An eoq model with stock dependent demand

Yugoslav Journal of Operations Research
Volume 20 (2010), Number 2, 237-247
DOI:10.2298/YJOR1002237I
AN EOQ MODEL WITH STOCK DEPENDENT DEMAND
AND IMPERFECT QUALITY ITEMS
Shibaji PANDA
Department of Mathematics, Bengal Institute of Technology, Kolkata, India.
shibaji_panda@yahoo.com
Received: July 2004 / Accepted: November 2010
Abstract: This paper deals with an economic order quantity model where demand is
stock dependent. Items received are not of perfect quality and each lot received contains
percentage defective imperfect quality items, which follow a probability distribution.
Two cases are considered. 1) Imperfect quality items are held in stock and sold in a
single batch after a 100 percent screening process. 2) A hundred percent screening
process is performed but the imperfect quality items are sold as soon as they are detected.
Approximate optimal solutions are derived in both cases. A numerical example is
provided in order to illustrate the development of the model. Sensitivity analysis is also
presented, indicating the effects of percentage imperfect quality items on the optimal
order quantity and total profit.
Keywords: Inventory, stock dependent demand, screening cost, imperfect quality items.
AMS Mathematics Subject Classification : 90B05
1. INTRODUCTION
The diversity of demand rate leads many departmental store managers to fall in
some confusing situation. It is observed that for some items the demand rate is a
combination of two parts. One is constant in nature and the other is proportional with the
amount of inventory displayed. According to Levin et. al. [9] ”at times, the presence of
inventory has a motivational effect on people around it. It is a common belief that large
piles of goods displayed in the departmental store will lead the customer to buy more.” It
was also investigated by Silver and Peterson [20] that retail level sales vary with the
amount of inventory displayed. These observations impressed many researchers to
investigate the modeling aspects of the demand phenomenon. The variability of demand

238 S., Panda / An EOQ Model With Stock Dependent Demand and Imperfect Quality Items
rate on the analysis of inventory system had been focused by researchers like Dutta and
Pal [7], Pal et al. [12], Phelps [13], Mondal and Phaujder [11], Baker and Urban [3],
Goswami and Choudhuri [8], Ritchie and Tsado [15], Silver [21], Silver et al. [22]. They
described the demand rate as inventory level dependent and also assumed that the amount
of inventory supplied is of good quality. This is somewhat unrealistic and disagrees with
the observations of the managers that for some inventory system all units received in a lot
are not of good quality. This inspired many researchers to study the effects of imperfect
quality items on EOQ in details. The effect of imperfect quality items on optimal order
quantity was first introduced in the model developed by Porteus [14]. In the classical
EOQ model he estimated the effect of defective items under the assumption that the
production process might go out of control, while producing one unit of the item, had the
probability q. Cheng [6] developed an inventory decision problem with demand
dependent unit production cost and imperfect production process. Rosenblatt and Lee
[16] formulated a model under the assumption that time between the beginning of the
production run i.e. in control state and until the process goes out of control was
exponential and the imperfect items were reworked instantaneously at a cost. They
concluded that the presence of defective items ensures the smaller lot sizes. Schwaller
[19] presented an extended EOQ model under the assumption that the defective items in a
lot received were of a known proportion and the fixed variable inspection cost were
included in removing the items. Zang and Gerchak [25] incorporated an EOQ model in
which random portion of units were defective and defective items could not be used but
had to be replaced by good items. Salemah and Jaber [18] discussed an EOQ model in
which they assumed that each lot received contains percentage defective items with
known probability distribution. They concluded that increment in average percentage
defective items motivates larger lot sizes. The effects of imperfect quality items in lot
sizing policy were noted and discussed by Anily [1], Urban [23,24], Lee and Rosenblatt
[17], Chakraborty and Stub [5], Ben Deya and Hariga [4]. However, all the models cited
above regarding the management of imperfect quality items in pure inventory scenario
as well as production inventory scenario were dealt with static demand rate. As a result
the lot sizing decision in pure inventory scenario under stock dependent demand structure
was ignored. In this paper we address the issue. It is assumed that under inventory level
dependent demand rate, the lot size received at the beginning of an order cycle contains
percentage defective items, which follow a probability distribution. A screening process
is performed to differentiate perfect and imperfect quality items. Two cases are
considered – (i) imperfect quality items are sold in a single batch after a hundred percent
screening process and (ii) imperfect quality items are sold as soon as they are detected.
The objectives in both cases are to maximize expected unit profit of the system.
2. ASSUMPTIONS AND NOTATIONS
We adopt the following assumptions and notations for the model to be
discussed.
• The demand rate R(q) is deterministic and depends on instantaneous level
of inventory q(t) at time t and is of the form R(q(t)) = a + bq(t). a > 0 is
the initial demand rate independent of stock level. 0 ≤ b ≤ 1 is the stock
sensitive parameter of demand. q(t) is the instantaneous inventory level at
time t.

