PRODUCTION & MANUFACTURING | RESEARCH ARTICLE
An EOQ model for imperfect quality items
with partial backordering under screening errors
Ehsan Sharifi, Mohammad Ali Sobhanallahi, Abolfazl Mirzazadeh and Sonia Shabani
Cogent Engineering (2015), 2: 994258
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Sharifi et al., Cogent Engineering (2015), 2: 994258
http://dx.doi.org/10.1080/23311916.2014.994258
PRODUCTION & MANUFACTURING | RESEARCH ARTICLE
An EOQ model for imperfect quality items
with partial backordering under screening errors
Ehsan Sharifi1*, Mohammad Ali Sobhanallahi1, Abolfazl Mirzazadeh1 and Sonia Shabani1
Received: 08 April 2014
Accepted: 18 November 2014
Published: 08 January 2015
*Corresponding author: Ehsan Sharifi,
Department of Industrial Engineering,
College of Engineering, University of
Kharazmi, Mofatteh Ave., Tehran, Iran
E-mail: [email protected]
Reviewing editor:
Zude Zhou, Wuhan University of
Technology, China
Additional information is available at
the end of the article
Abstract: In practice, when a lot size received, an inspection process is ­necessary
to identify the defective items. In addition, the inspection process itself is not
­error-free and it may contain misclassification errors. In this paper, an economic
order quantity model for imperfect quality items with partial backordering under
screening errors is studied. The objective is to maximize the expected annual profit
by optimizing the order size and the maximum number of backorder units. Also, the
aim of this paper is to develop a general and practical model that is more realistic
in the competitive commercial situations. For authenticity of the developed model,
a case study and a numerical example are illustrated, and the sensitivity analysis is
also carried out.
Subjects: Engineering & Technology; Industrial Engineering & Manufacturing; Operations
Research; Manufacturing Engineering; Production Engineering
Keywords: economic order quantity; imperfect quality; screening errors; partial backordering
1. Introduction
In the classical economic order quantity (EOQ) models, the items received are implicitly assumed to be
with perfect quality. This approach is idealistic, but in the practical situation it is unreliable to assume
100% of ordered items are perfect. Hence, many researchers come up with a number of more practical
and realistic EOQ models, which assume items are imperfect. Porteus (1986) surveyed the influence of
defective items on the basic EOQ model. Rosenblatt and Lee (1986) assumed that the time between
the in-control and the out-of-control state of a process follows an exponential distribution and that
the defective items are reworked instantaneously and suggested producing in smaller lots when the
process is not perfect. Lee and Rosenblatt (1987) studied a joint lot sizing and inspection policy for an
EOQ model with a fixed percentage of defective products. Yoo, Kim, and Park (2009) developed an
economic production quantity model for imperfect quality items and two way imperfect inspections.
Papachristos and Konstantaras (2006) investigated the disposal time of the imperfect items. They
ABOUT THE AUTHOR
PUBLIC INTEREST STATEMENT
Ehsan Sharifi received his BSc in 2010 at Ferdowsi
University in Mashhad, Iran. He continued his
education in Industrial Engineering and received
his MSc degree in 2013 from Kharazmi University
in Tehran, Iran. His interests are in fuzzy sets and
its applications, operations and supply chain
management and optimization.
Control of inventory, which typically represents
45–90% of all expenses for business, is needed
to ensure that the business has the right goods
on hand to avoid stockouts, to prevent shrinkage
(spoilage/theft) and to provide proper accounting.
One of the most important problems in inventory
control is how to assign the order quantity.
Economic order quantity (EOQ) models provide
the best way to minimize the costs. In this paper,
an EOQ model is developed for imperfect quality
items with considering some constraints that may
happen in real situations.
© 2015 The Author(s). This open access article is distributed under a Creative Commons Attribution
(CC-BY) 4.0 license.
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discussed the issue of non-shortages in inventory models where the proportion of defective items
was a random variable. Chang and Ho (2010) revisited the work of Wee, Yu, and Chen (2007) by
using renewal reward theorem to derive the expected profit per unit time for their model.
