Research Journal of Applied Sciences, Engineering and Technology 4(20): 3896-3904, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 18, 2011
Accepted: April 23, 2012
Published: October 15, 2012
A Two-Warehouse Inventory Model with Imperfect Quality and Inspection Errors
Tie Wang
School of Management, Shanghai University, Shanghai, 200444, China
School of Mathematics, Liaoning University, Shenyang, Liaoning 110036, China
Abstract: In this study, we establish a new inventory model with two warehouses, imperfect quality and
inspection errors simultaneously. The mathematical model by maximizing the annual total profit and the
solution procedure are developed. As a byproduct, we correct some technical error in developing the optimal
ordering policies in the above two papers. Moreover, we find a mild condition satisfied by most common
distributions to make the ETPU(y) concavity. The Proposition 1 is used to determine the optimal solution of
ETPU(y).
Keywords: EOQ, imperfect quality, misclassification errors, two-warehouse
INTRODUCTION
The traditional EOQ (economic order quantity)
model is a crucial building block of the deterministic
inventory theory because of its simple and elegant
structure as well as its rich managerial insights. However,
some unrealistic assumptions make it not conform to
actual inventories; the model must be extended or altered.
Porteus (1986) investigated the influence of defective
items on the traditional EOQ model. He assumed that
there is a fixed probability that the process would go outof-control. 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. They
suggested producing in smaller lots when the process is
not perfect. In a later study, (Lee and Rosenblatt, 1987)
studied a joint lot sizing and inspection policy for an EOQ
model with a fixed percentage of defective products.
Salameh and Jaber (2000) extended the traditional EOQ
model by accounting for imperfect quality items and
considered the issue that poor-quality items are sold as a
single batch by the end of the 100% screening process.
Along this line, (Papachristos and Konstantaras, 2006)
discussed the non-shortages in inventory models where
the proportion of defective items is a random variable.
They proposed an alternative to the Salameh and Jaber
(2000) by speculating on the timing of withdrawing and
selling the imperfect lot, (Wee et al., 2007) developed an
optimal inventory model for items with imperfect quality
and shortage backordering by assuming that all customers
are willing to wait for new supply when there is a
shortage. (Maddah and Jaber, 2008) rectified a flaw
(Salameh and Jaber, 2000) by the renewal-reward theorem
and leaded to simple expressions of the optimal order
quantity and expected profit per unit time. Chang and Ho
(2010) revisited Wee et al. (2007) and applied the wellknown renewal-reward theorem to obtain a new expected
net profit per unit time and derived the exact closed-form
solutions to determine the optimal lot size, backordering
quantity and maximum expected net profit per unit time
without differential calculus. Khan et al. (2011) explored
an EOQ model with imperfective items and an imperfect
inspection process That is, the inspector may commit
errors while screening. The probability of
misclassification errors is assumed to be known. The
inspection process would consist of three costs:
C
C
C
Cost of inspection
Cost of Type I errors
Cost of Type II errors
(Sarker and Kindi, 2006) developed an inventory
model from the Buyer’s perspective, to determine the
optimal ordering policies in response to a discount offer
settled by the vendor for five possible cases. Goyal and
Jaber (2006) and Cardenas-Barron (2009) extended and
corrected (Sarker and Kindi, 2006). Cardenas-Barron
(2009) investigated an inventory model for imperfective
items under a one-time-only discount, where the
defectives can be screened out by a 100% screening
process and then can be sold in a single batch by the end
of the 100% screening process. The optimal order policies
associated with three kinds of effective times of the
reduced price are obtained. Recently, Yang (2004)
considered two-warehouse models for deteriorating items
with shortages under inflation. Yang (2006) extended the
above study by incorporating partial backlogging and then
compared the two two-warehouse inventory models based
on the minimum cost approach. Zhou and Yang (2005)
presented a two-warehouse inventory model for items
with stock-level-dependent demand rate. Moreover, (Lee
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Res. J. Appl. Sci. Eng. Technol., 4(20): 3896-3904, 2012
and Hsu, 2009) developed a two-warehouse inventory
model for deteriorating items with time-dependent
demand. Chung et al. (2009) investigated a new inventory
model with two warehouses and imperfect quality
simultaneously. The mathematical model by maximizing
the annual total profit and the solution procedure were
developed.
The above study mentioned did not consider the
inventory model with two warehouses, imperfect quality
and inspection errors simultaneously. Based on
(Chung et al., 2009; Khan et al., 2011), this study tries to
develop an inventory model to incorporate concepts of
two warehouses, imperfect quality and inspection errors
to establish a new economic production quantity model.
Consequently, the inventory model in this study is more
practical than the traditional EOQ model. Of course, this
study generalizes (Chung et al., 2009; Khan et al., 2011).
