Development of a fuzzy economic order qu (1)

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Journal of Intelligent & Fuzzy Systems 28 (2015) 2377–2389
DOI:10.3233/IFS-141519
IOS Press

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Development of a fuzzy economic order
quantity model for imperfect quality items
using the learning effect on fuzzy parameters
Nima Kazemi, Ezutah Udoncy Olugu∗ , Salwa Hanim Abdul-Rashid and Raja Ariffin Bin Raja Ghazilla

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Center for Product Design and Manufacturing (CPDM), Department of Mechanical Engineering,
Faculty of Engineering, University of Malaya, Kuala Lumpur, Wilayah Persekutuan, Malaysia

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Abstract. This paper develops an inventory model for items with imperfect quality in a fuzzy environment by assuming that

learning occurs in setting the fuzzy parameters. This implies that inventory planners collect information about the inventory
system and build up knowledge from previous shipments, and thus learning process occurs in estimating the fuzzy parameters.
So, it is hypothesized that the fuzziness associated with all fuzzy inventory parameters is reduced with the help of the knowledge
acquired by the inventory planners. In doing so, the study developed a total profit function with fuzzy parameter, where triangular
fuzzy number is used to quantify the fuzziness of the parameters. Next, the learning curve is incorporated into the fuzzy model
to account for the learning in fuzziness. Subsequently, the optimal policy, including the batch size and the total profit are derived
using the classical approach. Finally, numerical examples and a comparison among the fuzzy learning, fuzzy and crisp cases are
provided to highlight the importance of using learning in fuzzy model.

1. Introduction

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Keywords: EOQ model, fuzzy set theory, imperfect quality, inventory control, learning

In today’s highly competitive market, where companies are under pressure to reduce the time they need
for distributing their products and services, the importance of inventory management has become highly
inevitable for organizations. In order to be successful
in facing this challenge, organizations need to continually adopt proper techniques of keeping and managing

their inventories. Choosing the right policy for managing inventories has always been a great challenge
to manufacturing companies. Since the early of 20th
century, when the foundation of the earlier economic
order quantity (EOQ) and economic production quantity
models (EPQ) was laid, numerous mathematical models
∗ Corresponding

author. Ezutah Udoncy Olugu, Center for Product Design and Manufacturing (CPDM), Department of Mechanical
Engineering, Faculty of Engineering, University of Malaya, 50603
Kuala Lumpur, Wilayah Persekutuan, Malaysia. Tel.: +60379675212;
Fax: +60379675317; E-mail: olugu@um.edu.my.

emerged with the objective to assist organizations in
better planning of their inventories. Although there has
been a significant endeavor by scholars and practitioners
to provide more practical versions of inventory models, these models have significant shortcomings with
real-world problems. For example, one of the assumptions of the classical models is that the supply process
is of perfect quality, whereas it is a common occurrence
that buyers receive batches containing certain fraction
of imperfect quality items (see, e.g., [1, 2]). Thus, it is

essential to develop models that release the unrealistic
assumptions of the conventional models and consider
imperfect quality in produced or received batches.
Another unworkable assumption of the most inventory systems is that the data available to the decision
makers are constant during the planning horizon. However, the available data may vary from time to time,
which makes the inventory system’s modeling more
cumbersome. That is, one or some inventory values may
not remain constant during planning horizon, and could

1064-1246/15/$35.00 © 2015 – IOS Press and the authors. All rights reserved

N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality

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ues (which shows the amount of fuzziness) is not simple
since decision makers are not either acquainted well
with the data or do not have much information about
the process. However, during the planning period, as the
time passes and more information about the properties

of decision data are collected and analyzed, the decision
makers could then be able to enhance the accuracy of data
estimation, and thus could reduce the amount of uncertainty they are facing. It is obvious that learning is a very
useful tool for decision makers to reduce the impact of
imprecision on the quality of their decisions.
A closer look at the literature shows that in spite of
several models which were developed under fuzzy conditions, only a few numbers of studies were devoted to
model human factor role in the problem, while the entire
focus was mainly on modeling the fuzziness associated
with the planning problem. Of the entire aforementioned
studies, only one study can be found (see section 2) that
modeled the human role and its impact on fuzziness
modeling in inventory planning. However, the important
role that human plays in an inventory planning process and the high proportion of human work either in
gathering or collecting, processing and revising inventory data highlights the influence that human has on
inventory systems. Thus, it is apparent that the developed inventory models present an imperfect picture of
real-world’s inventory planning problems, which influence the planning outcome. In addition, by considering
human interaction with the inventory system, it is obvious that assuming a constant amount of fuzziness in
every phase of the planning is completely an unrealistic assumption as the amount of uncertainty a decision
maker encounters may vary over time. In order to present

a better representation of reality and close this research
gap, this paper develops a model with imperfect quality
and fuzzy parameters which learning occurs in setting
fuzziness values. For this purpose, it is assumed that
annual demand, holding cost, set up cost, selling price
of defective items and percentage of defective items
are fuzzy numbers, and therefore their value fluctuates
between a lower and upper bound. It is subsequently
assumed that the amount of fuzziness is subjected to
learning and the fuzzy total profit function with learning
in fuzziness is thus developed using fuzzy mathematics.