In addition, traditional inventory models supposed that there is no fault in the screening process
that identifies the defective items and inspectors are error-free. So, defective items could be
screened without any inspection. To deal with this unreliable assumption, many researches
focused on screening process of imperfect quality items. Raouf, Jain, and Sathe (1983) studied
human error in inspection planning for the first time. Salameh and Jaber (2000) developed an EOQ
model for imperfect quality items and considered that poor quality items are sold as a single batch
by the end of the 100% screening process. Goyal and Cárdenas-Barrón (2002) suggested a simpler
approach to the Salameh and Jaber’s model (2000). They suggested repeating the cycle of
inspection to ensure the product quality, and determined an optimal number of inspection cycles
based on the cost of inspection and misclassifications. Duffuaa and Khan (2005) suggested an
inspection plan for these critical components where an inspector can commit a number of
misclassifications. Maddah, Salameh, and Moussawi-Haidar (2010) extended Salameh and Jaber’s
(2000) model by assuming that the inspection process is not error-free. They assumed that
inspection process could fail to be perfect by two types of errors (Type I and Type II). Khan, Jaber,
Guiffrida, and Zolfaghari (2011) presented a review of the extensions of a modified EOQ model
for imperfect quality items. Khan and Jaber (2011) also extended the work of Salameh and Jaber
(2000) by assuming that the screening process is not error-free. They developed an EOQ model
for items with imperfect quality and inspection errors, but without any shortages. Hsu and Hsu
(2013) developed an EOQ model for imperfect quality items with screening errors and fully
backorder shortage and sales return.
On the other hand, many researchers considered full backorder shortages in the EOQ models but
in the competitive commercial situations, customers are not willing to wait for the next delivery
when a shortage occurs. So, it is profitable for the company to allow partial backorders. Rezaei
(2005) developed an EOQ model with backorder for imperfect quality items. Yu, Wee, and Chen
(2005) discussed an optimal ordering policy for a deteriorating item with imperfect quality and
partial backordering in the production process. Wee, Yu, and Wang (2006) developed an inventory
model for deteriorating items with imperfect quality and shortage backordering considerations.
Wee et al. (2007) extended Salameh and Jaber’s model (2000) by considering permissible shortage
backordering and the effect of varying backordered cost values. Eroglu and Ozdemir (2007) also
extended Salameh and Jaber’s model (2000) by considering fully backorder shortage. Roy, Sana, and
Chaudhuri (2011) developed the model of Salameh and Jaber (2000) for the case where a buyer’s
cycle starts with shortages that may have occurred due to lead-time or labour problems. Shabani,
Mirzazadeh, and Sharifi (2014) developed an inventory model with fuzzy deterioration and fully
backlogged shortage under inflation. To the author’s knowledge, there is no EOQ model for imperfect quality items with inspection errors that considered partial backorder. Thus, in this paper, we
developed an EOQ model for items with imperfect quality and partial backordering under screening
errors. The analysis shows that our model is a generalization of the models in current literatures.
The rest of the paper is organized as follows. In Section 2, the notation and model description are
introduced. In Section 3, a mathematical model is developed. In Section 4, a special case is exhibited.
Section 5 provides numerical example and sensitivity analysis to illustrate important aspects of the
model. In Section 6, a case study is done and finally, a general conclusion and future directions of
the present study are provided in Section 7.
2. Notation and model description
Consider a lot size y is being replenished instantaneously. It is assumed that each lot contains
a fixed proportion p of defective items. So, the lot size y contains defective items of py and
non-defective items of (1 − p)y. In addition, each lot is screened by an inspector with a screening
rate x. The screening process of an entire lot is not perfect and it generates misclassification errors,
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that is a proportion α of non-defective items are classified to be defective and a proportion β of
defective items are classified to be non-defective. Also, it is assumed that items with poor quality
are kept in stock and sold before receiving the next shipment as a single batch at a discounted
price.