• Percentage defective item in each lot is p, with known probability density
function f(.).
• A hundred percent screening process is performed at a rate x units per unit
time.
• Shortages are not allowed.
• The time horizon of the inventory system is infinite. Only a typical planning
schedule of length T is considered, all remaining cycles are identical.
• A is the set up cost per cycle. C1 is the holding cost/unit/unit time. C2 is the
unit purchase cost of the product. d is the screening cost per unit. u and v
are the unit selling price of perfect and imperfect quality items respectively.
3. MODEL FORMULATION
At the beginning of the order cycle Q units of inventory are received in stock.
As time progresses inventory level decreases due to quantity demand and screening to
differentiate perfect and imperfect quality items. The time required to perform a hundred
percent screening process is given by
Ttt x
Q
≤≤=′ (1)
After time t′ the inventory level decreases due to the quantity demanded and
ultimately it reaches the zero level at the end of the cycle time T. The instantaneous states
of q(t) over the cycle time T is governed by the following first order linear differential
equations
'
tt0,))((
dt
dq(t)
≤≤−−= xptqR (2)
Ttt,))((
dt
dq(t) '
≤≤−−= xptqR (3)
with the initial and boundary conditions q(0) = Q and q(T) = 0 respectively. Solving (2)
and (3) we get
( ) t0,e
b
a
b
a
e
b
xpa
Q
b
xpa
-q(t) t-Tbtb
′≤≤+−=⎟
⎠
⎞
⎜
⎝
⎛ +
++
+
= ′′−
t (4)
( ) Ttte
b
a
q(t) t-Tb
≤≤′+−=
b
a
(5)
At time t = t′ from (4) and (5) we have
( )t-Tbtb
e
b
a
b
a
e
b
xpa
Q
b
xpa
- ′′−
+−=⎟
⎠
⎞
⎜
⎝
⎛ +
++
+
Using (1) and simplifying, cycle time T can be found as,

e
a
xp
a
xpbQa
ln
b
1
T x
bQ
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
++
= (6)
The number of perfect quality item is Q(1 − p). Since shortages are not allowed,
the number of perfect quality item is at least equal to the demand during the screening
time t′. Demand during screening is
pQe1
b
xpa
QR(q(t))dtD x
bQ
t
0
s −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−⎟
⎠
⎞
⎜
⎝
⎛ +
+=∫=
−′
(7)
Thus, simplifying (7) it is found that p will be restricted by the following
relation
'x
bQ
p
x
a
)1e(/p =−−≤ xbQ (8)
3.1. Case -1
For this case we assume that imperfect quality items are sold in a single batch
after a hundred percent screening process. Then, holding cost of inventory for the
replenishment cycle is
∫
′
+⎟
⎠
⎞
⎜
⎝
⎛ ′
−′+∫
′
=
T
t
(t)dtq1C
2
tpQ
tpQ1C
t
0
dtq(t)1CHC
Substituting (1) and simplifying we have
b
aT
2x
pQ
)p1(
b
Q
CHC
2
1
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−+−= (9)
Total profit of the system consists of sales volume of perfect and imperfect
quality items, set up cost, screening cost, purchase cost and holding cost and is given by
HC)dQC2Q(A-vQpp)Q-u(1TP(Q) ++++= (10)
Now the total relevant profit per unit time can be obtained as
T
HC)dQQC(A-Qvpp)Qu(1
T
TP(Q)
TPU(Q) 2 ++++−
== (11)
The expected total profit per unit time is, therefore, given by
dp(p)f
T
HC)dQQC(A-Qvpp)Qu(1
ETPU(Q)
p
0
2
∫ ⎥
⎦
⎤
⎢
⎣
⎡ ++++−
=
′
(12)

