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PRODUCTION & MANUFACTURING | RESEARCH ARTICLE

An EOQ model for imperfect quality items with partial backordering under screening errorsEhsan Sharifi, Mohammad Ali Sobhanallahi, Abolfazl Mirzazadeh and Sonia Shabani

Cogent Engineering (2015), 2: 994258

Sharifi et al., Cogent Engineering (2015), 2: 994258http://dx.doi.org/10.1080/23311916.2014.994258

PRODUCTION & MANUFACTURING | RESEARCH ARTICLE

An EOQ model for imperfect quality items with partial backordering under screening errorsEhsan Sharifi1*, Mohammad Ali Sobhanallahi1, Abolfazl Mirzazadeh1 and Sonia Shabani1

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. IntroductionIn 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

*Corresponding author: Ehsan Sharifi, Department of Industrial Engineering, College of Engineering, University of Kharazmi, Mofatteh Ave., Tehran, IranE-mail: ehsan_sharifi_57@yahoo.com

Reviewing editor:Zude Zhou, Wuhan University of Technology, China

Additional information is available at the end of the article

ABOUT THE AUTHOREhsan 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.

PUBLIC INTEREST STATEMENTControl 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.

Received: 08 April 2014Accepted: 18 November 2014Published: 08 January 2015

© 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 imper-fect 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 descriptionConsider 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 modelFigure 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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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−�)+p�

]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:

(1)T=y

[(1−p)(1−�)+p�

]D

=y(Ax+D)

Dx

Figure 1. Behaviour of the inventory level over time.

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where

Referring to Figure 1, the findings are as follows:

According to Equation 4, the value of z2 is obtained as follows:

And using Equation 6 to calculate the value of z1:

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:

TC(y, B) is the sum of ordering cost, purchasing cost, screening cost, backordering cost, lost sale cost and holding cost, one has:

By using Equations 3–9, TC(y, B) can be calculated as:

(2)A=(1−p)(1−�)+p�−D

x

(3)t1=�B

�D=B

D

(4)t2=�B

Ax=

y−z2[

(1−p)(1−�)+p�]x=y−z

2

Ax+D

(5)t3=y

x

(6)t3− t

2=z2−z

1−B

1

D

(7)B1=y

[(1−p)�+p(1−�)

]=y

(1−

D

x−A

)

(8)z2=y−

B[(1−p)(1−�)+p�

]

(1−p)(1−�)+p�− D

x

=y−B(Ax+D)

Ax

(9)z1=Ay−B−BD

(1−�)

Ax

(10)TR(y,B)= sy(1−p)(1−�)+vy(1−p)�+vyp

(11)TC(y,B)=cy+k+dy+cr(1−p)y�+capy�+cB

(t1+ t

2)�B

2+cL

t1(1−�)B

2

+h

{(y+z

2)t2

2+(t3− t

2)(z

2+z

1+B

1)

2+z1(T− t

1− t

3)

2

}

(12)

TC(y,B)=cy+k+dy+cr(1−p)y�+capy�+cB(Ax+�D)B

AxD

�B

2+cL

B2

D

(1−�)

2

+h

2

{[2y−B

(Ax+D

Ax

)](�B

Ax

)+

(Ay−�B

Ax

)[(2−

D

x

)y−B

(2+D

(2−�

Ax

))]

+

[y(Ax+D)

Dx−B

D−y

x

] [Ay−B−

BD(1−�)

Ax

]}

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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:

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:

From Equation 1, the expected cycle length is:

Using the renewal reward theorem (Chang & Ho, 2010), the expected annual profit is:

(13)

TP(y,B)=TR(y,B)−TC(y,B)= sy(1−p)(1−�)+vy(1−p)�+vyp−cy−k−dy−cr(1−p)y�

−capy�−cB(Ax+�D)

AxD

�B2

2−cL

B2

D

(1−�)

2

−h

2

{[2y−B

(Ax+D

Ax

)](�B

Ax

)+

(Ay−�B

Ax

)[(2−

D

x

)y−B

(2+D

(2−�

Ax

))]

+

[y (Ax+D)

Dx−B

D−y

x

] [Ay−B−

BD(1−�)

Ax

]}

(14)

E[TP(y,B)]

=E[TR(y,B)

]−E

[TC(y,B)

]= sy (1−E[p]) (1−E [�])+vy (1−E[p]) E [�]+vyE[p]−cy−k−dy−cry (1−E[p]) E [�]

−cayE[p]E [�]−cB(E[A]x+E[�]D)

E[A]xD

E[�]B2

2−cL

B2

D

(1−E[�])

2

−h

2

{[2y−B

(E[A]x+D

E[A]x

)](E[�]B

E[A]x

)+

(E[A]y−E[�]B

E[A]x

)

×

[(2−

D

x

)y−B

(2+D

(2−E[�]