As a byproduct, we correct some technical error in
developing the optimal ordering policies in the above two
studies.
This study incorporates the concepts of the basic two
warehouses, imperfect quality and inspection errors. The
expected total profit per unit time function ETPU(y) is
concave. The expected total profit per unit time function
ETPU(y) is piecewise concave. Moreover, the expected
total profit per unit time function ETPU(y) in this study is
not piecewise concave in general. In addition, we find a
mild condition satisfied by most common distributions to
make the ETPU(y) concavity. The Proposition 1 is used
to determine the optimal solution of ETPU(y).
NOTATIONS AND ASSUMPTIONS
It is assumed that the retailer owns a warehouse
(denoted by OW) with a fixed capacity of w units and any
quantity exceeding this should be stored in a rented
warehouse (denoted by RW), which is assumed to be
available with abundant space. The transportation cost
which transfers from RW to OW is included in holding
cost of the RW. So it is not considered by our study
separately. At the beginning of the period, the lot size y
enters the system with a purchasing price of c per unit and
an ordering cost of K. Out of these y units, w units are
kept in OW and y-w units in RW. (If y#w, y units are only
kept in OW.) The products of the OW are sold only after
consuming the products kept in RW. All of products
ordered arrive at the same time. They are screened with
unit inspection cost d in the OW and RW simultaneously
by the inspectors. It is assumed that each lot received
contains fixed percentage p of defective items. The
inspection process of the lot is conducted at a rate of x
units/unit time. The inspectors screen out the defective
items from the lot with fixed rate of misclassification.
That is, a proportion m1 of no defective items are
classified to be defective and a proportion m2 of defective
items are classified to be no defective. It is assumed that
p, m1, m2 have independent and identical distribution
function and the probability density functions, f(p), f(m1)
and f(m2) are known. The selling price of good-quality
item classified by the inspectors is s per unit. The
inspection process of the lot is conducted at a rate of x
units per unit time. It is assumed that the items that are
returned from the market are stored with those that are
classified as defective by the inspectors. They are all sold
as a single batch at the end each cycle at a discounted
price of v/unit. After inspection time, no holding cost is
considered for imperfect items classified by inspectors.
This cost was scaled accordingly. There are also no
holding costs for the returned products.
The behavior of the inventory level may be illustrated
in Fig. 1, where T is the cycle length, y (1-p) m1+py (1-m2)
is the number of defective items classified by the
inspectors withdraw from inventory, tR is the total
inspection time of the RW, tw the total inspection time of
the OW and to the time to use up of the RW. Figure 1 and
2 reveal tR = (y-w)/x and tw = w/x.
THE MODEL
From the above assumptions about the model and
Fig. 1 and 2, we know the consumption process continue
at the demand rate until the end of cycle time T. Due to
inspection error, some of the items used to fulfill the
demand are defective. These defective items are later
returned to the inventory. To avoid shortages, it is
assumed that the number of no defective items at least
equal to the adjusted demand, that is the sum of the actual
demand and items that are replaced for ones returned
(pm2y) from the market over T.
Thus,
y  y (1  p)m1  yp(1  m2 )  DT  pm2 y
y (1  p)(1  m2 )  DT
So, for the limit case, the cycle length can be written as:
3897
T
y (1  p)(1  m2 )
D
(1)
It should note here that this behavior was suggested
by Khan et al. (2011).
Define NR(y, p) and NO(y, p) as the number of no
defective items with respect to warehouses RW and OW,
respectively. There are two cases occur.
Case 1: Suppose y<w. Then
Res. J. Appl. Sci. Eng. Technol., 4(20): 3896-3904, 2012
Fig. 1: The model of products in stock of two warehouses if t0$ tw
Fig. 2: The model of products in stock of two warehouses if tO< tw
No( y , p)  y  y (1  p)m1  yp(1  m2 )
(2)
To avoid shortages, it is assumed that NO(y, p) are at
least equal to, the sum of the actual demand during
3898
times y/x and items that are replaced for ones
returned from the market during time T, that is:
No( y, p)  D
y
 ypm2
x
Res. J. Appl. Sci. Eng. Technol., 4(20): 3896-3904, 2012
Substituting (2) into (3). We have:
(1  p)(1  m1 ) 
ETPU ( y ).  E[
D
.
x
About the computations of ETPU(y) there are
two cases to occur.
Case 2: Suppose y$w. Then:
Case (I): Suppose y<w.
N R ( y , p)  y  w  ( y  w)(1  p)m1
 ( y  w) p(1  m2 )
(3)
N o (Y , P)  w  w(1  P)m1  wp(1  m2 )
(4)
This case is same to the model presented in
Khan et al. (2011) study. But, there is a
mathematical error in (2) of their paper for
holding cost per cycle, so the total cost per
cycle, the total profit per cycle, average profit
per cycle and optimal order size. The corrected
total cost per cycle should be
TC1(y) = procurement cost + screening cost +
holding cost:
To avoid shortages, it is assumed that NR(y, p)
and NO(y, p) are at least equal to, the sum of the
actual demand during times tR and tw and items
that are replaced for ones returned from the
market during time t0 and T-tO that is:
N R ( y , p)  Dt R  ( y  w) pm2
 K  cy  dy  cr y (1  p)m1  ca ypm2  hw [
(5)