where p′ is the upper bound of p, HC and T is given by (9) and (6) respectively.
Our problem is to determine the approximate optimal order quantity Q*
which
maximizes ETPU(Q) of the inventory system. The necessary condition for ETPU(Q) to
be maximum is
0ETPU(Q)
dQ
d
= (13)
Since the expression under the integration sign of equation (12) and
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
T
QTP )(
Q
are continuous (see Theorem 1 and Theorem 2 of Appendix) then by
Leibnitz rule [2] the differentiation under the sign of integration is permissible. Applying
Leibnitz rule, from (13) we have
0dp)p(f
T
T
Q
TP(Q)TP(Q)
Q
T
ETPU(Q)
dQ
d p
0
2
=∫
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
∂
∂
−
∂
∂
=
′
(14)
In the limit as b → 0 and p → 0 the above equation leads to
1
*
C
2Aa
Q →
which is the classical EOQ as expected.
However, it is difficult to find an explicit expression for Q*
by solving the
integral equation (14). Solving (14) by the following solution procedure one can easily
find Q*
, the approximate optimal order quantity. Then the corresponding average
expected total profit ETPU*
, cycle time T*
and probability bound p*
′ can be found from
(12), (6) and (8), respectively.
3.2. Case - 2
In this case we assume that hundred percent screening process is performed but
the imperfect quality items are sold as soon as they are detected.
The holding cost is given by
b
aT
)p1(
b
Q
Cq(t)dtCq(t)dtCHC 1
T
t
1
t
0
1 ⎥
⎦
⎤
⎢
⎣
⎡
−−=∫+∫=
′
′
(15)
Replacing this expression for HC in (12) and proceeding in the same way as in
case - 1, the value of Q*
, ETPU*
and T*
can be determined numerically.

A solution procedure
The problem is to find Q for which
∫ ==
′p
0
0dpQ)G(p,I
Assume that ε is the error tolerance. n1 and n2 are the number of iterations to
evaluate the integral value and to find the optimal value of Q*
. Let Q be the initial
approximation for the optimal order quantity. Then it is convenient to proceed in the
following way.
Step 1 : Input parameter values, n1 , n2 , Q = 0
Step 2 : , i = 0 and j = 0
Step 3 :Taking a step length i
p
2
'
use Trapezoidal rule to evaluate
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= jTI i
p
,
2
'
Step 4 : If the absolute ε≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
jT i
p
,
2
'
then
go to Step 6.
else
i = i+1
evaluate
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= jTI
i
p
,
2
'
Step 5 : if the absolute ε≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
jT i
p
,
2
'
then
go to Step 6
else
k = j+1
use Richardson’s extrapolation technique
14
,
2
,
2
4
,
2
1
−
⎟
⎠
⎞
⎜
⎝
⎛
−⎟
⎠
⎞
⎜
⎝
⎛
=⎟
⎠
⎞
⎜
⎝
⎛ −
k
ii
k
i
j
h
Tj
h
T
k
h
T
if the absolute ε≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
kT i
p
,
2
'
then
go to Step 6
else
m = m+1, j = j+1
if m ≤ n1 then
go to Step 5
else
l = l+1
if l ≤ n2 then
Q = Q+r
go to Step 2
else