E[A]x

))]

+

[y (E[A]x+D)

Dx−B

D−y

x

] [E[A]y−B−

BD (1−E[�])

E[A]x

]}

(15)E(T)=y

[(1−E(p)) (1−E(�))+E(p)E(�)

]D

=y(E(A)x+D)

Dx

(16)

E[TPU(y,B)]=E[TP(y,B)]

E[T]=

Dx

E[A]x+D

×

�s (1−E[p]) (1−E[�])+v(1−E[p])E[�]+vE[p]−c−

k

y

−d−caE[p]E[�]−cr (1−E[p]) E[�]�−cB

E[�] (E[A]x+E[�]D)B2

2E[A] (E[A]x+D) y

−cL(1−E[�])

2

B2x

y (E[A]x+D)−h

2

��2D

(E[A]x+D)−

BD

E[A]xy

�E[�]B

E[A]

+D��2−

D

x

�y−B

�2+D

�2−E[�]

E[A]x

���

(E[A]x+D)−E[�]B

�2−

D

x

�D

E[A] (E[A]x+D)

+E[�]B2D

⎛⎜⎜⎜⎝

2+D�2−E[�]

E[A]x

�

E[A] (E[A]x+D) y

⎞⎟⎟⎟⎠+

�E[A]y−B

�1+

d(1−E[�])

E[A]x

��

×

�1−

Bx

y(E[A]x+D)−

D

E[A]x+D

��

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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:

Taking the second derivative, we have:

(17)

�E[TPU(y,B)

]�y

=−1

y2

{−

kDx

E[A]x+D−cBE[�]B

2 (E[A]x+E[�]D)

2E[A] (E[A]x+D)

−cL(1−E[�])

2

B2x

(E[A]x+D)+h

2

E[�]B2D

E[A2

]x−h2

E[�]B2D

E[A] (E[A]x+D)

×

(2+D

(2−E[�]

E[A]x

))−h

2

B2x

E[A]x+D

(1+D

1−E[�]

E[A]x

)}

−h

2

D

(E[A]x+D)

[2−

D

x

]−h

2E[A]+

h

2

E[A]D

E[A]x+D

(18)

�E[TPU(y,B)]

�B

=B

{−cB

E[�](E[A]x+E[�]D)

E[A] (E[A]x+D) y−cL

(1−E[�]) x

y(E[A]x+D)−

hE[�]D

E[A](E[A]x+D)y

×

(2+D

(2−E[�]

E[A]x

))+hDE[�]

E[A2]xy−

hx

y (E[A]x+D)

(1+D

(1−E[�]

E[A]x

))}

−hDE[�]

E[A](E[A]x+D)+

hD

2 (E[A]x+D)

(2+D

(2−E[�]

E[A]x

))

+h

2E[A] (E[A]x+D)E[�]D

(2−

D

x

)+h

2

E[A]x

(E[A]x+D)

+h

2

(1+D

1−E[�]

E[A]x

)−h

2

(1+D

1−E[�]

E[A]x

)(D

E[A]x+D

)

(19)

�2TPU(y,B)

�y2=−

2

y3 (E[A]x+D)

×

[kDx+

(cBE[�] (E[A]x+E[�]D)

2E[A]+cL

(1−E[�])

2x

+h

2x

(1+D

1−E[�]

E[A]x

)+h

2

E[�]D

E[A2]x(E[A]x+D−E[�]D)

)B2]

(20)

�2TPU(y,B)

�B2=−

1

y (E[A]x+D)

×

{cBE[�] (E[A]x+E[�]D)

E[A]+cL (1−E[�]) x

+hE[�]D

E[A]

(2+D

(2−E[�]

E[A]x

))+hx

(1+D

(1−E[�]

E[A]x

))

−hDE[�]

E[A2]x(E[A]x+D)

}

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and

Since 0 < μ < 1 and A, k, D, x, cL, cB > 0 we have 𝜕2[ETPU(y,B)]∕𝜕y2<0, 𝜕2[ETPU(y,B)]∕𝜕B2<0 and [�2TPU(y,B)∕�B�y

]2−[�2TPU(y,B)∕�y2

] [�2TPU(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:

Thus, we have:

(21)

�2TPU(y,B)

�B�y=−

B

y2 (E[A]x+D)

×

�cBE[�] (E[A]x+E[�]D)

E[A]+cL (1−E[�]) x+

hE[�]D

E[A]

×

�2+D

�2−E[�]

E[A]x

��+hx

�1+D

�1−E[�]

E[A]x

��−hDE[�]

E�A2

�x(E[A]x+D)

⎫⎪⎬⎪⎭

(22)

(�2TPU(y,B)

�B�y

)2

−

(�2TPU(y,B)

�y2

)(�2TPU(y,B)

�B2

)

