulated with a combination of backlogged and lost opportunity of sales
[14]. A balancing will be created between two most significant factors: Cost and Time. The variety of time
cycle is assessed by minimum total cost. An inventory model for perishable goods under shortage, demand
is considered time varying by various authors [15], [16], [17]. The inventory model developed by [14] has
been modified and extended for real problem [18]. They has accompaniment the shortcoming by inserting
both the cost lost sales and non-stable purchase costs to their models. Also applied an Economic Order
Quantity model in their research and used backlogged with imperfect quality items [19], [20]. An
Economic Order Quantity (EOQ) model for non-instantaneous has been perishable goods with inventory-
level dependent demand under shortages with partial backlogging [21], [22].
A lot of work accomplished by numerous researchers in the area of inventory by taking different typed of
demand. Two-Ware house soft computing-based model with deteriorating products with partially
backlogged [23], [24], [25]. A deterministic EOQ model has developed, in which firstly demand is
considered as partially backordered, and then extended it’s to full backordering [26], [27]. Then, in the year
2009 proposed An Economic Order Quantity Model for deteriorating products with power demand [28].
Also, partial backordering is taken and used deterioration as a specific form of Weibull Density function.
The development a model in Inventory Production Control Problem with backorder and shortages are
acceptable [29]. They conclude that when demand rate per unit time and replenishment price are increased
then order quantity, relevant total amount and order quantity will also be improved. Holding cost
considered as a linear function of time. An enhanced inventory control model with partial backordered,
where deterioration rate is assumed time varying and demand is stock dependent [30]. An accurate
inventory model formulated for deteriorating commodities in 2014 [31]. The backlogging rate is supposed
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5. to be a decreasing function of time for the next replenishment. Demand is taken quadratic and partially
backlogged.
An inventory model developed and purpose to find high quality provided goes to reliable products and
stock to retain the customer balance of availability satisfaction can be the key factor for operating the
inventory management of a company that in turn can further increase your market share, productivity and
viability’s [32]. Here, shortage is allowed and taken partially backlogged. The backlogged item will be
replenished in the next cycle. Demand is taken time based in their model. A model has been implemented
quality items with deterioration rate under trade-credit policies by taking selling rate dependent on demand
under shortages with fully backordered [33]. Deteriorating products with different price has been dependent
demand under shortage with fully backlogged conditions [34], [35].
An inventory model for deteriorating products has been establishment, demand is assumed constant [36].
Partially backlog shortages are also considered and are intended to decrease the total inventory cost in a
finite planning horizon. A periodic model in which quantity discount pricing as well as partial backlogging
are to be taken at what time, the items in stock deteriorates over time [37]. Deterioration rate has been
considered as linear time dependent. Focused on to formulate a deteriorating EOQ inventory model and
demand is considered in different forms like price with negative power of constant, linear price with stock
dependent and also exponential function of price. Shortages are permissible and fully backordered [38].
LITERATURE REVIEW OF FUZZY INVENTORY MODEL WITH
SHORTAGES
Figure. 1 displays the basic outlines of fuzzy model. Virtually all models in fuzzy environment follow the
above methodologies as a basic procedure to apply the fuzziness in their model while using distinct
methods and strategies.
FIGURE 1 Classification of Fuzzy Inventory model
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6. A backlogged inventory model has including Fuzzy Order Quantity and order quantity as triangular fuzzy
number, and to Defuzzifier the values he applied centroid method [39]. Numerous researchers had
considered several classes of fuzzy inventory control models.
The contributions in the area of fuzzy inventory control theory with their works over the period
1996-2020 are summarized as follows. An EOQ (Economics Order Quantity) models with backlogged in
fuzziness with use of fuzzy extension principle, and he examined an inventory model along with fuzzy cost
and also with fuzzy requirement [34]. A problem in inventory with backorder in fuzzy sense and also
solved the model in fuzzy environment by assuming the backlogged quantity as triangular fuzzy number
[40].
Establishment and extended of inventory model in fuzzy environment with backorder [41]. The authors
focusing on total demand which is basically centered on interval valued fuzzy set. Demand is Fuzzifier by
centroid method and considered as triangular fuzzy numbers. An inventory model has derived in fuzzy
environment with backorder [42]. Shortage quantity and order quantity is Fuzzifier by authors and taken as
triangular fuzzy numbers. Further, he found the centroid and membership function of the fuzzy total cost.