The following nomenclature is used throughout the paper:
y
order size for each cycle
D
demand rate
x
screening rate
c
unit purchasing cost
k
fixed ordering cost
p
the defective percentage in y
s
unit selling price of a non-defective item
v
unit selling price of a defective item, v < s
d
unit screening cost
B
maximum backorder level
B1
number of items that are classified as defective by inspector
h
unit holding cost
t
cycle length
t1
time to build up a backorder level of B units
t2
time to eliminate the backorder level of B units
t3
time to screen y units ordered per cycle
α
Type I error (classifying a non-defective item as defective)
β
Type II error (classifying a defective item as non-defective)
μ
fraction of demand backordered during a stock out
f(p)
the probability density function of p
f(α)
the probability density function of α
f(β)
the probability density function of β
f(μ)
the probability density function of μ
cr
cost of rejecting a non-defective item
ca
cost of accepting a defective item
cL
cost of lost sale per unit
cB
cost of backorder per unit
3. Mathematical model
Figure 1 shows the behaviour of the inventory model. When the order quantity received, inventory
level increases. The maximum level of inventory is y. Due to response to the prior backorder, the
screening process and consumption of inventory begin simultaneously. As mentioned earlier, lot size
y contains defective items of py along with non-defective items of (1 − p)y. In inspection process,
there are four possibilities. Those are: Case (1) a non-defective item is classified as non-defective.
Case (2) a non-defective item is classified as defective. Case (3) a defective item is classified as
non-defective and Case (4) a defective item is classified as defective. So, the number of items going
into different categories following these cases is given by:
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Figure 1. Behaviour of the
inventory level over time.
Case (1): y.(1 − p).(1 − α)
Case (2): y.(1 − p).α
Case (3): y.p.β
Case (4): y.p.(1 − β)
With considering inspection errors, the number of items that are non-defective and the inspectors
have chosen them as non-defective are (1 − p)(1 − α)y, and the number of items that are defective
and the inspectors have chosen them as non-defective are ypβ. Totally, the whole items that are
chosen as non-defective by inspectors are ((1 − p)(1 − α) + pβ)y. These items are required to satisfy
[
]
backorders with the rate of (1 − p)(1 − 𝛼) + p𝛽 x − D during time t2.
Then, the screening and consumption processes continue until time t3 and the process ends after.
All the defective items (B1) are subtracted from inventory and the remaining items (Z1) meet the
demand with the rate of D. When the inventory level reaches zero, during the period t1, the shortages
consist of a combination of backorder and lost sales occur, since μ is the percentage of demand
backordered during this time.
In a cycle, considering the demand is met from perfect items, the cycle length can be calculated
as:
T =y
[
]
(1 − p)(1 − 𝛼) + p𝛽
D
=y
(Ax + D)
Dx
(1)
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where
A = (1 − p)(1 − 𝛼) + p𝛽 −
D
x
(2)
Referring to Figure 1, the findings are as follows:
t1 =
𝜇B B
=
𝜇D D
(3)
t2 =
y − z2
y − z2
𝜇B
=[
] =
Ax
Ax
+D
(1 − p)(1 − 𝛼) + p𝛽 x
(4)
t3 =
y
x
(5)
z − z − B1
t3 − t2 = 2 1
D
(6)
(
)
[
]
D
B1 = y (1 − p)𝛼 + p(1 − 𝛽) = y 1 − − A
x
(7)
According to Equation 4, the value of z2 is obtained as follows:
z2 = y −
[
]
B (1 − p)(1 − 𝛼) + p𝛽
(1 − p)(1 − 𝛼) + p𝛽 −
D
x
=y −B
(Ax + D)
Ax
(8)
And using Equation 6 to calculate the value of z1:
z1 = Ay − B − BD
(1 − 𝜇)
Ax
(9)
We define TR(y, B) and TC(y, B) as the total revenue and the total cost per cycle, respectively.
