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Int. J. Production Economics 135 (2012) 345–352
Contents lists available at SciVerse ScienceDirect
Int. J. Production Economics
journal homepage: www.elsevier.com/locate/ijpe
Disaggregation and consolidation of imperfect quality shipments in an
extended EPQ model
Ali Yassine n, Bacel Maddah, Moueen Salameh
American University of Beirut, Faculty of Engineering and Architecture, Engineering Management Program, Beirut, Lebanon
a r t i c l e i n f o
abstract
Article history:
Received 19 February 2010
Accepted 9 August 2011
Available online 17 August 2011
We consider a standard economic production quantity (EPQ) model. Due to manufacturing variability, a
fraction P of the produced inventory will have imperfect quality, where P is a random variable with a
known distribution. We consider a 100% inspection policy and further assume that the inspection rate
is larger than that of production. Thus, all imperfect quality items will be detected by the end of the
production cycle. For such an augmented EPQ model, we first derive the new optimal production
quantity assuming that the imperfect quality items are salvaged once at the end of every production
cycle. Then, we extend this base model to allow for disaggregating the shipments of imperfect quality
items during a single production run. Finally, we consider aggregating (or consolidating) the shipments
of imperfect items over multiple production runs. Under both scenarios we derive closed-form
expressions for both the economic production quantity and the batching policy, and show that our
desegregation/consolidation schemes can lead to significant cost savings over the base model.
& 2011 Elsevier B.V. All rights reserved.
Keywords:
Economic production quantity (EPQ)
Lot sizing
Imperfect quality
Screening
Batching
Aggregation
Consolidation
1. Introduction
The economic production quantity (EPQ) model is the oldest
and most useful in production and inventory management. The
main assumptions of the EPQ model are: deterministic and constant production rate, deterministic and constant demand rate,
perfect production line (i.e., no production defects or machine
breakdowns), a fixed setup cost per production run, a fixed holding
cost per item per unit time, and no shortage or backordering is
allowed (Zipkin, 2000). The main objective of the model is to find
the economic production quantity that minimizes the total system
cost per unit time, which results from a tradeoff between production setup costs and inventory holding costs.
In spite of the above simplistic assumptions, the EPQ model has
been the basis for building more complex and realistic production
models. Numerous extensions have been proposed and analyzed
including imperfect production quality (Yano and Lee, 1995),
random machine breakdown (Makis and Fung, 1998), inspection
and maintenance policies (Salameh and Jaber, 1997), and linear
demand trend (Goswami and Chaudhuri, 1991), to name a few.
In this paper, we relax a common simplifying assumption in the
EPQ model that relates to production quality. In many manufacturing industries, production yields are less than perfect (for example,
due to manufacturing variability, machine age, etc.) and thus a
n
Corresponding author.
E-mail address: [email protected] (A. Yassine).
0925-5273/$ - see front matter & 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijpe.2011.08.010
fraction of the produced items may be of imperfect quality. We
also assume a 100% inspection during production to insure that all
imperfect quality items are identified as produced. The imperfect
quality items are stored until the end of a production run, where
they are then sold lump sum at a discounted price. Under such
conditions, we determine the new optimal production quantity.
Furthermore, we extend this base model to allow for disaggregation and consolidation of imperfect quality shipments during a
single production run and over multiple production runs, respectively. Under both scenarios we derive closed-form expressions for
both the economic production quantity and the batching policy
and show that our desegregation/consolidation schemes can lead
to significant cost reduction over the base model. For instance,
under specific production conditions (e.g., low production setup
cost and high shipment cost), consolidation could potentially result
in savings in excess of 15% compared to the base model. Alternatively, under opposite conditions, disaggregation has a modest
cost savings of about 3% over the base model.
The rest of the paper proceeds as follows. In Section 2, we review
major EPQ extensions in the literature and relate them to the
current proposed model. In Section 3, we state model assumptions
and derive the optimal production quantity that minimizes total
system cost per unit time. In Section 4, we derive the optimal
production quantity and optimal batch size for salvaging imperfect
quality items within a single production run and state the conditions for which this policy is optimal. Section 5 presents the case for
consolidating the sale of imperfect quality items across multiple
production runs and determines the conditions for which this would
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A. Yassine et al. / Int. J. Production Economics 135 (2012) 345–352
be the optimal policy. In Section 6, we present several numerical
examples along with sensitivity analyses. Finally, we present a
summary of the model and contributions along with our concluding
remarks in Section 7.
2. Literature review
A typical assumption in EPQ models is that produced items are
of perfect quality. In reality, and due to limitation of quality
control procedures, among other factors, items of imperfect
quality are often present (see, for example, Yano and Lee, 1995).
This fact has been recognized by many researchers who study
‘‘random yield’’ models considering uncertainty in the supply
process due to quality problems.