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adopt different values from cycle to cycle. This happens
frequently in real-world inventory problem when, for
example, companies have modification in their product,

or they encounter a situation that market demand alters
regularly. Hence, this raises the degree of uncertainty
that inventory planners should take into account when
planning. Having this dynamic situation makes inventory planning more troublesome, as decision makers
are unable to define exact values for input data. In such
cases, it is possible that the decision objects have a fluctuation from their bases or could be defined orally, such
as: “ordering cost is substantially less than x” or “set
up cost is approximately of value y”. Fuzzy theory has
been recognized as a useful tool to tackle this kind of
imprecision that allows converting the oral expression
or approximate estimation to a mathematical relation
(e.g., [3, 4]). These mathematical expressions could
be combined into the inventory problems and could be
helpful in providing a flexible model, which facilitates
modeling imprecise data.
Although fuzzy set theory provides an efficient tool
to either model various types of uncertainties mathematically or deal with various sources of uncertainties
in inventory management, the uncertainty associated
with the inventory system could decrease performance
of the system, blur data estimation process and increase

the complexity of planning. On the other hand, with reference to the literature on inventory management, the
higher level of uncertainty could be so costly for firms
and could thus increase their total cost of inventory system, which is the reason why firms undoubtedly try
to avoid it. These intelligibly indicate that it is crucial
to develop inventory models to tackle the uncertainty
of inventory systems in an appropriate manner. This
will definitely aid organizations to avoid making wrong
and costly decisions. However, this research topic still
remains as one of the challenging problems in the inventory management literature.
Learning concept, which occurs in every process as
a consequence of practice, is found to be a useful tool
to improve performance (see [5]). One of the areas that
learning could help to improve the performance is the
decision making process, particularly when decision
makers should reconsider or revise their decision in the
latter steps according to the knowledge they acquired
from the earlier steps. As stated above, in fuzzy inventory
models, there are abundant situations in which an exact
value cannot be determined for parameters of the model,
but instead a specific range (termed as spread values)

can be defined so that inventory parameters fluctuate.
At the initial planning stage, estimating the varying val-

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2. Literature review
Despite their simplicity, the EOQ and EPQ models
are still, surprisingly, the cornerstone of many inventory
models and are widely used in both theoretical and prac-

N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality

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the percentage of defective items and demand simultaneously. Vijayan and Kumaran [18] studied another
form of the EOQ model termed as the Economic Order

Time (EOT) model and performed two different policies of fuzziness. In the first policy, they assumed that
all the parameters of the EOT model were imprecise, but
could be described by trapezoidal and triangular fuzzy
numbers and then obtained the optimal policy using the
Lagrangian method. In the next step, they examined
the parameters and variable of the model under fuzzy
sense and again trapezoidal and triangular fuzzy numbers were applied to model the fuzziness. Bj¨ork [19]
developed an EOQ model with backorders where the
lead time (and consequently the maximum inventory
level) and total demand were assumed to be triangular
fuzzy numbers. Using an analytical solution, an optimal order quantity is derived for the model proposed.
A similar topic was treated by Kazemi et al. [20], who
investigated the classical model with backorders with a
different defuzzification method to that of Bj¨ork [19].
The author proposed an analytical solution for solving the fully-fuzzy model, where the model was tested
for triangular and trapezoidal fuzzy numbers. Bj¨ork
[21] considered a simpler problem to that of Kazemi
et al. [20] and proposed a multi-item EOQ model with
fuzzy cycle time. In a recent paper, Shekarian et al. [22]
developed a fuzzified version of a lot-sizing model for

a single-stage production system with defective items
and rework, which defective items are immediately
reworked within the same cycle. They assumed that the
rate of defects and demand rate were triangular fuzzy
number and therefore used two defuzzification methods
to derive the crisp total cost function.
The models discussed so far treated human capabilities as a constant factor, e.g., by assuming that the
fuzziness associated with inventory data adopts a constant value over the planning horizon, and ignored the
fact that human capabilities are subject to change by
passing time, and that this element could in turn influence estimating the fuzzy parameters. The only work
which is an exception to this line of research is that
of Glock et al. [3], who investigated the possibility of
reducing the fuzziness in demand in an EOQ model
using the learning curve. Through a numerical example,
the authors suggested that when the amount of uncertainty is rather high in demand prediction, it is better for
the buyer to increase the frequency of orders by ordering
small quantities to the supplier. Hence, this paper counters an assumption in previous works and contributes to
the area of fuzzy inventory by the assumption that the
planners could use their knowledge in setting the fuzzy

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tical aspects. However, some of their assumptions are
never met in practice [6]. This inspired several authors
to present the improved version of the basic models to
give them a touch of reality. One of the limitations of the
EOQ or EPQ models, which has received considerable
attention by researchers, is that the items in a produced
or received batch are not of perfect quality. Several works
can be found in the literature that addressed this problem.
For example, Chan et al. [7] developed an EPQ model
with imperfect quality items which are sold at a lower
price at the end of the production period or cycle. Jamal
et al. [8] developed an EPQ model that produces defective items which are reworked to good ones during the
same production cycle. C´ardenas-Barr´on [9] developed
an EPQ model for determining the economic production
quantity and size of backorders for a system that generates imperfect quality (defective) items with planned
backorders. Defective items are reworked to as-good-asnew condition in the same production cycle. One of the
prominent works along this line of research is the model
of Salameh and Jaber [10], who assumed that shipments
contain a random percentage of defective items. Upon
receiving batches, they undergo 100% inspections with a
rate faster than demand rate and imperfect quality items
are withdrawn and sold as a single batch by the end of
the screening process. The work of Salameh and Jaber
[10] has received increasing attention by researchers.
Hence, lots of researchers have presented the extension
or modification of this model. For an extensive review
of the models that deal with the model of Salameh and
Jaber [10] the readers are referred to Khan et al. [11]. In
addition, some papers which also appeared after 2011
are shortly noted in Jaber et al. [12]. None of the above
reviewed models considered uncertainty in their model.
However, as discussed in the previous section, assuming
deterministic values in an uncertain decision situation
may lead to erroneous inventory policies.
Fuzzy set theory is recognized as a proper method
in dealing with uncertainty. Since the development of
fuzzy set theory by Zadeh [13], it has been one of
the interesting area of research for scholars [14–16].
Reviewing the literature shows that numerous studies
have thus far been conducted to investigate the application of fuzzy set theory in inventory management.
One of the main research streams in this line of thought
has been extending the classical models into a fuzzy
environment enabling the models to tackle the fuzziness. For example, Chang [17] presented an extended
version of the work of Salameh and Jaber [10] in two
different models, where the first one fuzzified the percentage of defective items and the second one fuzzified

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N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality

˜ is called a triangular fuzzy number
A fuzzy number A
(TFN) and is denoted by (l, m, n) if it has the following
piecewise linear membership function

o ≤ x ≤ n,
otherwise.