N o ( y , p)  Dt 0  wpm2
TP
]
T
( y  Dt y  y (1  p)m1  yp(1  m2 ))
Dt y2
2
(T  t y ) 
ypm2
T ].
2
(8)
(6)
Since ty = y/x, (8) can be written as follows:
Substituting (4) and (5) in (6) and (7) and
replacing tR and tw by (y-w)/x, respectively. We
have:
TC1( y )  K  cy  dy  cr y (1  p)m1  ca ypm2
 2(m1  p)  2 p(m1  m2 )  pm2 y 2 p(1  m1)m2 


hw y 2 
x
2D



2  (1  p)(1  m1)(1  p)(1  m1)  pm2



D


D
(1  p)(1  m1 ) 
x
The total profit per cycle can now be written as
the difference between the total revenue and the
total cost per cycle, that is:
Combining the above arguments, we conclude:
(1  p)(1  m1 ) 
D
x
(9)
(7)
TP1( y )  TR( y )  TC1( y ) sy (1  p)(1  m1 )
 sypm2  vy (1  p)m1  vyp  {K  cy  dy
All of products are stored in OW and RW,
respectively and screen them in OW and RW
simultaneously at the same time. In addition,
TR(y) denote the sum of total sales revenue of no
defective quality classified by the inspectors,
imperfect quality items classified by the
inspectors and returned items per cycle; TC(y),
TP(y) and ETPU(y) denote the sum of total cost
per cycle and the profit per cycle and average
profit per cycle, respectively.
Then,
hw y 2
2
 2(m1  p)  pm2  2 p(m1  m2 )

D

 cr y (1  p)m1  ca ypm 

(1  p)(1  m1)((1  p)(1  m1 )  pm2 ) 

D

 hw
(10)
y 2 p(1  p)(1  m1)m2 

2D

To obtain the exact closed-form solution to
determine the optimal lot size without
differential calculus, we can use the renewalreward theorem to derive the expected net
profit per unit time ETPU1(y) as the following:
TP(y) = TP(y). ! TC(y)
TR( y ).  sy (1  p)(1  m1 )  sypm2 )
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Res. J. Appl. Sci. Eng. Technol., 4(20): 3896-3904, 2012
TC A ( y )  K  cy  dy  cr y (1  p)m1  ca ypm2
 TP ( y )  E[TP1 (Y )
ETPU1( y )  E  1 