TR(y, B) is the sum of total sales volume of good quality and the imperfect quality items. One has:
TR(y, B) = sy(1 − p)(1 − 𝛼) + vy(1 − p)𝛼 + vyp
(10)
TC(y, B) is the sum of ordering cost, purchasing cost, screening cost, backordering cost, lost sale cost
and holding cost, one has:
(t + t )𝜇B
t (1 − 𝜇)B
+ cL 1
TC(y, B) = cy + k + dy + cr (1 − p)y𝛼 + ca py𝛽 + cB 1 2
2
2
}
{
(y + z2 )t2 (t3 − t2 )(z2 + z1 + B1 ) z1 (T − t1 − t3 )
+
+
+h
2
2
2
(11)
By using Equations 3–9, TC(y, B) can be calculated as:
(Ax + 𝜇D)B 𝜇B
B2 (1 − 𝜇)
+ cL
TC(y, B) = cy + k + dy + cr (1 − p)y𝛼 + ca py𝛽 + cB
AxD
D
2
{[
(
)] ( ) (
) [(
) 2 (
(
))]
Ay − 𝜇B
𝜇B
2−𝜇
Ax + D
D
h
2y − B
+
2−
y −B 2+D
+
x
Ax
Ax
Ax
Ax
2
][
[
]}
y(Ax + D) B y
BD(1 − 𝜇)
+
− −
Ay − B −
Dx
D x
Ax
(12)
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The total profit per cycle can now be written as the difference between the total revenue and total
cost per cycle, that is:
TP(y, B) = TR(y, B) − TC(y, B) = sy(1 − p)(1 − 𝛼) + vy(1 − p)𝛼 + vyp − cy − k − dy − cr (1 − p)y𝛼
(Ax + 𝜇D) 𝜇B2
B2 (1 − 𝜇)
− cL
− ca py𝛽 − cB
AxD
2 ( D
)
(
(
))]
{[
(
)]
) (2
) [(
Ay − 𝜇B
2−𝜇
𝜇B
Ax + D
h
D
y −B 2+D
−
2y − B
+
2−
x
Ax
Ax
Ax
Ax
2
][
]}
[
y (Ax + D) B y
BD(1 − 𝜇)
− −
Ay − B −
+
Dx
D x
Ax
(13)
Since p, α, β and μ are random variables with probability density functions f (p), f (α), f (β) and f(μ).
The expected total profit can be formulated as:
E[TP(y, B)]
[
]
[
]
= E TR(y, B) − E TC(y, B)
= sy (1 − E[p]) (1 − E [𝛼]) + vy (1 − E[p]) E [𝛼] + vyE[p] − cy − k − dy − cr y (1 − E[p]) E [𝛼]
(E[A]x + E[𝜇]D) E[𝜇]B2
B2 (1 − E[𝜇])
− cL
− ca yE[p]E [𝛽] − cB
E[A]xD
D
2
2
{[
(
)] (
) (
)
E[A]y
−
E[𝜇]B
E[𝜇]B
E[A]x + D
h
−
2y − B
+
E[A]x
E[A]x
E[A]x
2
)
(
(
[(
))]
2
−
E[𝜇]
D
y −B 2+D
× 2−
x
E[A]x
]}
][
[
y (E[A]x + D) B y
BD (1 − E[𝜇])
− −
E[A]y − B −
+
Dx
D x
E[A]x
(14)
From Equation 1, the expected cycle length is:
E(T) = y
[
]
(1 − E(p)) (1 − E(𝛼)) + E(p)E(𝛽)
D
=y
(E(A)x + D)
Dx
(15)
Using the renewal reward theorem (Chang & Ho, 2010), the expected annual profit is:
E[TP(y, B)]
Dx
=
E[T]
E[A]x + D
�
k
× s (1 − E[p]) (1 − E[𝛼]) + v(1 − E[p])E[𝛼] + vE[p] − c −
y
E[TPU(y, B)] =
�
E[𝜇] (E[A]x + E[𝜇]D) B2
−d − ca E[p]E[𝛽] − cr (1 − E[p]) E[𝛼] − cB
2E[A] (E[A]x + D) y
��
�
2
E[𝜇]B
(1 − E[𝜇])
2D
BD
B x
h
−
− cL
−
y (E[A]x + D) 2
(E[A]x + D) E[A]xy E[A]
2
��
�
�
�
���
�
�
D
D 2 − Dx y − B 2 + D 2−E[𝜇]
D
2
−
E[A]x
x
+
− E[𝜇]B
E[A] (E[A]x + D)
(E[A]x + D)
�
�
⎞ �
⎛
�
��
2 + D 2−E[𝜇]
⎟
E[A]x
d(1 − E[𝜇])
2 ⎜
+ E[𝜇]B D ⎜
⎟ + E[A]y − B 1 + E[A]x