(1)

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3.1. Function principle

l ≤ x ≤ m,

OR

µA˜ (x) =
⎧
x−1
⎪
θ(x) = m−l
,
⎪
⎨
n−x
π(x) = n−o ,
⎪
⎪
⎩
0,

In this paper, Graded Mean Integration Representation (GMIR) method is applied to transform the fuzzy
total profit function to its corresponding crisp function. The main reason is due to the non-linear nature
of the function that was used in this paper. In fact, the
fuzzy total profit function in this paper consists a couple of fuzzy multiplication and division terms, and since
GMIR method keeps the shape of membership function,
it is a proper choice to defuzzify the fuzzy profit function of the model. In the following section, the GMIR
method introduced by Chen and Hsieh [25] is described.
˜ in Equation (1), let θ −1 and
For the fuzzy number A
π−1 be the inverse functions of θ and π, respectively.
−1
−1
˜ is ρ(θ (ρ)+π (ρ))
The graded mean ρ-level value of A
2
˜ is calculated as:
and the GMIR of the fuzzy number A




 wA
1
ρ θ −1 (ρ) + π−1 (ρ)
˜ =
τ(A)
dρ
ρ dρ
2
0
0
 wA 

θ −1 (ρ) + π−1 (ρ) dρ,
=
(2)

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3. Preliminaries

3.2. Defuzzification

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parameters, and the fuzziness associated with inventory data could in turn be reduced. To this aim, this
paper applies the concept of learning in fuzziness to an
EOQ model with imperfect quality, which has enjoyed
increasing attention in recent years. To the best of our
knowledge, there is no inventory model with imperfect
quality in the literature that applies learning in fuzziness. This is the limitation that this paper addresses.
The model developed in this paper is the extension of
Salameh and Jaber [10] to the fuzzy-learning environment. The next section will review some basics and
definitions of fuzzy set theory, which will be applied
through the paper.

To perform fuzzy arithmetical operations by TFN,
Function Principle proposed by Chen [23] is used.
Function Principle is a suitable method for performing
the operations of complex models to prevent arriving
at a degenerated solution. This method will be so helpful in handling the fuzzy operations, especially when
the crisp model comprises terms of multiple operations of fuzzy numbers. Furthermore, the type of fuzzy
membership function will be kept constant during the
operations, which helps to avoid facing further complexity by arithmetical operations (e.g., [24]). Now,
˜ = (l1 , m1 , n1 ) and B
˜ = (l2 , m2 , n2 ) are
assume A
two positive TFNs and α be a real number. Based on
the Functional Principle, the operations of the fuzzy
˜ and B
˜ are as the following:
numbers A
˜ = (αl1 , αm1 , αn1 ) and if α < 0
i. If α ≥ 0 αA
˜
αA = (αn1 , αm1 , αl1 ).
˜ +B
˜ = (l1 + l2 , m1 + m2 , n1 + n2 ).
ii. A
˜
˜ = (l1 − n2 , m1 − m2 , n1 − l2 ).
iii. A − B
˜
˜ = (l1 l2 , m1 m2 , n1 n2 ).
iv. A × B


˜
m1 n1
A
v. B˜ = nl12 , m
,
l
2
2

0

where 0 < ρ ≤ wA and 0 < wA ≤ 1.
˜ = (l, m, n) be a TFN. The graded
Now assume A
˜ can be calculated
mean integration representation of A
by formula (2) which is as


 1
1
−1 (ρ) + π −1 (ρ))
ρ(θ
˜ =
τ(A)
dρ
2
0
0
ρdρ = (l + 4m + n)/6.

(3)

3.3. Learning
Learning is defined as the improvement in a process which is obtained as a result of practice [26].
Early investigations in learning theory disclosed that
the amount of time required to perform a task declines
at a decreasing rate as experience with the task increased
[26]. The rate at which cumulative experience allows an
employee to improve tasks can be represented using a
graphical representation termed as the learning curve.
The learning curve is also a mathematical function that
can be used to estimate the progress of a person as
they learn to do their work. The earlier study on the
effect of learning in an industrial setting was due to
Wright [27], who studied the impact of learning on production costs in the aircraft industry. Since the work
of Wright [27], learning theory has received increasing attention by researchers and practitioners in various

N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality

Q
h
x
D
ν
s

(4)

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yi = y1 i−β

where yi is the performance at the time of ith shipment, y1 is the performance at the beginning of the
planning period, i is the number of shipments and β is
learning exponent. The learning exponent can be calculated using the equation β = − log(δ)/ log(2), where
δ is learning rate and can be described as a percentage
ranging from 50% to 100%. If the learning rate is 100%,
the learning exponent adopts 0 in equation while in case
the learning rate is 50% the learning exponent is equal
to 1. Figure 1 illustrates the behavior of the learning
curve in Equation (4).
4. The model
The following notations are being made for developing the mathematical models:
p
d
A
γ


ui

4.1. Fuzzy modeling of the EOQ model with
imperfect quality

In this paper, a fuzzy EOQ model is developed for
a buyer, who receives lot size containing the percentage of defectives γ, with a known probability density
function, f (γ). This model is the fuzzy version of the
model of Salameh and Jaber [10]. They modeled the
EOQ model with imperfect quality under the following set of assumptions: 1- the orders are replenished
instantaneously 2- upon receiving the order, a 100%
screening process of the lot is conducted 3- Items
of poor quality are kept in stock and sold prior to
receiving the next shipment as a single batch at a discounted price. In order to prevent shortages, Salameh
and Jaber [10] assumed that the number of good items
in each order is at least equal to the demand during the
screening time; that is N(γ, y) = y − γy ≥ Dt. In this
paper, it is assumed that decision makers are unable
to state deterministic values for some parameters of
the inventory model because of the uncertainty they
encounter, yet could specify a triangular fuzzy membership function for each uncertain parameter. For this
purpose, let us assume that the fraction of defective
items, demand rate, buyer’s ordering cost per cycle,
buyer’s holding cost per unit per year and screening
cost per unit are uncertain. This is a true image of reality as these parameters are very difficult or in many
situations impossible to estimate in the real world. It is
assumed that decision makers can define membership
functions for the aforementioned imprecise parameters using the triangular fuzzy number, which can be
represented as