 T  E [T ]
( y  w) 2 2(m1  p)  p(m1  m2 )  m1 p
[

2
x
(1  p)(1  m1 )(1  p)(1  m1 )  2 pm2 )
]
D
hR
s1  E[ p])(1  E[m1]  ( s  ca ) E[ p]E[m2 ] 
(v  cr )(1  E[ p]) E[m1]  vE[ p]  (d  c)
D
1  E[ p]1  E[m1]

w2
((1  p)m1  p(1  m2 )) 
x
w((1  p)(1  m1  pm2 )(2 y  w)(1  p)(1  m1 )
]
2D
w 2 pm2 (1  P)(1  m1 )
hw [
]
2D
hw [
2( E[m1 ]  E[ p]  E[ p])  E[ p]E[m2 ]
{
hw y
KD

y (1  E[ p])(1  E[m1 ])
2
[D
 2 E[ p]( E[m1 ]  E[m2 ])
x (1  E[ p])(1  E[m1 ])
E[(1  p) 2 ](1  E[m1 ])2 E[ p(1  p)]E[m2 ]
}
1  E [ p]
G
 C  (  yH )
y
Case B: Suppose tO<tw. Then
TCB(y) = procurement cost + screening cost +
holding cost:
(11)
K  cy  dy  cr y (1  p)m1  ca ypm2  hR [
Equation (10) and (11) are corrected total profit per
cycle and expected annual profit for Eq. (6) and (9)
in Khan et al. (2011). Note that when m1 = m2 = 0,
Eq. (11) gives the exact expression for Eq. (8) of
Chung et al. (2009) as:
ETPU1 ( y )  D
s(1  E[ p]  vE[ p]  (d  c)
1  E [ p]

2 E [ p]
KD
h y
E[1  p]2  

 w D


(
1
[
]
2
(
1
[
])
1  E[ p]  
y

E
p
x

E
p



(12)
Which is same to the Eq. (6) presented in (Maddah
and Jaber, 2008).
Case (II): Suppose y$w. There are two cases to be
discussed.
Case A: Suppose tO$tw. Then
TCA(y) = procurement cost + screening cost +
holding cost:
 K  cy  dy  cr y (1  p)m1  ca ypm2  hR [
Dt
2
(14)
DR2
2
( y  w) pm2
tO 
2
y  w  Dt R  ( y  w)(1  p)m1  ( y  w) p(1  m2 )
(t O  t R )]
2
 ( y  w  Dt R )t R 
D( t w  t O ) 2
 ( w  D(t w  t O ))(t w  t O ) 
2
w  D(t w  t O )  w(1  p)m1  wp(1  m2 ))
(T  t w )
2
wpm2

(T  t O )]
2
hw [ wt O 
(15)
Since tO = (y-w) (1-p) (1-m1)/D, tR = (y-w)/x, tw = w/x
and T = y (1-p)(1-m1)/D, (15) can be written as
follows:
TC B ( y )  K  cy  dy  cr y (1  p)m1  ca ypm2 
( y  w) 2 2(m1  p)  p(m1  m2 )  m1 p
[

2
x
(1  p)(1  m1 ((1  p)(1  m1 )  2 pm2 )
]
D
hR
w 2 2(1  p)m1  p(1  m2 )  p
(

2
x
w((1  p)(1  m1 )(2 y  w)  ypm2 ))(1  p)(1  m1 )
]
2D
2
w pm2 (1  p)(1  m1 )
hw [
]
2D
hw [
2
R
( y  w) pm2
to 
2
y  w  Dt R  ( y  w)(1  p)m1  ( y  w) p(1  m2 )
(tO  t R )]
2
 ( y  w  Dt R )t R 
So, we conclude:
hw [ wt w  ( w  w(1  p)m1  wp(1  m2 ))(t O  t R ) 
(13)
( w  w(1  p)m1  wp(1  m2 ))
wpm2
(T  t O ) 
(T  t O )]
2
2
TC2 ( y )  TC A ( y ) I [ t) t
Since tO = (y-w) (1-p) (1-m1)/D, tR = (y-w)/x, tw = w/x
and T = y(1-p)(1-m1)/D, (13) can be written as
follows TCA(y):
hR
w
]
 TC B ( y ) I [ tO  t w ]
 K  cy  dy  cr y (1  p)m1  ca ypm2 
3900
( y  w) 2 2(m1  p)  p(m1  m2 )  m1 p
[