An inventory model with backlogged in fuzzy environment and applied fuzzy on storing cost, on back-
order cost, on order quantity, cost of hiring an order as well as on shortage cost [43]. For Defuzzification,
signed distance method is used, from that they obtain approximate total cost of an inventory in fuzzy sense.
A mixture for inventory model and relating variable lead-time with backlogged has lost sales in fuzzy
environment [44]. Firstly, they used random lead-time demand to be a fuzzy random variable and then the
total demand is also Fuzzifier which is assumed as triangular fuzzy number. After that, authors evaluate all
cost in fuzzy sense with the help of centroid method of Defuzzification. The extended the work of Ouyang
and Chuang and assumed lead time, review period as decision variables in crisp sense [45]. The Fuzzifier
back order rate, probable demand shortage and expected demand in this model. To Defuzzifier the cost
signed distance formula is all plied and given an estimation of entire probable yearly cost in the fuzzy
environment and also compare the results of crisp and fuzzy models by giving an example. Construct an
inventory model in which partial demand is backlogged and the rest part is vanished throughout stock out
time [46]. These parameters are measured under fuzziness. To signify these characteristics trapezoidal
fuzzy numbers are used and find fuzzy cost.
An EOQ model for goods with imperfect quality and shortage is acceptable and completely backlogged
[47], [48]. In Gani model the inventory parameters (demand cost, ordering cost, defective rate, holding
cost, shortage cost)is considered as triangular fuzzy numbers. To Defuzzifier inventory parameters graded
mean integration, signed distance methods are applied to Defuzzifier the values respectively.
A model with fuzzy parameters and decision variables along with backorder has implemented in 2010 [49].
An inventory model has implemented in fuzzy environment for decomposable products where demand rises
with time and shortages are allowable and fully backlogged [50], [51], [52]. Researchers used different
methods to Defuzzifier the values. They minimize the cost and maximize the profit in fuzzy environment
with different parameters and compared the result of different Defuzzification methods [53].
An EOQ model has used in fuzzy environment in which deteriorating items dependent on time and
unfulfilled demand is partially backordered [54]. The inventory parameters (ordering cost, unit cost,
holding cost) have been fixed constant. Deterioration rate considered as fuzzy number in place of crisp
number. The development of two models in fuzzy environment completed in 2013 in which first model
input parameter is Fuzzifier and decision variables considered as crisp variables [55]. But in second model,
input parameters as well as decision variable are Fuzzifier. For Defuzzification graded mean integration
method is applied. A mathematical model developed and used in fuzzy environment for finding the least
total cost of multi-products with multi objectives [56]. Focused on permitting shortages for each item and
considered as fully backlogged, model could be capable to reduce manufacture cost due to increase in
demand thus increasing order quantity. A fuzzy model with partially backlog for deteriorating item and
model the demand rate is two parameters Weibull Distribution; inventory parameters are taken as triangular
number [57].
An inventory model under shortage with fully backlogged in fuzzy sense and demand is taken
deterministic and uniform [58]. Inventory parameters are taken as Triangular fuzzy number, parabolic
fuzzy number, Trapezoidal fuzzy number. The authors Defuzzifier these parameters by signed distance and
also by graded mean integration method. The formulation and implementation of the fuzzy set theory to the
inventory problem done and studied about perishable items with power demand rate where shortage is
acceptable and taken as partially backlogged [59]. The cost parameters are considered as triangular fuzzy
numbers. After that, model is Fuzzifier and for this they used signed distance, centroid, and graded mean
040023-5

7. integration methods. A fuzzy inventory model for deteriorating products has been proposed and its demand
in exponential and shortage is allowable under fully backlogged condition [60]. The authors translate the
inventory model into fuzzy inventory model. The inventory parameters are supposed as trapezoidal fuzzy
numbers. And to Defuzzifier the parameters signed distance method is used. Demand is price dependent by
assuming extreme interest earning rate on prefixed sales revenue [53]. Under Function Principle
arithmetic’s operations are defined. After Defuzzifier the parameters they found the optimal estimate of
inventory which maximize the total profit.