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fields like manufacturing, healthcare, energy, military,
information technologies, education, design, and banking [28, 29]. Since then, several learning curve models
were developed that have different forms [29–33]. All
the models developed have commonly argued that the
performance improves with the repetition of a task.
Of all the available models, the Wright’s [27] learning curve is found to be the most popular and basic
model due to its mathematical simplicity and ability
to fit well a wide range of learning data [28, 31, 33,
34]. For a comprehensive review of learning models
and their application readers may refer to Jaber [35]
and Anzanello and Fogliatto [29]. The learning curve,
which is used in this paper, is that of Wright [27] which
is of the form

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Fig. 1. The behavior of the Wright’s learning curve.

purchasing price of unit product
screening cost per unit
buyer’s ordering cost per cycle
fraction of defective items

lot size per cycle
buyer’s holding cost per unit of time
screening rate
demand rate per unit of time
unit selling price of defective items
unit selling price of non-defectives
(good) items
probability density function of γ
number of good items in each order
screening time
cycle length
the lower deviation value (spread) value
for ith parameter i = D, h, γ, d, A
the upper deviation value (spread) value
for ith parameter i = D, h, γ, d, A

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f (γ)
N(γ, y)
t
T

li

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N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality

˜ = (D −
lD , D, D +
uD ),
D

−

h˜ = (h −
lh , h, h +
uh ),

(A −
lA , A, A +
uA )


1
1
1
,
,
1 − γ +
lγ 1 − γ 1 − γ −
uγ

γ˜ = (γ −
lγ , γ, γ +
uγ ),
d˜ = (d −
ld , d, d +
ud ),
˜ = (A −
lA , A, A +
uA ),
A

(5)

1
(D −
lD , D, D +
uD )
Q

Q
(h −
lh , h, h +
uh )
2
Q
+ (h −
lh , h, h +
uh )(γ −
lγ ,
2
γ, γ +
uγ )

˜
˜
˜ − ν + hQ )
T PU(Q)
= D(s
x
˜
˜
˜ − hQ − p − d˜ − A )( 1 )
+D(ν
x
Q 1 − γ˜
˜
˜
hQ(1
− γ)
2

(6)

Replacing triangular fuzzy numbers in Equation (6)
will result in the following fuzzy total profit function

+

Q
(D −
lD , D, D +
uD )
x
(h −
lh , h, h +
uh )

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˜
TPU(Q)
= (s − ν) (D −
lD , D, D +
uD )

+ν(D −
lD , D, D +
uD )


1
1
1
,
,
1 − γ +
lγ 1 − γ 1 − γ −
uγ
−

Q
(D −
lD , D, D +
uD )
x
(h −
lh , h, h +
uh )


1
1
1
,
,
1 − γ +
lγ 1 − γ 1 − γ −
uγ

−p(D −
lD , D, D +
uD )


1
1
1
,
,
1 − γ +
lγ 1 − γ 1 − γ −
uγ

+ν(D −
lD )

OR

−

−

1
1 − γ +
lγ

1
Q
(D −
lD ) (h −
lh )
x
1 − γ +
lγ

−p(D −
lD )

1
1 − γ +
lγ

−(D −
lD )(d −
ld )
−

1
1 − γ +
lγ

1
1
(D −
lD )(A −
lA )
Q
1 − γ +
lγ


Q
Q
(h −
lh ) + (h −
lh )(γ −
lγ )
2
2

4
Dυ
QDh
DQh
+ D(s − υ) +
+
−
6
x
1−γ
x(1 − γ)
−


Dd
pD
AD
Qh(1 − γ)
−
−
−
−
(1 − γ) (1 − γ) Q(1 − γ)
2

1
Q
+ (s − υ)(D +
uD )+ (D+
uD )(h +
uh )
6
x
+ν(D+
uD )

−(D −
lD , D, D +
uD )
(d −
ld , d, d +
ud )


1
1
1
,
,
1 − γ +
lγ 1 − γ 1 − γ −
uγ

(7)

Using Function Principle defined in Section 3.1, the
graded mean integration (defuzzified) value of the fuzzy
total profit function in Equation (7) is given by


˜
T PU(Q)
=

Q
1
(D −
lD ) (s − ν) + (D −
lD ) (h −
lh )
6
x

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Here,
li , i = D, h, γ, d, A and
ui , i =
D, h, γ, d, A represent the values that a parameter
can deviate from its base value, which are called
fuzziness values or spread values. The fuzzy form of
the EOQ model with imperfect quality in Salameh and
Jaber [10] is given as

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−

−

1
1 − γ −
uγ

1
Q
(D +
uD )(h +
uh )
x
1 − γ −
uγ

−p(D +
uD )

1
1 − γ −
uγ

N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality

−(D +
uD )(d +
ud )

1
1 − γ −
ur

−

1
1
(D +
uD )(A +
uA )
Q
1 − γ −
uγ

−

Q
Q
(h +
uh ) + (h +
uh )(γ +
uγ )
2
2

˜
=
τL (T PU(Q))

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1
(s − ν) (D −
lD, 1 i−β )
6

Q
(D −
lD, 1 i−β )(h −
lh, 1 i−β )
x
1
+ν(D −
lD, 1 i−β )
1 − γ +
lγ, 1 i−β
+

(8)

˜
τ(T PU(Q))
is considered as the estimated value or
crisp value of fuzzy total profit per unit time. In the
next section, the fuzzy EOQ model presented in this
section will be developed to account for learning in
fuzziness.