x
2
(16)
Res. J. Appl. Sci. Eng. Technol., 4(20): 3896-3904, 2012
(1  p)(1  m1 )((1  p)(1  m1 )  2 pm2 )
]
D
w2
hw I [ tO  t w ] [
((1  p)m1  p(1  m2 )) 
x
w((1  p)(1  m1 )  pm2 (2 y  w)(1  p)(1  m1 )
]
2D
2( E[m1 ]  E[ p])  E[ p]( E[m1 ]  E[m2 ]
hR w
E (1  p) 2 E (1  m1 ) 2 ]
E[ p(1  p) E[m2 ]
2
)
(1  E[ p])(1  E[m1 ]
1  E [ p]

hw w 2 E[(1  p) 2 [ E[(1  m1 ) 2 ] Ep(1  p) E[m2 ]

(
(1  E[ p])(1  E[m1 ])
2
1  E [ p]
E[ p(1  p) I [ tO  t w ] ]E[m2 ]


1  E [ p]
KD
1

{ (
y (1  E[ p])(1  E[m2 ])
w 2 2(1  p)m1  p(1  m2 )  p

hw I [ tO  tw ] [
(
2
x
w((1  p)(1  m1 )(2 y  w)  ypm2 ))(1  p)(1  m1 )
]
2D
(17)
w 2 pm2 (1  p)(1  m1 )
hw [
]
2D

2( E[m1 ]  E[ p])  E[ p]( E[m1 ]  E[m2 ]
Note that when m1 = m2 = 0, TCA(y) = TCB(y) and (17)
will be reduced to Eq. (13) presented in Chung et al.
(2009).
 E[m1 ]E[ p]
hR w 2
[D
x (1  E[ p])(1  E[m1 ]
2

Consequently:
(1  E[ p]) E[m1 ]  E[ p](1  E[m2 ]

x (1  E[ p])(1  E[m1 ]
E[ pI [ tO  t w ] E[m2 ]
E[ p(1  p) E[m2 ]
D

2 x (1  E[ p])(1  E[m1 ]
2(1  E[ p])
hw y 2
[
2
hR
E[(1  p) 2 ]E[(1  m1 ) 2
])  y [
2(1  E[ p](1  E[m1 ])
2
2( E[m1 ]  E[ p])  E[ p]( E[m1 ]  [m2 ])
 E[m1 ]E[ p]
D
x (1  E[ p](1  E[m1 ]

 2 m1  p  p m1  m2   m1 p
hR



x

 (1  p)(1  m1 )(1  p)(1  m1 )  2 pm2 ) 


D


 y  w2 
2
 w2

((1  p)m1  p(1  m2 )) 


x

hw I[ tO  t w ] 
 w((1  p)(1  m1 )  (2 y  w)(1  p)(1  m1 ) 


2D


 w2  2(1  p)m1  p(1  m2 )  p 







2
x


hw I[ tO  t w ]
 w((1  p)(1  m )(2 y  w)  ypm ))(1  p)(1  m ) 
1
2
1


2D


 w2 pm2 (1  p)(1  m1)  
hw 

2D

 
E[(1  p) 2 E[(1  m1 ) 2
E[ p(1  p) E[m2 ]
2
]
(1  E[ p])(1  E[m1 ])
1  E [ p]
 hw w 2 [ D
TP2 ( y )  TR( y )  TC2 ( y )  sy (1  p)(1  m1 )
 sypm2  vy (1  p)m1  vyp  {K  cy  dy
 cr y (1  p)m1  ca ypm2 
 E[m1 ]  E[ p]
x (1  E[ p](1  E[m1 ])
E[(1  p) 2 ]E[(1  m1 ) 2
E[ p(1  p) E[m2 ]
2
(1  E[ p])(1  E[m1 ])
1  E [ p]
J
 I  (  yL)
y

Note that when m1 = m2 = 0, Eq. (19) gives the
corrected expression for Eq. (18) of Chung et al.
(2009) as:
(18)
ETPU 2 ( y )  D
To obtain the exact closed-form solution to determine
the optimal lot size, we can use the renewal-reward
theorem to derive the expected net profit per unit
time ETPU2(y) as:
hR w( D
hw w
s(1  E[ p])  vE[ p]  (d  c)
(1  E[ p])(1  E[m1 ]
E[(1  p) 2 ]
2 E [ p]