An idea about an inventory model has used for deteriorating goods and in their proposed model they
established an Economic Order Quantity (EOQ) model for deteriorating commodities with linear demand
rates in fuzzy environments [61]. Shortage is permitted and partially backorder. They applied graded mean
integration, centroid, and signed distance methods to Defuzzifier the values, which minimizing the cost. A
formulation of fuzzy inventory control model for time dependent Weibull deterioration and quadratic
demand rate [62], [63], [64]. Shortages are allowable, considered as partially backlogged. Also, in 2017 an
inventory model under shortage with completely backlogged in fuzzy sense has been established [65], [66],
[67], [68]. The backorder cost, carrying cost, and ordering cost are taken as triangular, trapezoidal, and
pentagonal fuzzy numbers. Further, signed distance method is applied for Defuzzification. A paper of
inventory model with Imperfect products under backlog, allowed proportionate discount in fuzzy
environment [69], [70]. They discussed equally crisp model and fuzzy EOQ model, with acceptable
proportionate discounts, dependent on the quantity of defective products. They applied fuzzy on defective
rate considered as triangular fuzzy number. For Defuzzification signed distance method is applied and
estimate total cost of an inventory. A fuzzy inventory model of two wares under shortage is developed and
used in fuzzy EOQ model for retailer by considering different parameters like a shortage, cost of ordering,
and cost of deteriorating by apply the method of graded mean integration [65], [66]. The analysis of the all
literature used in this study has been shown in table 1.
On the basis of this carefully review on fuzzy economic order quantity model under shortages this study
advocate that, for the Fuzzification centroid method used by 26 %, signed distance method used by 26%
and graded mean integration method used by majority of the researches can be seen in (Fig.2a). It has been
observed that Fuzzy Economic order quantity inventory model by Hexagonal fuzzy number (3%),
Pentagonal fuzzy number (6%) in Fig. 2b, the Triangular Fuzzy Number is used by preponderance
researcher 66% and followed by Trapezoidal fuzzy number 25%. The different parameters used by
researchers have shown in Fig. 2c. The fuzzy holding cost parameter and fuzzy shortage cost have been
used most of the authors and followed by fuzzy purchase cost then deterioration and ordering cost.
FIGURE 2 Analysis of (a) Fuzzy Inventory Parameters (b) Solution Method and (c) Defuzzification
Method
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8. CONCLUSIONS
Although fuzzy set theory has a lot of applications in different fields, it is not yet familiar to people how it
can be used in different products to minimize their cost that are presently existing in the market. Scientific
meaning of the term fuzzy is still fuzzy for many people. It is essential that these people identify where and
how fuzzy set theory can be applied. Fuzzy set theory assists to study fuzzy inventory models since it has
capability to enumerate vagueness and fuzziness without using randomness This brilliant issue defines
many vital research developments in real-life applications and thereby encourages others people to explore
research and growth in fuzzy set theory. There are many applications of fuzzy, which are pending for
research and developed. Over the past few years, inventory control models in fuzzy environment have
acknowledged attention because of providing more accessible information and handling the ambiguity. In
this paper, the participation of researchers in numerous classes of inventory models in fuzzy environment in
the last few years has been summarized. We thoroughly studied and examined the research papers in the
particular research area to classify the most important innovations, progresses and highlight the research
gaps. Each model either of inventory or fuzzy inventory based on some hypothesis has its aids and drawbacks.
The highlighting in each analysis was to classify how the fuzziness is applied in the development of the
inventory model. This paper may turn as a tool to encourage all researchers who are working or will be
working in the area of fuzzy inventory control for emerging innovative inventory models and also, it assists
as a good reference. A table 1 is prepared to examine the effort done by researchers in the field of fuzzy
inventory models with shortages either partially backlogged or fully backlogged. With the help of this
collected data, we can easily understand and differentiate their work.
TABLE 1 Different parameters and methods used Fuzzy modeling
Author's
Name
Demand
Type
Fuzzy Inventory Parameters Solution
Method
Defuzzifica
tion
Method
EOQ
Model
Deterioration
Fuzzy
Deterioration
Rate
Fuzzy
Lead
Time
Fuzzy
Demand
Cost
Fuzzy
Holding
Cost
Fuzzy
Purchase
Cost
Ordering
Cost
Fuzzy
Shortage
Cost
Fuzzy
Selling
Price
Cost
Fuzzy
Backorder
Cost
Fuzzy
Order
Quantity
Fuzzy
Stock
Quantity
Fuzzy
Storage
Quantity
Lost
Sales
Triangular
Fuzzy
Number
Trapezoidal
Fuzzy
Number
Pentagonal
Fuzzy
Number
Hexagonal
Fuzzy
Number
Centroid
Method
Signed
Distance
Graded
Mean
integration
Yao and
Lee (1996)
[39]
Cons
tant
dema
nd
√ √
√ √
Chang et
al. (1998)
[40]
Fixed
dema
nd
√ √ √ √
Yao and
Su (2000)
[41]
Fixed
dema
nd
√ √ √ √
Kweimei
Wu et al.