Q
(D −
lD, 1 i−β )(h −
lh, 1 i−β )
x
1
1 − γ +
lγ, 1 i−β

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−

−p(D −
lD, 1 i−β )
4.2. Incorporating learning into fuzzy model

1
1 − γ +
lγ, 1 i−β


lj, i =



i=1


lj, i

lj, 1

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In this section, we model the decision maker’s learning in estimating the fuzziness values and develop a
model with the assumption that decision makers learn
with time when adopting a fuzziness value for the
parameters. In addition, we assume that decision maker
could use their knowledge to reduce the fuzziness of
the data. In order to investigate the effect of learning in
fuzziness on inventory policy, we suppose that buildup
of knowledge occurs with the number of shipments.
We consider the learning to follow the Wright’s [27]
power learning curve given in Equation (4). If learning affects the fuzzy parameters and if their value
changes according to the number of shipment, then for
j = D, A, h, d, γ the value of jth upper and lower
fuzziness parameter at the time of ith shipment will be
as


uj, i
i=1

uj, i =
(9)
−β
i>1

uj, 1 i

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−(D −
lD, 1 i−β ) (d −
ld, 1 i−β )

i−β

i>1

(10)

Where β is the learning exponent defined in Equation
(4). Equations (9) and (10) show that for the first shipment the fuzziness values take their initial values (
lj, 1
for lower fuzziness and
uj, 1 for upper one), which
are the maximum values during the planning horizon.
These values are usually determined by the decision
makers. After the first shipment, buildup of knowledge
through learning occurs and the value of the lower and
upper fuzziness parameters reduces exponentially. Consequently, the total fuzzy profit function with learning
˜
(τL (T PU(Q)))
for ith shipment i ≥ 1 is given as:

1
1 − γ +
lγ, 1 i−β

−

1
(D −
lD, 1 i−β ) (A −
lA, 1 i−β )
Q
1
1 − γ +
lγ, 1 i−β

−

+

Q
Q
(h −
lh, 1 i−β ) + (h −
lh, 1 i−β )
2
2

(γ −
lγ, 1 i−β )

Dν
DQh
4
QDh
+
D(s − ν) +
−
6
x
1−γ
x(1 − γ)

Dd
AD
Qh(1 − γ)
pD
−
−
−
(1 − γ) (1 − γ) Q(1 − γ)
2

1
+ (s − ν) (D +
uD, 1 i−β )
6

−

Q
(D +
uD, 1 i−β ) (h +
uh, 1 i−β )
x
1
+ν(D +
uD, 1 i−β )
1 − γ −
uγ, 1 i−β
+

−

Q
(D +
uD, 1 i−β ) (h +
uh, 1 i−β )
x
1
1 − γ −
uγ, 1 i−β

−p(D +
uD, 1 i−β )

1
1 − γ −
uγ, 1 i−β

−(D +
uD, 1 i−β ) (d +
ud, 1 i−β )



2384

N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality

1
1 − γ −
ur, 1 i−β

1
1 − γ −
uγ, 1 i−β

1
+ (h +
uh, 1 i−β ) (γ +
uγ, 1 i−β )
2

1
− (D +
uD, 1 i−β ) (A +
uA, 1 i−β )
Q

Q
Q
(h +
uh, 1 i−β ) + (h +
uh, 1 i−β )
2
2

(γ +
uγ, 1 i−β )

(11)

Before obtaining the optimal policy for the model, we
need to verify that the total profit per unit time given in
Equation (11) is concave. This can be proved by taking
the first and second partial derivatives with respect to Q.
Taking the partial derivatives with respect to Q yields:

(h −
lh, 1 i−β )

+

AU
TH

1
− (D −
lD, 1 i−β ) (h −
lh, 1 i−β )
x
1
1 − γ +
lγ, 1 i−β

1
(D −
lD, 1 i−β ) (A −
lA, 1 i−β )
Q2
1
1 − γ +
lγ, 1 i−β

1
1
− (h −
lh, 1 i−β ) + (h −
lh, 1 i−β )
2
2

(γ −
lγ, 1 i−β )



2 Dh
Dh
AD
h(1 − γ)
+
−
+
−
3 x
x(1 − γ) Q2 (1 − γ)
2

1 1
+
(D +
uD, 1 i−β )(h +
uh, 1 i−β )
6 x
1
− (D +
uD, 1 i−β ) (h +
uh, 1 i−β )
x
1
1 − γ −
uγ, 1 i−β
+

1
(D +
uD, 1 i−β ) (A +
uA, 1 i−β )
Q2

1
(1 − γ +
lγ,1 i−β )
+(D +
uD, 1 i−β ) (A +
uA, 1 i−β )
1
(1 − γ +
uγ,1 i−β )

4AD
+
(1 − γ)

(13)

Since the first multiplication term in Equation (13) is
negative for all values of Q, the function will be strictly
concave if and only if the second multiplication term
will be greater or equal to zero. One can easily conclude
that the second expression in Equation (13) is the summation of three positive terms, so it will be greater or
equal zero. As a result, the function in Equation (10) is
strictly negative for all values that Q adopts. Now, the
objective is to maximize the estimation of fuzzy total
˜
profit per unit time with learning. Since τL (T PU(Q))
is concave, the optimal lot size could be obtained by
setting the partial derivative (first grade) equal to zero.
This yields

OR


˜
∂τL (T PU(Q))
1 1
=
(D −
lD, 1 i−β )
∂Q
6 x

PY

−

(12)

˜
∂τ2L (T PU(Q))
=
∂Q2

1
− 3 (D −
lD,1 i−β ) (A −
lA,1 i−β )
6Q

CO

1
1 − γ −
uγ, 1 i−β



Q=



Y
X

(14)

Where
Y=

1
[(D −
lD,1 i−β ) (A −
lA, 1 i−β )
6
4AD
1
+
−β
1 − γ +
lγ, 1 i
(1 − γ)