)
x (1  E[ p])(1  E[m1 ] (1  E[ p])
E[(1  p) 2
KD
1
{ (

y 1  E[ p])
(1  E[ p]
hR w 2
E[(1  p) 2 ]
2 E [ p]
[D

]
x (1  E[ p]) (1  E[ p])
2
TP2 ( y ) E[TP2 ( y )]

T
E[T ]
s(1  E[ p](1  E[m1 ])  ( s  ca ) E[ p]E[m2 ]) 
ETPU 2 ( y )  E
D
(19)
 hw w 2 [ D
(v  cr )(1  E[ p]) E[m1 ]  vE[ p]  (d  c))

1  E[ p])1  E[m1 ])
y
3901
E [ p]
E[(1  p) 2 ]

]) 
x (1  E[ p]) 2(1  E[ p])
hR
E[1  p) 2
2 E [ p]

[D
]
x (1  E[ p]) (1  E[ p]
2
(20)
Res. J. Appl. Sci. Eng. Technol., 4(20): 3896-3904, 2012
E[ pI [ tO  tw ] ]  EpI
Combining (11) and (20), we have:
 ETPU1 ( y )
ETPU ( y )  
 ETPU 2 ( y )
if y  w
(21),(22)
if y  w
 E[ pF (1 
Dw
]
x ( y  w )(1 p )
]
Dw
]
x ( y  w)(1  p)
Taking derivative of ETPU2(y) with respect y gives:
and,
ETPU1(w) = ETPU2(w)
(23)
ETPU1 ( w)  C  2 GH
When the two terms related to y in (11) are equal,
implying:
y1* 
G
H

2 KD
(1  E[ p])(1  E[m1 ]
2( E[m1 ]  E[ p])
Dw
)]
x ( y  w)(1  p)
J
dETPU 2 ( y ) 
 2
2
x (1  E[ p]( y  w)
y
p
Dw
)]
DwE[m2 ]E
f (1 
1
1 p
x ( y  w)(1  p)

 L
y
2 x 2 (1  E[ p](1  E[m1 ]( y  w) 2
DwE[m2 ]E[ pf (1 
Using the arithmetic geometric mean inequality
(AM-GM) theorem:
lim
 

Dw
DwE[m2 ]E  pf  1 

x
(
y

w
)(
1

p
)
 
 
x (1  E[ p])( y  w) 2
(25)

J
y2

(28)
Let m1 = m2 = 0, (28) gives the exact expression for
(25) of Chung et al. (2009) as:
The (23) is the corrected mathematical expression for
(10) of (Khan et al., 2011). Note that when m1 = m2 =
0, (23) gives the exact expression for (24) of (Chung
et al., 2009) as:


 p


Dw
DwE[m2 ]E 
f 1

p
x
y
w
p
(
)(
)
1
1




 
1

 L  0.
y
2 x 2 (1  E[ p])(1  E[m1 ]( y  w) 2
and (11) reduces to equality, that is, the maximum
profit is:
2 KD
[
]
2
E
p

hw [ D
 E (1  p) 2
x

dETPU 2 ( y )
dETPU 2 ( y )
 0 lim
  L  0.
y  0 
dy
dy
Hence there must exit a root for d ETPU2(y)/dy = 0 at
least, so y2* must satisfy the following equation:
(24)
ETPU 1* ( y )C  2 GH
(27)
From the assumptions about the system parameters,
we know:
y  0
 2 E[ p]( E[m1 ]  E[m2 ]) E (1  p) 2 ](1  E[m1 ])
 E[ p]E[m2 ]
 2 E[ p(1  p)]E[m2 ]
hw [ D

x (1  E[ p])(1  E[m1 ])
1  E [ p]
y1** 
[ m1  1
2 E [ p]
KD
h w
E (1  p) 2 ] 
 R D


1  E [ p]
2  x (1  E[ p]
1  E[ p] 
y2** 
.
(26)
This is same to the result obtained by Maddah and
Jaber (2008).
Note that,

E [ p]

hw w2  D
 x (1  E[ p])
2 E [ p]
hR 

D
2  x  (1  E[ p])
E[(1  p) 2 

1  E[ p] 
(29)
Taking derivative of ETPU2(y) with respect y again
gives:
2
d ETPU 2 ( y )
dy 2



Dw
E[ p(1  p) I [ tO  t w ] ]  E  p(1  p) I  m1  1 

x
(
y

w
)(
1

p
)