write the method does not converge
Step 6 : write output Q = Q*
, T*
and ETPU(Q*
)
4. NUMERICAL EXAMPLE
In this section we present computational results that yields some insight
behaviour of Q*
, T*
and ETPU*
as p and b varies. The parameter values are taken as a =
60000 units, C1 = $5 units per year, A = $100 per year, x = 1 unit per minute, d = $0.5
per unit, C2 = $25 per unit, u = $50 per unit, v = $20 per unit. We assume that the
percentage defective random variable p follows uniform distribution having the
distribution function f(p) = 25 for 0≤p≤0.04 and 0 elsewhere.
Table1. Effects of b and p on Q*
, T*
, ETPU*
and probability bound p′*
for Model A.
b p T*
Q*
ETPU*
($) p′*
0.02 0.04 0.02605 1629.14 1455897.00 0.657
0.05 0.02617 1653.27 1453850.00 0.657
0.06 0.02635 1680.34 1433496.00 0.657
0.07 0.02666 1702.59 1421712.00 0.657
0.08 0.02679 1722.01 1390249.00 0.657
0.04 0.04 0.02735 1732.86 1456277.00 0.657
0.05 0.02742 1745.03 1454532.00 0.657
0.06 0.02747 1754.49 1433562.00 0.657
0.07 0.02761 1770.48 1421860.00 0.657
0.08 0.02771 1784.67 1390942.00 0.657
Table1 and Table2 present the effects of inventory dependent demand parameter
b and the percentage defective factor p on the approximate optimal solution Q*
, T*
and
ETPU*
. It has been examined in both cases separately so that the sufficient condition for
maximization of ETPU is satisfied. It is observed from Table1 and Table2 that
• ETPU*
is higher when the imperfect quality items are sold as soon as they
are detected instead of holding them in stock. What is quite obvious and
reasonable explanation is that single batch imperfect quality items selling
requires xpQC 2/2
1 extra holding cost.
• Q*
increases as p increases i.e. optimal order quantity increases as the
percentage defective item increases. But expected average profit decreases
as percentage of defective item increases because imperfect quality items
are sold at a lower price than perfect quality items.
• For a particular b and p, Q*
of case-1 is higher than that of case-2 but
ETPU*
of case-1 is lesser than that of case-2. That is, if the percentage
defective item is the same, then the inventory level is higher if the defective
items are sold in a single batch after a full screening process than if the
defective items are sold as soon as they are detected, though the expected
total profit is less in former case.
• p is bounded by 0.657 which is within the range 0≤ p ≤ 1, to avoid
shortages.

• For a particular p increments of b indicates the increment of ETPU*
. That
is, expected total profit increases as demand increases.
5. CONCLUSION
In this article an inventory model is developed, where demand is stock
dependent and a random percentage of items are defective. Full screening is performed to
differentiate perfect and imperfect quality items. In case-1, it is assumed that the
imperfect quality items are held in stock during screening and are sold in a single batch
when the screening process is completed. Whereas, in the second case it is assumed that
imperfect quality items are sold as soon as they are detected. These types of assumptions
are quite appropriate when the selling prices of perfect and imperfect quality items are
different and the goodwill is a matter of consideration. Retailers in supermarkets and
many departmental store managers face this kind of problem while selling products like
polymer equipments, cotton garments, electronic devices etc., whose quality should be
tested before sale. And these products are not only costly but are also long term usable
Table 2. Effects of b and p on Q*
, T*
, ETPU*
and probability bound p′*
for Model A.
b p T*
Q*
ETPU*
($) p′*
0.02 0.04 0.02592 1620.53 1456036.00 0.657
0.05 0.02601 1643.69 1454612.00 0.657
0.06 0.02610 1666.80 1433573.00 0.657
0.07 0.02617 1674.62 1421805.00 0.657
0.08 0.02627 1685.22 1391021.00 0.657
0.04 0.04 0.02800 1751.47 1456287.00 0.657
0.05 0.02805 1773.14 1454598.00 0.657
0.06 0.02813 1776.19 1433573.00 0.657
0.07 0.02827 1780.02 1421895.00 0.657
0.08 0.02835 1809.76 1391081.00 0.657
products. Otherwise, it results in customer dissatisfaction and hence decrement of
goodwill. This article suggests some interesting characteristics regarding the management
of random mixture of perfect and imperfect quality items lot-size. (i) Inventory level
increases as the percentage defective item increases. (ii) It is expected that imperfect
quality items should be sold as soon as they are detected instead of holding them in stock.
Not only that, single batch selling of defective items produces lesser profit for higher
level of inventory than the detection and sell process. The model can be extended by
applying different types of demand rate like exponential demand, different power
demand patterns and random behaviour of imperfect quality items which follow various
probability distributions. Such types of problems are not only quite natural but also have
a great deal of practical importance.