+(D +
uD, 1 i−β ) (A +
uA, 1 i−β )
1
]
1 − γ +
lγ, 1 i−β
X=

1 1
[− (D −
lD, 1 i−β ) (h −
lh, 1 i−β )
6 x

2385

N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality

1
+ (D −
lD, 1 i−β ) (h −
lh, 1 i−β )
x
1
− (h −
lh, 1 i−β ) (γ −
lγ, 1 i−β − 1)]
2
Dh
h(1 − γ)
2 Dh
−
−
]
− [
3 x
x(1 − γ)
2

(γ +
lγ, 1 i−β )

5. Numerical example

AU
TH

OR

It can be noted that when there is no learning, β = 0,
the total profit function in Equation (11) and optimal
order quantity in Equation (14) reduce to that of fuzzy
model developed in section 4.1. Besides, when there
is no imprecision in estimating the parameters, then
deviation values will be equal to zero, that is
lj = 0
and
uj = 0, j = D, A h d γ. In this case, the total
profit per unit time and optimal lot size lead to optimal
policy derived in Salameh and Jaber [10].

CO

1
1
+ (h +
uh, 1 i−β ) − (h +
uh, 1 i−β )
2
2

PY

1 1
+ [− (D +
uD, 1 i−β ) (h +
uh,1 i−β )]
6 x
1
+ (D +
uD, 1 i−β ) (h +
uh, 1 i−β )
x
1
1 − γ −
uγ, 1 i−β

cost A = $100/cycle, holding cost h = $5/unit/year,
screening rate x = 1 unit/min (equivalently, x = 175,200
units/year), screening cost d = $0.5/unit, purchase
cost p = $25/unit, selling price of good-quality items
s = $50/unit, and selling price of imperfect quality
items ν = $20/unit. The values in which the fuzzy
parameters can fluctuate are respectively set as (5000,
10000), (6, 5), (0.1, 0.35), (0.0005, 0.15), (1, 2) for
D, A, d, γ, h. In order to facilitate the computation
process, the formulas were written in Microsoft Excel
2010. To investigate the optimal policy of the model
with learning in fuzziness, the impacts of different
learning rates on the optimal order quantity and total
profit per unit time are examined, with the results summarized in Table 1. The learning exponents are set
from 0.862 (very slow learning) to 0.074 (very fast
learning). For discussion on various learning rates in
industry reader may refer to Jaber [26]. The optimal
policy was derived for different total number of shipments in a cycle, which adopts 5, 10, 15 and 20. The
obtained results show that, comparing with the crisp
model (see Table 2 and also Salameh and Jaber [10]),
the optimal policy of the learning-fuzzy model is fairly
sensitive to learning rate. Even though the change in
policy is not much significant, it can be inferred from
the result that using learning change the optimal policy
of the model so that when learning increases both optimal values increase as a result. For example, when the
number of orders is 5 and learning exponent shifts from
0.862 to 0.074, the optimal order quantity increases
from 1438.11 to 1450.87. For other shipments, the same
pattern can be observed. Comparing the total profit with
crisp model also indicates that there is a slight change
in the learning-fuzzy model; however learning-fuzzy
model generates lower profit. Generally, the result may
suggests that provided that learning occurs in inventory
system, the deviation generated by the learning model
could change the policy, and consequently ignoring

In this section, an example is given to illustrate the
behavior of the model presented in Section 4.2. The
input parameters of numerical example are adopted
from Salameh and Jaber [10]. Consider an inventory model with crisp parameters having the following
values: demand rate D = 50, 000 units/year, ordering

Table 1
Optimal policy with different learning rate
i=5

β
0.862
0.737
0.621
0.515
0.415
0.322
0.234
0.152
0.074

i = 10

i = 15

i = 20

Q∗

˜
τ(T PU(Q))

Q∗

˜
τ(T PU(Q))

Q∗

˜
τ(T PU(Q))

Q∗

˜
τ(T PU(Q))

1438.11
1439.03
1440.10
1441.35
1442.80
1444.45
1446.34
1448.47
1450.87

1,203,877.78
1,204,360.20
1,204,867.17
1,205,388.28
1,205,910.70
1,206,418.77
1,206,893.57
1,207,312.34
1,207,647.78

1436.40
1437.08
1437.94
1439.04
1440.40
1442.10
1444.19
1446.75
1449.84

1,202,830.03
1,203,271.60
1,203,784.39
1,204,363.41
1,204,997.64
1,205,667.98
1,206,344.58
1,206,983.26
1,207,520.61

1435.83
1436.38
1437.12
1438.09
1439.36
1441.01
1443.14
1445.85
1449.27

1,202,427.34
1,202,816.97
1,203,295.73
1,203,865.95
1,204,523.02
1,205,251.80
1,206,021.70
1,206,779.58
1,207,439.46

1435.54
1436.02
1436.67
1437.55
1438.75
1440.34
1442.46
1445.25
1448.88

1,202,210.87
1,202,560.02
1,203,006.22
1,203,557.80
1,204,216.28
1,204,971.77
1,205,796.04
1,206,632.26
1,207,379.31

2386

N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality
Table 2
Comparing crisp, fuzzy and fuzzy learning model


l


u

5000
6
0.3
0.01
1

3000
8
0.2
0.03
2
5
0.415
1423.65
1,196,968.71
10
0.515
1,426.24
1,198,111.14
15
0.515
1,427.47
1,198,634.91
20
0.515
1428.24
1,198,954.34


l


u

5000
6
0.3
0.01
1

3000
8
0.2
0.03
2
5
0.234
1,420.30
1,195,389.97
10
0.234
1,422.26
1,196,329.95
15
0.234
1,423.30
1,196,807.90
20
0.234
1423.98
1,197,118.61


l

3000
8
0.2
0.03
2
5
0.074
1,418.17
119,4321.97
10
0.074
1,417.41
1,193,926.07
15
0.074
1,417.86
1,194,160.39
20
0.074
1418.17
1,194,321.97

1455
1450
1445
1440
1435
1430
1425
0.862 0.737
0.621 0.515
0.415 0.322
Learning
Exponent

0.234

0.152

20
15
10
5
0.074

Fig. 2. Three dimensional graph of the optimal order quantity, learning exponent and number of shipments.