⎜ E[A] (E[A]x + D) y ⎟
⎠
⎝
��
�
D
Bx
−
× 1−
y(E[A]x + D) E[A]x + D
(16)
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Our objective is to maximize the expected net profit. By taking the first derivative of E[TPU(y, B)] with
respect to y and B, we have:
[
]
𝜕E TPU(y, B)
𝜕y
{
1
=− 2
y
−
c E[𝜇]B2 (E[A]x + E[𝜇]D)
kDx
− B
E[A]x + D
2E[A] (E[A]x + D)
2
E[𝜇]B2 D
(1 − E[𝜇])
B2 x
h E[𝜇]B D
+
[ ] − h2
E[A] (E[A]x + D)
(E[A]x + D) 2 E A2 x
2
(
(
))
(
)}
2 − E[𝜇]
1 − E[𝜇]
h B2 x
× 2+D
−
1+D
E[A]x
E[A]x
2 E[A]x + D
]
[
h
D
h E[A]D
h
D
− E[A] +
2−
−
x
2 (E[A]x + D)
2
2 E[A]x + D
− cL
𝜕E[TPU(y, B)]
𝜕B
{
hE[𝜇]D
E[𝜇](E[A]x + E[𝜇]D)
(1 − E[𝜇]) x
= B −cB
− cL
−
E[A] (E[A]x + D) y
y(E[A]x + D) E[A](E[A]x + D)y
(
(
))
(
(
))}
2 − E[𝜇]
hDE[𝜇]
1 − E[𝜇]
hx
−
× 2+D
+
1+D
E[A]x
E[A]x
E[A2 ]xy y (E[A]x + D)
(
(
))
hDE[𝜇]
2 − E[𝜇]
hD
+
−
2+D
E[A](E[A]x + D) 2 (E[A]x + D)
E[A]x
)
(
h E[A]x
h
D
+
+
E[𝜇]D 2 −
x
2E[A] (E[A]x + D)
2 (E[A]x + D)
)
)(
)
(
(
1
−
E[𝜇]
1
−
E[𝜇]
D
h
h
+
−
1+D
1+D
E[A]x
E[A]x
E[A]x + D
2
2
(17)
(18)
Taking the second derivative, we have:
𝜕 2 TPU(y, B)
𝜕y 2
𝜕 2 TPU(y, B)
𝜕B
2
2
y 3 (E[A]x + D)
[
(
E[𝜇] (E[A]x + E[𝜇]D)
(1 − E[𝜇])
+ cL
x
× kDx + cB
2E[A]
2
(
) ]
)
1 − E[𝜇]
h
h E[𝜇]D
+
D
−
E[𝜇]D)
+ x 1+D
B2
+
(E[A]x
E[A]x
2
2 E[A2 ]x
=−
1
y (E[A]x + D)
{
E[𝜇] (E[A]x + E[𝜇]D)
+ cL (1 − E[𝜇]) x
× cB
E[A]
(
(
))
(
(
))
2 − E[𝜇]
1 − E[𝜇]
hE[𝜇]D
2+D
+ hx 1 + D
+
E[A]
E[A]x
E[A]x
}
hDE[𝜇]
−
(E[A]x + D)
E[A2 ]x
(19)
=−
(20)
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𝜕 2 TPU(y, B)
B
=− 2
𝜕B𝜕y
y (E[A]x + D)
�
E[𝜇] (E[A]x + E[𝜇]D)
hE[𝜇]D
+ cL (1 − E[𝜇]) x +
× cB
E[A]
E[A]
⎫
�
�
��
�
�
��
⎪
2 − E[𝜇]
1 − E[𝜇]
hDE[𝜇]
× 2+D
+ hx 1 + D
− � � (E[A]x + D)⎬
E[A]x
E[A]x
⎪
E A2 x
⎭
(21)
and
(
𝜕 2 TPU(y, B)
𝜕B𝜕y
)2 (
−
𝜕 2 TPU(y, B)
𝜕 2 TPU(y, B)
𝜕y 2
[
B2
)(
)
𝜕B2
E[𝜇] (E[A]x + E[𝜇]D)
+ cL (1 − E[𝜇]) x
E[A]
y (E[A]x + D)
(
(
))
(
(
))
hE[𝜇]D
2 − E[𝜇]
1 − E[𝜇]
+
2+D
+ hx 1 + D
E[A]
E[A]x
E[A]x
](
)
hDE[𝜇]
−
(E[A]x + D) −2kDxB2
E[A2 ]x
[
E[𝜇] (E[A]x + E[𝜇]D)
B2
= 4
+ cL (1 − E[𝜇]) x
cB
2
E[A]
y (E[A]x + D)
(
(
))
(
(
))] (
)
hE[𝜇]D
1 − E[𝜇]
1 − E[𝜇]
+
1+D
+ hx 1 + D
−2kDxB2
E[A]
E[A]x
E[A]x
=
2
4
cB
(22)
2
2
2
2
Since 0 < μ < 1 and A, k, D, x, cL, cB > 0 we have 𝜕 [ETPU(y, B)]∕𝜕y < 0, 𝜕 [ETPU(y, B)]∕𝜕B < 0 and
[
]2 [
][
]
𝜕 2 TPU(y, B)∕𝜕B𝜕y − 𝜕 2 TPU(y, B)∕𝜕y 2 𝜕 2 TPU(y, B)∕𝜕B2 ≤ 0 which implies that the function
ETPU (y, B) is strictly concave. Thus, the optimal order size that represents the maximum annual
profit is determined by setting the first derivative equal to zero.