=B2

y4 (E[A]x+D)2

[cBE[�] (E[A]x+E[�]D)

E[A]+cL (1−E[�]) x

+hE[�]D

E[A]

(2+D

(2−E[�]

E[A]x

))+hx

(1+D

(1−E[�]

E[A]x

))

−hDE[�]

E[A2]x(E[A]x+D)

] (−2kDxB2

)

=B2

y4 (E[A]x+D)2

[cBE[�] (E[A]x+E[�]D)

E[A]+cL (1−E[�]) x

+hE[�]D

E[A]

(1+D

(1−E[�]

E[A]x

))+hx

(1+D

(1−E[�]

E[A]x

))](−2kDxB2

)

(23)

y=

√√√√√√kDx+

[cB

E[�](E[A]x+E[�]D)

2E[A]+cL

(1−E[�])

2x+ h

2x(1+D 1−E[�]

E[A]x

)+

h

2

E[�]D

E[A2]x(E[A]x+D−E[�]D)

]B2

h

2

[D(2−

D

x

)+E[A2]x

]

(24)B=

h

2

[2D

(1+D

(1−E[�]

E[A]x

))+E[A]x

(2+D

(1−E[�]

E[A]x

))]

cBE[�](E[A]x+E[�]D)

E[A]+cL (1−E[�]) x+

hE[�]D

E[A]

(2+D

(2−E[�]

E[A]x

))

+hx(1+D

(1−E[�]

E[A]x

))−

hDE[�]

E[A2]x(E[A]x+D)y

(25)y=

√√√√kDx+M1B2

h

2M2

(26)B=

h

2M3

2M1

y=hM

3

4M1

y

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that M1, M2 and M3 are variables for simplification.

Finally,

4. Special caseTo 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:

M1 = ∞, M2 = ∞ and M3=2x+ Dx

D−x, which substituting this values in Equation 25 lead to reach to the

traditional EOQ formula.

5. Numerical examples and sensitivity analysisConsider 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

ca $ 500/unit

(27)

lM1=c

B

E[�] (E[A]x+E[�]D)

2E[A]+c

L

(1−E[�])

2x+

h

2x

(1+D

1−E[�]

E[A]x

)

+h

2

E[�]D

E[A2

]x(E[A]x+D−E[�]D)

(28)M2=D

(2−

D

x

)+E

[A2

]x

(29)M3=2D

(1+D

(1−E[�]

E[A]x

))+E[A]x

(2+D

(1−E[�]

E[A]x

))

(30)y∗ =

√√√√√√kDx+ h2

16

M3

2

M1

h

2M2

(31)B∗ =hM

3

4M1

y∗

(32)y∗ =

√2kD

h

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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:

Then we have:

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.

f (p)=

{1

p1

0≤p≤p1

0 Otherwisef (�)=

{1

�1

0≤�≤�1

0 Otherwise

f (�)=

{1

�1

0≤�≤�1

0 Otherwisef (�)=

{1

�1

0≤�≤�1

0 Otherwise

E(p)=∫

p1

0

pf (p)dp=p1

2, E (�)=

�1

2, E (�)=

�1

2, E (�)=

�1

2

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 solu-tion. 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 y* B* ETPU(y*, B*)0.02 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

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

Table 4. Effect of μ on ETPU(y, B) when the maximum number of backordering units is uniformly distributed between 0 and 1μ1 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 changesh 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 studyIn 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 com-pany 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 uni-form distribution with probability density functions as:

which

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.

f (p)=

{1

0.050≤p≤0.05

0 Otherwisef (�)=

{1

0.010≤�≤0.01

0 Otherwise

f (�)=

{1

0.010≤�≤0.01

0 Otherwisef (�)=

{1

0.40≤�≤0.4

0 Otherwise

E(p)=∫

0.05

0

pf (p)dp=0.025, E(�)=0.005, E(�)=0.005, E(�)=0.2

Figure 3. Expected annual profit for Sanayeh Dashboard Iran.

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FundingThe authors received no direct funding for this research.

Author detailsEhsan Sharifi1

E-mail: ehsan_sharifi_57@yahoo.comMohammad Ali Sobhanallahi1

E-mail: Sobhanallahi@yahoo.comAbolfazl Mirzazadeh1

E-mail: a.mirzazadeh@aut.ac.irSonia Shabani1

E-mail: Sonia_sh@rocketmail.com1 Department of Industrial Engineering, College of Engineering,

University of Kharazmi, Mofatteh Ave., Tehran, Iran.

Citation informationCite 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.