(2003) [42]
Const
ant
dema
nd
√ √ √ √
040023-7

9. Author's
Name
Dem
and
Type
Fuzzy Inventory Parameters Solution
Method
Defuzzifica
tion
Method
Chiang, JS.
Yao.
(2005) [43]
Const
ant
dema
nd
√ √ √ √ √ √ √ √
Chang et
al. (2006)
[44]
Fixed
dema
nd
√ √
`
√
`
YU -Jen
Lin (2008)
[45]
Const
ant
dema
nd
√ √ √ √
N. Kazemi
et al.
(2010) [49]
Fixed
dema
nd
√ √ √ √ √ √
Nagoor A
Gani, S.
Maheshwa
ri, (2010)
[47]
Const
ant
dema
nd
√ √ √ √ √ √ √
C. K. Jaggi
et al.
(2012) [50]
Time
varyi
ng
dema
nd
√ √ √ √ √ √ √ √ √
Sumana
Saha
(2012) [54]
Expo
nentia
l
dema
nd
√ √ √ √
Dutta and
Pavan
Kumar
(2013)
Const
ant
dema
nd
√ √ √ √ √ √ √
Adrijit
Gaur
Chandra
Mahata
Goswami
(2013)
Const
ant
dema
nd
√ √ √ √ √ √ √ √ √
`
√
Dutta, D.,
and Pavan
Kumar
(2013) [17]
Const
ant
dema
nd
√ √ √ √ √ √
Shanka
Ravr
Kumara
and A.
Goswamib
(2013) [48]
Annu
al
dema
nd
√ √ √ √
Nabendu
Sen, Sanju
kta
Malakar
(2015) [58]
Deter
minis
tic
dema
nd
√ √ √ √ √ √ √
040023-8

10. Author's
Name
Dem
and
Type
Fuzzy Inventory Parameters Solution
Method
Defuzzi
fication
Method
J. Sujatha
and P.
Parvathi
(2015) [57]
Weib
ull
type
dema
nd
√ √ √ √ √ √ √ √
N.Rajeswa
ri et al.
(2015) [59]
Powe
r
Dema
nd
√ √ √ √ √ √ √ √
D.
Sharmila
R.
Uthayaku
mar (2015)
[60]
Expo
nentia
l
dema
nd
√ √ √ √ √ √
Indrajit
Singha et
al. (2016)
[52]
Stock
depen
dent
dema
nd
√ √ √ √ √ √
J.Jayanthi
et.al (2017)
[67]
Const
ant
dema
nd
√ √ √ √ √
Rojalin
Patro et al.
(2017) [62]
Quad
ratic
dema
nd
and
time
depen
dent
√ √ √ √
Chandra K.
Jaggi
et.al.
(2017) [33]
Price
depen
dent
√ √ √ √
Dr. Mrs.P.
Parvathi,
D. Chitra
(2017)
Price
depen
dent
√ √ √
R .M.
Rajalaksh
mi,
G.Michael
Rosario
(2017) [65]
Const
ant
dema
nd
√ √ √ √ √ √ √ √
Neeraj
Kumar,San
jeyKumar
(2017) [61]
Linea
r
dema
nd
√ √ √ √ √ √ √ √ √
Rojalini
Patro et al.
(2018) [69]
Fixed
dema
nd
√ √ √ √
S.K.,
Indrajit
Singha,
(2019) [70]
Sellin
g
price
depen
dent
√ √ √ √ √ √ √
R.Arora(20
21) [71]
Constan
t
demand
√ √ √ √
040023-9

11. ACKNOWLEDGMENTS
Mrs. Preety Poswal is thankful for the K. R. Mangalam University for providing the necessary
facilities to conduct the research. Authors are very thankful for the anonymous reviewers for their
valuable suggestions.
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for variable lead time”, Journal of Operational Research Society, 47(6), 829-832(1996)
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backordering”, International Journal of Production Economics”, 59(1-3), 511–518, (1999)
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