After some basic manipulations, we have:
√
)
(
]
[
√
√ kDx + c E[𝜇](E[A]x+E[𝜇]D) + c (1−E[𝜇]) x + h x 1 + D 1−E[𝜇] + h E[𝜇]D (E[A]x + D − E[𝜇]D) B2
2
√
L
B
2E[A]
2
2
E[A]x
2 E[A ]x
y =√
[ (
)
]
√
h
D
2
D 2 − x + E[A ]x
2
(23)
h
2
[
(
(
(
(
))]
+ E[A]x 2 + D 1−E[𝜇]
E[A]x
B=
(
(
))
2−E[𝜇]
hE[𝜇]D
E[𝜇](E[A]x+E[𝜇]D)
−
E[𝜇])
x
+
+
c
2
+
D
cB
(1
L
E[A]
E[A]
E[A]x
))
(
(
hDE[𝜇]
1−E[𝜇]
− E A2 x (E[A]x + D)y
+hx 1 + D E[A]x
[ ]
2D 1 + D
1−E[𝜇]
E[A]x
))
(24)
Thus, we have:
√
√
√ kDx + M1 B2
y =√
h
M
2 2
B=
h
M
2 3
2M1
y=
hM3
y
4M1
(25)
(26)
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that M1, M2 and M3 are variables for simplification.
(
)
E[𝜇] (E[A]x + E[𝜇]D)
1 − E[𝜇]
(1 − E[𝜇])
h
+ cL
x+ x 1+D
lM1 = cB
E[A]x
2E[A]
2
2
h E[𝜇]D
+
[ ] (E[A]x + D − E[𝜇]D)
2 E A2 x
(
)
[ ]
D
M2 = D 2 −
+ E A2 x
x
(
(
M3 = 2D 1 + D
1 − E[𝜇]
E[A]x
))
(27)
(28)
(
(
))
1 − E[𝜇]
+ E[A]x 2 + D
E[A]x
(29)
Finally,
√
√
2
√
M 2
√ kDx + h 3
16 M1
√
y∗ = √
h
M
2 2
(30)
hM3 ∗
y
4M1
B∗ =
(31)
4. Special case
To simplify the complicated formulas and to indicate the authenticity of the proposed model, a
special case is considered in this study. Salameh and Jaber (2000) developed an EOQ model for
imperfect quality items that has known as a traditional EOQ model in the literature review. In this
section, our aim is release all the constraints in our model and eventually reach to the traditional
EOQ formulas that have been proved before.
In the model, when p = α = β = 0 and cB = cL = ∞, we have:
Dx
, which substituting this values in Equation 25 lead to reach to the
M1 = ∞, M2 = ∞ and M3 = 2x +
D−x
traditional EOQ formula.