ReferencesChang, H. C., & Ho, C. H. (2010). Exact closed-form solutions for

“optimal inventory model for items with imperfect quality and shortage backordering”. Omega, 38, 233–237. http://dx.doi.org/10.1016/j.omega.2009.09.006

Duffuaa, S. O., & Khan, M. (2005). Impact of inspection errors on the performance measures of a general repeat inspection plan. International Journal of Production Research, 43, 4945–4967. http://dx.doi.org/10.1080/00207540412331325413

Eroglu, A., & Ozdemir, G. (2007). An economic order quantity model with defective items and shortages. International Journal of Production Economics, 106, 544–549. http://dx.doi.org/10.1016/j.ijpe.2006.06.015

Goyal, S. K., & Cárdenas-Barrón, L. E. (2002). Note on: Economic production quantity model for items with imperfect quality—a practical approach. International Journal of Production Economics, 77, 85–87. http://dx.doi.org/10.1016/S0925-5273(01)00203-1

Hsu, J. T., & Hsu, L. F. (2013). An EOQ model with imperfect quality items, inspection errors, shortage backordering and sales returns. International Journal of Production Economics, 143, 162–170. http://dx.doi.org/10.1016/j.ijpe.2012.12.025

Khan, M., & Jaber, M. Y. (2011). An economic order quantity (EOQ) for items with imperfect quality and inspection errors. International Journal of Production Economics, 133, 113–118. http://dx.doi.org/10.1016/j.ijpe.2010.01.023

Khan, M., Jaber, M. Y., Guiffrida, A. L., & Zolfaghari, S. (2011). A review of the extensions of a modified EOQ model for imperfect quality items. International Journal of Production Economics, 132, 1–12. http://dx.doi.org/10.1016/j.ijpe.2011.03.009

Lee, H. L., & Rosenblatt, M. J. (1987). Simultaneous determination of production cycle and inspection schedules in a production system. Management Science, 33, 1125–1136. http://dx.doi.org/10.1287/mnsc.33.9.1125

Maddah, B., Salameh, M. K., & Moussawi-Haidar, L. (2010). Order overlapping: A practical approach for preventing shortages during screening. Computers and Industrial Engineering, 58, 691–695. http://dx.doi.org/10.1016/j.cie.2010.01.014

Papachristos, S., & Konstantaras, I. (2006). Economic ordering quantity models for items with imperfect quality. International Journal of Production Economics, 100, 148–154. http://dx.doi.org/10.1016/j.ijpe.2004.11.004

Porteus, E. L. (1986). Optimal lot sizing, process quality improvement and setup cost reduction. Operations Research, 34, 137–144. http://dx.doi.org/10.1287/opre.34.1.137

Raouf, A., Jain, J. K., & Sathe, P. T. (1983). A cost-minimization model for multicharacteristic component inspection. IIE Transactions, 15, 187–194. http://dx.doi.org/10.1080/05695558308974633

Rezaei, J. (2005, September 11–13). Economic order quantity model with backorder for imperfect quality items. In Proceedings of IEEE International Engineering Management Conference (pp. 466–470). St. John’s, Newfoundland, Canada.

Rosenblatt, M. J., & Lee, H. L. (1986). Economic production cycles with imperfect production processes. IIE Transactions, 18, 48–55. http://dx.doi.org/10.1080/07408178608975329

Roy, M. D., Sana, S. S., & Chaudhuri, K. (2011). An economic order quantity model of imperfect quality items with partial backlogging. International Journal of Systems Science, 42, 1409–1419. doi:10.1080/00207720903576498

Salameh, M. K., & Jaber, M. Y. (2000). Economic production quantity model for items with imperfect quality. International Journal of Production Economics, 64, 59–64. http://dx.doi.org/10.1016/S0925-5273(99)00044-4

Shabani, S., Mirzazadeh, A., & Sharifi, E. (2014). An inventory model with fuzzy deterioration and fully backlogged shortage under inflation. SOP Transactions on Applied Mathematics, 1, 1161–1171.

Wee, H. M., Yu, J., & Chen, M. C. (2007). Optimal inventory model for items with imperfect quality and shortage

7. ConclusionIn 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.

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backordering. Omega, 35, 7–11. http://dx.doi.org/10.1016/j.omega.2005.01.019

Wee, H. M., Yu, J. C. P., & Wang, K. J. (2006). An integrated production inventory model for deteriorating items with imperfect quality and shortage backordering considerations. Lecture Notes in Computer Science, 3982, 885–897. http://dx.doi.org/10.1007/11751595

Yoo, S. H., Kim, D., & Park, M. S. (2009). Economic production quantity model with imperfect quality items, two way

imperfect inspection and sales return. International Journal of Production Economics, 121, 255–265. http://dx.doi.org/10.1016/j.ijpe.2009.05.008

Yu, J. C. R., Wee, H. M., & Chen, J. M. (2005). Optimal ordering policy for a deteriorating item with imperfect quality and partial backordering. Journal of the Chinese Institute of Industrial Engineers, 22, 509–520. http://dx.doi.org/10.1080/10170660509509319