E (1  p) 2 ] 

2(1  E[ p] 
 2
 

Dw
DwE [m2  E  pf  1 

x ( y  x )(1  p)  
 
x (1  E[ p])( y  w) 3
 p


Dw
DwE[m2 ]E 
f 1

x ( y  x )(1  p)  
 1  p 
 2
3
2
y ( y  w)
2 x (1  E[ p])(1  E[m1])
3y  w


Dw
)
 E  p(1  p) F (1 
x ( y  w)(1  p) 

1
J
2 3 2
y
y
and
3902
Dw
p
f 1
)]
x ( y  w)(1  p)
1 p
2 x 2 (1  E[ p])(1  E[m1 ])( y  w) 2
DwE[m2 ]E
Res. J. Appl. Sci. Eng. Technol., 4(20): 3896-3904, 2012
p
Dw
)]
f ' (1 
1 p
x ( y  w)(1  p)
2
4
x (1  E[ p]( y  w)
D 2 w 2 E[m2 ]E

1
y



p
Dw
D2 w2 E [m2 ]E 
f ' 1 

2
x ( y  w)(1  p)  

 (1  p)
3
3
2 x (1  E[ p])( y  w)
(30)
From the (30), we can not determine the conavity of
ETPU2(y). But, we notice that d2 ETPU2(y)/dy2<0
when f’(x)#0. This condition is satisfied for many
usually used distributions such as normal
distribution, exponential distribution and uniform
distribution and so on.
In the following discussion, we assume f’(x)#0. Note
that I>0 and J>0 because of hR is more than hw. Both
ETPU1(y) and ETPU2(y) are concave. So, y1* and y2*
are the possible optimal solutions for ETPU(y). Let
yopt* represent the optimal solution of ETPU(y).
Because y1*<w and y2*$w must be satisfied, so we
have the following proposition.
Proposition 1: Under f’(x)#0 three cases may occur:
C
if y1*<w and y2*$w, then yopt* = y1* or y2* such that
ETPU (yopt*) = max{ ETPU1(y1*), ETPU2(y2*)}
C
C
if y1*<w and y2*<w, then yopt* = y1*
if y1*$w and y2*$w, then yopt* = y2*
CONCLUSION
This study incorporates the concepts of the basic two
warehouses, imperfect quality and inspection errors to
generalize Chung et al. (2009) and Khan et al. (2011).
The expected total profit per unit time function ETPU(y)
in Salameh and Jaber (2000) is concave. The expected
total profit per unit time function ETPU(y) in Chung et al.
(2009) is piecewise concave. However, the expected total
profit per unit time function ETPU(y) in this study is not
piecewise concave in general. But, we find a mild
condition satisfied by most common distributions to make
the ETPU(y) concavity. The Proposition 1 is used to
determine the optimal solution of ETPU(y).
NOTATIONS
D
w
z
y
c
K
s
v
Number of units demanded per year
Storage capacity in OW, fixed
Storage capacity in RW
Order quantity
Unit variable cost
Fixed ordering cost
Unit selling price of items of good quality
Unit selling price of defective items, v<c
3903
x
Screening rate
d
Unit screening cost
hR,hw Holding cost for items in the RW and OW,
respectively, hR$hw
T
The cycle length
Probability of Type I error (classifying a no
m1
defective item as defective)
Probability of Type II error (classifying a defective
m2
item as no defective)
p
Probability that an item is defective
Inspection time of the RWtwinspection time of the
tR
OW
Time to use up of the RW
tO
f(p) Probability density function of p
f(m1) Probability density function of m1
f(m2) Probability density function of m2
BR1 Number of items that are classified as defective in
rented warehouse
Number of items that are classified as defective in
B01
own warehouse
BR2 Number of defective items that are returnedfrom
the market in rented warehouse
Number of defective items that are returnedfrom
B02
the market in own warehouse
Cost of accepting a defective item
ca
Cost of rejecting a non defective item
cr
TR(y) The sum of total revenue of good quality and
imperfect quality items per cycle
TC(y) The sum of total costs per cycle
yopt* He optimal solution such that ETPU (yopt*) will be
a maximum
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