√
∗
y =
2kD
h
(32)
5. Numerical examples and sensitivity analysis
Consider an inventory model with these parameters:
D
50,000 units/year
x
17,5200 units/year
c
$ 25/unit
k
$ 100/cycle
s
$ 50/unit
v
$ 20/unit
d
$ 0.5/unit
h
$ 5/unit
cr
$ 100/unit
c a
$ 500/unit
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cL
$ 15/unit
cB
$ 10/unit
In addition, we suppose the defective percentage, inspection errors and the fraction of backordered
demand follow a uniform distribution with probability density functions as:
{
f (p) =
{
f (𝛽) =
1
p1
0 ≤ p ≤ p1
0
Otherwise
0 ≤ 𝛽 ≤ 𝛽1
1
𝛽1
0
{
f (𝛼) =
{
f (𝜇) =
Otherwise
1
𝛼1
0 ≤ 𝛼 ≤ 𝛼1
0
Otherwise
0 ≤ 𝜇 ≤ 𝜇1
1
𝜇1
0
Otherwise
Then we have:
p1
E(p) =
∫0
pf (p)dp =
p1
2
, E (𝛼) =
𝛼1
2
, E (𝛽) =
𝛽1
2
,
E (𝜇) =
𝜇1
2
Now if p1 = α1 = β1 = 0.04 and μ1 = 1, then we have E(p) = E(α) = E(β) = 0.02 and E(μ) = 0.5. By using the
above parameters, the optimum values of solution are calculated as: y* = 1,740.913 units,
B* = 455.156 units and ETPU(y*, B*) = 1,094,770. In addition, the three-dimensional graph (Figure 2)
represents that the expected annual profit is concave and there exist unique solutions of y and B
that maximize the expected annual profit. For authenticity of the developed model, the sensitivity
analysis of the parameters is needed.
When defective rate increases, the number of items that are chosen as non-defective decrease.
So, for compensate this deficiency, the number of order size increases. With considering the
increment in order size, the number of backorder units decreases and as a result, the expected
annual profit decreases. Table 1 shows the optimal solutions for different defective probabilities. The
changes in variables show that the behavior of our model is correct.
Figure 2. Expected annual profit
is a concave function of the y
and B.
Table 1. Effect of p on ETPU(y, B) when the defective probability p is uniformly distributed
between 0 and p1
p1
y*
B*
ETPU(y*, B*)
0.02
1,730.768
457.238
1,103,176
0.04
1,740.913
455.156
1,094,770
0.06
1,751.064
453.020
1,086,195
0.08
1,761.217
450.828
1,077,444
0.1
1,771.367
448.580
1,068,513
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From Table 2, one can see that the probability of Type I error has a similar effect as the defective
probability does on the optimal solution, but in Table 3, the probability of a Type II error has a reverse
impact as both the defective probability and the probability of Type I error do on the optimal solution. When Type II error increases, the items that are defective but they are chosen as non-defective
increase, so the order size must decrease and the maximum number of backordering units ­increases.
Furthermore, the expected annual profit decreases as the probability of Type II error increases.
When the fraction of backordering units increases, the maximum number of backordering units
decreases and the expected annual profit become smaller. Table 4 shows the changes in μ.
As shown in Table 5, when the holding cost increases, the order size decreases. It is logical,
­because our aim is to decrease the total cost. So, with considering the increment in holding cost, the
Table 2. Effect of α on ETPU(y, B) when the probability of Type I error is uniformly distributed
between 0 and α1
α1
0.02
y*
B*
ETPU(y*, B*)
1,730.557
457.281
1,149,311
0.04
1,740.913
455.156
1,094,770
0.06
1,751.276
452.975
1,039,106
0.08
1,761.640
450.736
982,281
0.1
1,772.002
448.438
924,260
Table 3. Effect of β on ETPU(y, B) when the probability of Type II error is uniformly distributed
between 0 and β1
y*
B*
ETPU(y*, B*)
0.02
1,741.125
455.112
1,100,204
0.04
1,740.913
455.156
1,094,770
0.06
1,740.702
455.200
1,089,339
0.08
1,740.491
455.244
1,083,910
0.1
1,740.279
455.288
1,078,483
β1
Table 4. Effect of μ on ETPU(y, B) when the maximum number of backordering units is uniformly
distributed between 0 and 1
μ1
y*
B*
ETPU(y*, B*)
0
1,815.622
502.532
1,095,016
0.2
1,802.618
496.679
1,094,975
0.4
1,788.479
488.941
1,094,929
0.6
1,773.360
479.373
1,094,879
0.8
1,757.440
468.068
1,094,827
1
1,740.913
455.156
1,094,770
Table 5. Effect of h on ETPU(y, B) when the holding cost changes
h
y*
B*
ETPU(y*, B*)
1
3,397.993
247.086
1,097,686
2
2,498.795
331.069
1,096,584
3
2,113.631
385.739
1,095,825
4
1,890.145
425.197
1,095,242
5
1,740.913
455.156
1,094,770
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order size must decrease. In addition, when the holding cost increases, the maximum number of
backordering units increases, because of decrement in order size and as a result, the expected
annual cost decreases.