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APPENDIX
The expression under the integral of (12) on simplification yields
4
2
32
A
T
QAQAA
T
TP
+
++−
=
where A2, A3, A4 are given by A2 = u − up + vp − C2 − d +( C1p/b) −(C1/b), A3 = −(C1p/2x)
and A4 = (C1a/b)
The expression under the integral sign of equation (14) on simplification gives
1
2
2
2
32132
BT
B)QAQAA(Q)BAT(A
Q)G(p,
++−−+
=
Where B1 and B2 are given by
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
=−++= x
bQ
2
x
bQ
1 pe-1B,xpexpbQaB
Where T is given by equation (6).
Lemma 1: 0
a
B1
>
Proof: Since a > 0 it is sufficient to show that B1 > 0 that is
0xpexpbQa x
bQ
>−++
⎟
⎠
⎞
⎜
⎝
⎛
If possible let B1 < 0 that is
0xpexpbQa x
bQ
<−++
⎟
⎠
⎞
⎜
⎝
⎛
Which on simplification yields
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−+>
⎟
⎠
⎞
⎜
⎝
⎛
1ex/)(p x
bQ
bQa (16)

Since a > 0, x > 0, and 01e x
bQ
>−
⎟
⎠
⎞
⎜
⎝
⎛
, therefore, it follows that
.01ex/ x
bQ
>
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
a Thus equation (16) contradicts the assumption of no shortages
found by the equation (8). Hence B1 > 0 and the proof follows.
Lemma 2: B1≠a
Proof: If B1 = a then B1 − a = 0, that is,
0xpexpbQ x
bQ
=−+
⎟
⎠
⎞
⎜
⎝
⎛
Which on simplification yields,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−=
⎟
⎠
⎞
⎜
⎝
⎛
1ex/p x
bQ
bQ
Since a > 0, x > 0 and 01e x
bQ
>−
⎟
⎠
⎞
⎜
⎝
⎛
so equation (19) is a contradiction to the
equation (8)
and which implies that B1≠ a.
Theorem 1:
T
TP
is a continuous function of Q and p (0 ≤ p ≤1).
Proof:
T
TP
is a rational function of p and Q. So it will be discontinues if the denominator
T vanishes or T is undefined. Now T vanishes if
0ln
b
1 1
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
a
B
But by Lemma 1 it is found that 0ln
b
1 1
>⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
a
B
, that is, T ≠ 0. T is undefined if
11
=
a
B
but by Lemma 2 B1≠ a and hence
T
TP
is a continuous function for any value of
Q and p, (0 ≤ p ≤ 1).
Theorem 2: G(p,Q) is a continuous function of Q and p (0 ≤ p ≤ 1).
Proof: G(p,Q) is a rational function of p and Q. So it will be discontinues if the
denominator T2
B1 vanishes or undefined. But the Lemma 1 indicates that B1 > 0 and by
Theorem 1 it is found that T ≠ 0. Thus T2
B1 ≠ 0 and is defined. Hence the theorem
follows.

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