The literature on inventory models with random yield or imperfect quality is abundant (see, for example, Yano and Lee, 1995 for a
detailed review). Most relevant are works with inventory models
under random yield within an EOQ or EPQ settings. Examples of
studies on the EOQ model with random yield include Shih (1980),
Kalro and Gohil (1982), and Porteus (1986). More recently, Salameh
and Jaber (2000) develop an EOQ model assuming that identifying
imperfect quality items requires screening which takes place for
some time after an order is received. Imperfect quality items are then
assumed to be sold as a single lot for a reduced price at the end of the
screening process. Salameh and Jaber’s (2000) model has been
refined and extended by many authors (e.g., Goyal and CardenasBarron, 2002; Huang, 2004; Maddah and Jaber, 2008; Papachristos
and Konstantaras, 2006). Examples of studies on the EPQ model with
random yield include Chan et al. (2003), Hou (2007), and Rosenblatt
and Lee (1986) to name a few. Chan et al. (2003) consider an EPQ
model that integrates rework, lower pricing, and scrapping (at no
cost) of imperfect quality items. Hou (2007) considers an EPQ model
with imperfect production processes, in which the setup cost and
process quality are functions of capital expenditure. Rosenblatt and
Lee (1986) consider an EPQ model where the production process may
go out of control after an exponentially distributed time and starts
producing defective items. Unlike our proposed research, all these
papers ignore the cost structure of imperfect quality items.
Few recent papers incorporate a cost structure of imperfect
quality items in random yield models. For example, Maddah et al.
(2009) assume that imperfect items have their own cost structure
as well as their own demand and study the effect of these
additional considerations on the ordering policy in an EOQ setting.
Maddah and Jaber (2008) and Maddah et al. (2010), under two
different models of random supply, account for the holding and
shipping cost of imperfect items and explore the benefits of
consolidation of imperfect quality batches from different orders
within an EOQ setting also. In addition, Hayek and Salameh (2001)
investigate the possibility of reworking imperfect quality items
during production runs of an EPQ system while modeling holding
and rework costs for these items. Recently, Konstantaras et al.
(2007) consider similar reworking and holding costs within an EOQ
system where orders are inspected in a warehouse upon receipt
and then sent in batches to be sold at another location. However,
none of these papers explore the potential benefits for the EPQ
model from economies of scale in shipping the imperfect quality
items through disaggregating (into smaller batches within a single
production run) or consolidation (across multiple production runs)
of imperfect quality items, which is proposed in our paper.
3. EPQ with random yield: base model
In this paper, we consider a production facility that produces
items at a constant production rate a, unit production cost c, and
production setup cost K. The stock is depleted at a constant
demand rate b. In addition, to the conventional EPQ assumptions
detailed in Section 1, not all produced items are of perfect quality.
It is assumed that the production line has a fraction P of total
production of imperfect quality during any production cycle.
Items of imperfect quality are identified through a 100% screening
process. It is assumed that an item is inspected immediately after
the completion of its production. Furthermore, to ensure that all
demand for perfect items is met it is assumed that b r (1–P)a.1
Imperfect quality items are salvaged in one batch at the end of a
production cycle. This assumption will be relaxed in the next two
sections. Fig. 1 shows a schematic of the inventory level in our
proposed model and Table 1 provides the taxonomy. It is helpful
to split the overall production system into its two parts: the
perfect quality items and the imperfect quality items, as shown in
the lower two portions of Fig. 1, respectively.
It is worth noting that when considering perfect quality items
alone, our system resembles that of a standard EPQ model (see
Fig. 1b). Also we note that since P is a random variable, the actual
amount of imperfect quality items may differ from one production
cycle to the next (see Fig. 1c); however, its expected value is the
same in each cycle. It is to be noted that the inventory build-up in
both Figs. 1a and b is actually not linear because P is random,
which implies that the detection rates of perfect and imperfect
items are not uniform. However, we make the approximating
assumption that the build-up is linear, which is quite common in
similar settings. (e.g., for the classical EPQ system the actual
inventory build-up is seldom linear because it is unlikely to have
a demand with a perfectly uniform rate, and such a linearity
assumption is tacitly made).
For the proposed model, the maximum inventory level, z, as a
function of the production quantity, y, is
b
z ¼ 1
y:
ð1Þ
a
The maximum inventory of perfect items is
b
w ¼ ð1PÞ y:
a
ð2Þ
The cycle time, T, is
T¼
y
b
ð1PÞ:
ð3Þ
The inventory cost per ordering cycle, Ch(y), is the sum of the
corresponding costs of perfect and imperfect items; that is,
Ch(y)¼h(wT/2þPy2/(2a)), where w is as given by Eq. (2). Upon
simplification
1 1
1
P2 2
P þ
ð4Þ
y :
Ch ðyÞ ¼ h
b a 2
2b
The total relevant cost per ordering production cycle is
TC(y)¼KþCh(y). Noting that the process generating the profit is
renewal (with renewal points at the start of each production cycle),
then the expected total cost per unit time, can be calculated using
the renewal-reward theorem (Ross, 1996), as follows2:
h
i
2
2
K þ h b1 1a 12 m þ m 2þbs y2
E½TCðyÞ
¼
:
ð5Þ
ETCUðyÞ ¼
E½T
ðy=bÞð1mÞ
Additionally, we assume that P follows a known distribution with
known mean m and variance s2. Therefore, the expected total cost per
1
If P has a support (a, b), then this assumption holds if b r (1 b)a.