1208000
1206000
1204000
1202000
1200000
1198000
0.862 0.737

15
0.621 0.515
0.415 0.322
0.234 0.152
Learning Ex
ponent

5
0.074

mbe
r

AU
TH

learning in inventory planning may result in improper
policy. Additionally, the results in Table 1 can give
general policy in learning-fuzzy model with imperfect
quality. That is, when the learning rate is slow decision maker should order smaller lot size to the supplier,
which incur lower total profit. In contrast, faster learning results in greater lot sizes with higher total profit.
Therefore, learning in fuzziness makes the buyer order
more and more, which tends to increase the total profit.
The importance of learning here also suggests that organizations should provide an environment to facilitate
learning in their inventory systems. As to the sensitivity of the model to the number of shipments, it is clear
from Table 1 that both optimal values tend to decrease
with number of shipment, at a constant learning rate. So,
the optimal order quantity and total profit are expected
to decrease when the number of shipments in a cycle
increases. Figures 2 and 3, which are plotted based on
the data tabulated in Table 1, depict the variation in optimal policy, while varying the other parameters of the
model.
The behavior of the learning-fuzzy model developed
in Section 4.2 is additionally investigated using several
fuzzy numbers. Table 2 compares the result of crisp,
fuzzy and fuzzy learning model for some arbitrary sets
of
and different number of shipments, ranging from 5
to 20. For the learning-fuzzy model, the learning parameters are respectively set at 0.862, 0.621, 0.415, 0.234,


u

5000
6
0.3
0.01
1

mber

3000
8
0.2
0.03
2
5
0.621
1,426.58
1,198,256.81
10
0.737
1,429.28
1,199,380.64
15
0.737
1,430.43
1,199,839.55
20
0.737
1429.83
1,199,601.18

ment
nu

5000
6
0.3
0.01
1

men
t nu

3000
8
0.2
0.03
2
5
0.862
1,429.06
1,199,290.56
10
0.86
1,431.49
1,200,255.54
15
0.86
1,432.39
1,200,599.81
20
0.86
1432.87
1,200,779.70

Sh ip

5000
6
0.3
0.01
1


u

Ship


l

PY

3000
8
0.2
0.03
2
NA
NA
1,414.66
1,192,435.13
NA
NA
1,414.66
1,192,435.13
NA
NA
1414.66
1,192,435.13
NA
NA
1414.66
1,192,435.13


u

CO

5000
6
0.3
0.01
1

Fuzzy learning model

l

Optimal order quantity

0
0
0
0
0
NA
NA
1,434.61
1,212,235
NA
NA
1,434.61
1,212,235
NA
NA
1434.61
1,212,235
NA
NA
1434.61
1,212,235


u

OR

0
0
0
0
0

Fuzzy model

l

profit

D
A
d
γ
h
i
β
Q∗
˜
τ(T PU(Q))
i
β
Q∗
˜
τ(T PU(Q))
i
β
Q∗
˜
τ(T PU(Q))
i
β
Q∗
˜
τ(T PU(Q))


u

Optimal total

Crisp model

l

Fig. 3. Three dimensional graph of the optimal total profit, learning
exponent and number of shipments.

and 0.074. With the help of results in Table 2, it can be
examined if learning in fuzziness can be suitably used to
reduce the fuzziness of the model. The obtained results
reveal that the optimal quantities of the learning-fuzzy

2387

N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality
Table 3
The impact of increasing the amount of fuzziness for demand on optimal policies
Lower and upper limits
3
3
3
3
3
3
3
3
3

Fuzzy learning model
˜
τ(T PU(Q))


uA


ld


ud


lγ


uγ


lh


uh


lD


uD

i

β

Q∗

4
4
4
4
4
4
4
4
4

0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04

0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03

0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003

0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004

0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4

0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3

300
700
1400
2300
3400
4700
5600
6200
7800

400
800
1700
3200
4500
6900
8400
8900
10200

15
15
15
15
15
15
15
15
15

0.621
0.621
0.621
0.621
0.621
0.621
0.621
0.621
0.621

1435.32
1435.32
1435.42
1435.69
1435.78
1436.28
1436.55
1436.51
1436.38

1,201,497.36
1,201,497.09
1,201,647.35
1,202,098.96
1,202,248.96
1,203,077.15
1,203,528.76
1,203,452.98
1,203,225.79

Fuzzy model
˜
τ(T PU(Q))

Q∗

1438.69
1438.76
1439.37
1441.00
1441.68
1444.60
1446.22
1446.08
1445.62

PY


lA

1,201,852.33
1,201,844.50
1,202,639.43
1,205,047.72
1,205,834.82
1,210,256.88
1,212,665.18
1,212,249.12
1,211,004.84

Table 4
The impact of increasing the amount of fuzziness for ordering cost on optimal policies
Lower and upper limits

uD


ld


ud


lγ


uγ


lh


uh


lA


uA

4300
4300
4300
4300
4300
4300
4300
4300
4300

3400
3400
3400
3400
3400
3400
3400
3400
3400

0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04

0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03

0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003

0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.004

0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4

0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3

1
4
5
6
8
9
13
17
19

2
3
7.5
9
11
12
18
21
25

i

β

CO


lD

OR

15
15
15
15
15
15
15
15
15

0.621
0.621
0.621
0.621
0.621
0.621
0.621
0.621
0.621

Fuzzy learning model
˜
τ(T PU(Q))

Q∗

1434.89
1434.46
1435.25
1435.37
1435.39
1435.39
1435.87
1435.67
1436.14

1,200,741.01
1,200,743.15
1,200,739.21
1,200,738.62
1,200,738.55
1,200,738.52
1,200,736.16
1,200,737.14
1,200,734.85