6. Case study
In this section, the model has employed one of the automotive supplier companies as a real case.
This supplier is Sanayeh Dashboard Iran, which is located in Tehran, Iran. The product of this company is Dashboard, which is produced using a frame with ABS/PVC covering and injection of the
Isocyanat and Polyol mixture (semi-rigid foam) into it. We want to determine the order policy for
Isocyanat. The data are gathered from financial, marketing and engineering departments of the
Sanayeh Dashboard Iran. The parameter values are summarized as follows (R is an abbreviation form
of Rial, Persian monetary unit):
D = 750,000 units/year, x = 1,036,800 units/year, c = 2,850 R/unit, k = 3,500 R/cycle, s = 5,500
R/unit, v = 2,200 R/unit, d = 0.5 R/unit, h = 200 R/unit, cr = 3,000 R/unit, ca = 8,500 R/unit, cL = 1,050
R/unit, cB = 500 R/unit,
The defective percentage, inspection errors and the fraction of backordered demand follow a uniform distribution with probability density functions as:
{
f (p) =
{
f (𝛽) =
1
0.05
0
1
0.01
0
{
0 ≤ p ≤ 0.05
Otherwise
f (𝛼) =
0 ≤ 𝛽 ≤ 0.01
Otherwise
f (𝜇) =
1
0.01
{
0
1
0.4
0
0 ≤ 𝛼 ≤ 0.01
Otherwise
0 ≤ 𝛼 ≤ 0.4
Otherwise
which
0.05
E(p) =
∫0
pf (p)dp = 0.025, E(𝛼) = 0.005,
E(𝛽) = 0.005, E(𝜇) = 0.2
By using above parameters, the optimum values of solution are calculated as:
y* = 13,881.463 units, B* = 3,984.091 units and ETPU(y*, B*) = 1,959,330,633.
Figure 3 shows the expected annual profit for Sanayeh Dashboard Iran.
Figure 3. Expected annual profit
for Sanayeh Dashboard Iran.
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7. Conclusion
In the practical situations, it is unreliable to assume 100% of ordered items are perfect and the
inspectors are error-free. In industry, these errors are incontrovertible, but it is important to use
methods to identify them. In this paper, we developed an EOQ model for imperfect quality items
with partial backordering and screening errors. The aim of this paper is to maximize the expected
annual profit by optimizing the order size and the maximum number of backordering units. In
addition, to verify the reliability of our work, we proved the proposed model is concave, we studied
a special case that indicates our model could be easily transformed to Salameh and Jaber’s (2000)
model that is a popular case in our research literature and we illustrate the utility of our model with
a numerical example and sensitivity analysis on the parameters. For future research, the proposed
model can be studied in a fuzzy environment. Deteriorating items could be added to the model
and other practicable parameters like inflation, delay in payment and sales return could be
considered.
Funding
The authors received no direct funding for this research.
Author details
Ehsan Sharifi1
E-mail: [email protected]
Mohammad Ali Sobhanallahi1
E-mail: [email protected]
Abolfazl Mirzazadeh1
E-mail: [email protected]
Sonia Shabani1
E-mail: [email protected]
1
Department of Industrial Engineering, College of Engineering,
University of Kharazmi, Mofatteh Ave., Tehran, Iran.
Citation information
Cite this article as: An EOQ model for imperfect quality
items with partial backordering under screening errors,
E. Sharifi, M.A. Sobhanallahi, A. Mirzazadeh & S. Shabani,
Cogent Engineering (2015), 2: 994258.
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