Q∗

Fuzzy model
˜
τ(T PU(Q))

1436.53
1438.00
1439.22
1440.06
1440.44
1440.63
1443.96
1443.45
1446.39

1,197,732.78
1,197,725.50
1,197,719.47
1,197,715.36
1,197,713.46
1,197,712.51
1,197,696.06
1,197,698.58
1,197,684.06

Table 5
The impact of increasing the amount of fuzziness for fraction of defective items on optimal policies
Lower and upper limits

uD


lA


uA


ld


ud

4300
4300
4300
4300
4300
4300
4300
4300
4300

3400
3400
3400
3400
3400
3400
3400
3400
3400

3
3
3
3
3
3
3
3
3

4
4
4
4
4
4
4
4
4

0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04

0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03


lh


uh

0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4

0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3

0.001
0.004
0.008
0.009
0.009
0.01
0.011
0.012
0.013

Fuzzy learning model
˜
τ(T PU(Q))


uγ

i

β

Q∗

0.002
0.003
0.006
0.008
0.01
0.011
0.012
0.013
0.013

15
15
15
15
15
15
15
15
15

0.621
0.621
0.621
0.621
0.621
0.621
0.621
0.621
0.621

1434.90
1434.84
1434.81
1434.84
1434.91
1434.91
1434.91
1434.91
1434.82


lγ

AU
TH


lD

model are closer to that of crisp model, irrespective of
the rate of learning rate. This result is expected since the
decision maker retains knowledge from previous shipments, and thus is able to estimate the parameters with
more preciseness. This finding proves that learning in
fuzziness is an appropriate tool to reduce the amount
of uncertainty. Another conclusion from sample data is
that the learning-fuzzy model produces greater lot size
with higher total profit in comparison to fuzzy model.
This could be also expected and be interpreted when
learning occur. In other words, learning in fuzziness
diminishes the amount of fuzziness decision makers
encounter, and therefore helps in making decision with
lower imprecision.

1,200,741.54
1,200,758.81
1,200,766.63
1,200,757.20
1,200,739.00
1,200,738.65
1,200,738.29
1,200,737.93
1,200,764.16

Q∗

Fuzzy model
˜
τ(T PU(Q))

1436.86
1436.57
1436.48
1436.69
1437.06
1437.09
1437.11
1437.14
1436.66

1,197,748.08
1,197,826.21
1,197,841.90
1,197,779.85
1,197,674.54
1,197,664.45
1,197,654.17
1,197,643.69
1,197,770.87

To gain further insights on how the amount of fuzziness affects optimal order quantity and optimal total
profit per unit time and to analyze how learning affects
each individual fuzzy parameter, the lower and upper
limit of one parameter are increased at a time while
keeping the lower and upper limit of other parameters unchanged and the outcomes for fuzzy learning
and fuzzy models are compared. The results for annual
demand, ordering cost and percentage of defective
items are reported in Tables 3–5, respectively. It should
be noted that in all three tables the learning exponent is assumed to be 0.621 while the total number
of shipments in a cycle is considered to be 15 in the
computations. It is clear from three tables that, at a fixed

N. Kazemi et al. / Development of a fuzzy economic order quantity model for imperfect quality

learning exponent, considering the higher amount of
fuzziness in the fuzzy learning model leads to smaller
lot size in comparison to the fuzzy model. This can be
inferred from all three parameters that are analyzed.
Moreover, looking at Table 3 it can be observed that
when the amount of fuzziness in demand is increased
fuzzy model produce more profit in comparison to the
fuzzy learning model. The computations for the other
two parameters give different results to that of demand.
In other words, the total profit generated by fuzzy learning model is higher than fuzzy model when the amount
of fuzziness in ordering cost and defective items is
increased.

model (for review of forgetting model see Anzanello
and Fogliatto [29]), which are direct extensions of the
learning curves of Wright [27]. Moreover, a more comprehensive study is needed to analyze the model with
the data of learning process gained from real-world
inventory problems. This seems to be an interesting
research, which makes the model more applicable in
real situations. It would also be beneficial to study the
effect of alternative learning curves on the model studied in this paper to account for the fact that different
forms of learning may occur in practice that cannot be
tackled by using Wright’s learning curve. Finally, the
fuzzy model with learning in fuzziness presented in this
paper could be applied to other inventory models with
fuzzy condition.

PY

2388

Acknowledgments
The authors wish to express their gratitude to University of Malaya for funding their research (Grant no.
RP018a-13aet).

AU
TH

OR

This paper extended a fuzzy EOQ model with
imperfect quality by assuming that learning occurs in
estimating fuzzy parameters, which is retained by the
number of shipments. The models that have touched
upon the area of fuzzy inventory management have not
considered learning in fuzzy parameters, which is a
research gap that this paper covered. Wherefore, learning is used in setting triangular fuzzy numbers and it is
hypothesized that the fuzziness reduced in conformance
with a learning curve, following the learning curves
proposed by Wright [27]. The model, consequently,
proposes optimal planning for learning in fuzziness
situation. To gain further insights on how learning in
fuzziness affects the model, the behavior of the model
studied is numerically investigated and compared to the
fuzzy and crisp case. Examining the effect of learning on the model showed that, as learning increases,
both optimal lot size and optimal profit increases. A
further insight from this example is that when the learning rate is slow smaller lot size should be ordered,
however, when the learning becomes faster the optimal policy is to order higher lot size from the supplier.
The comparison between the crisp, fuzzy and fuzzy
model demonstrated that the model with learning in
fuzziness could appropriately handle the imprecision
associated with the model since it could result to closer
values to the crisp case by reducing the amount of uncertainty. This might be an enticement to benefit from the
usefulness of learning in inventory planning when the
available data are imprecise.
The model developed in this paper could be extended
in several directions. It would be interesting to modify
the model to account for forgetting. If this is the case,
the model could be tested against the learn-forget curve

CO

6. Summary and conclusion

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