Many papers failed to use this theorem when calculating the expected total
profit per unit time, which resulted in an inaccurate expression for the optimal
order or production quantity (examples include: Salameh and Jaber, 2000; Chan
et al., 2003; Konstantaras et al., 2007; Eroglu and Ozdemir, 2007).
2
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A. Yassine et al. / Int. J. Production Economics 135 (2012) 345–352
347
Overall Inventory
Level
y
z
w
α
–β
α-β
t1=y/α
t2=(z-Py)/β
Time
T=(y/β)(1-P)
Perfect Quality
Inventory
w
–β
(1-P)α-β
Imperfect Quality
Inventory
Py
αP
Fig. 1. Production system under study: (a) overall inventory system, (b) accumulation of perfect quality items, and (c) accumulation of imperfect quality items.
Table 1
Taxonomy.
Model parameter
Description
a
Constant production rate
Constant demand rate
Random fraction of imperfect items with known mean, m, and variance, s2
Production quantity
Optimal production quantity for the base model
Production quantity when disaggregating shipments of imperfect quality items
Production quantity when aggregating shipments of imperfect quality items
Number of imperfect quality items shipments within a production cycle
Number of imperfect quality items shipments across multiple production cycles
Maximum inventory level
Maximum inventory level of perfect items
Production phase in a cycle
Consumption phase in a cycle
Cycle time
Shipping cycle
Production setup cost ($)
Shipment setup cost ($)
Holding cost per product per unit time
b
P
Y
ynb
yd
ya
nd
na
z
w
t1
t2
T
SC
K
Ks
h
unit time as a function of the production quantity (ETCU(y)) becomes:
ETCUðyÞ ¼
Kb
þ hb
ð1mÞy
1
b
1
a
2
1
m2 þ s
m þ
2
2b
y
:
1m
ð6Þ
Then, the optimal production quantity for the base model, ynb ,
which minimizes ETCU(y) is obtained from the first-order optimality condition, ð@ðETCUÞÞ=@y ¼ 0, as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi
u
K
2K b
u
i¼
yb ¼ t h
,
2
2
yh
h b1 1a 12 m þ m 2þbs
It isp
worth
noting that Eq. (7) reduces to a standard EPQ equation,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EPQ ¼ ð2K bÞ=ðhð1ðb=aÞÞÞ, when
m ¼ s2 ffi¼0 and reduces further to
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a standard EOQ equation, EOQ ¼ ð2K bÞ=h, when a tends to infinity.
Additionally, one could easily compare the EPQ to ynb in Eq. (7) and
note that ynb tends to be larger than the EPQ when the variability of P
is reasonably low.
ð7Þ
4. Disaggregating shipments of imperfect items with a single
production cycle
where y ¼ ð1ðb=aÞÞð12mÞ þ m2 þ s2 . The first-order condition is
necessary and sufficient for optimality since ETCU(y) is convex in
yððð@2 ETCUÞ=@y2 Þ ¼ ðð2K bÞ=hy3 Þ 40Þ.
In this section we relax the earlier assumption relating to the
shipment of imperfect quality items as a single batch at the end of
each production cycle. In certain circumstances (e.g., when the
inventory holding cost is large or space is scarce), it might be
n
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A. Yassine et al. / Int. J. Production Economics 135 (2012) 345–352
advantageous to disaggregate this single batch into multiple
smaller batches to be shipped multiple times during a production
run. Additionally, each shipment of imperfect quality items incurs
a fixed cost Ks. The inventory levels of this augmented model are
shown in Fig. 2. In this model, the manufacturer needs not only
decide on the economical production quantity, but also on how
many shipments of imperfect quality items to use during a single
production run, nd, in order to minimize the total expected cost
per unit time for the production-inventory system.
The upper part of Fig. 2 shows the accumulation of total
inventory along the timeline corresponding to the base model.
The middle part of the figure shows the total inventory level
when three shipments of imperfect quality items are made (i.e.,
nd ¼3) instead of a single batch at the end of a production cycle.
The lower part of the figure shows the corresponding imperfect
quality inventory.
Then, the total relevant costs per production cycle associated
with imperfect quality items (TCI) and perfect quality items (TCP),
are as follows:
renewal reward theorem, the total expected cost per unit time
ETCUd ðyd ,nd Þ ¼
hy2d
2
h
12m þ m2 þ s2
b
ðyd =bÞð1mÞ
Py2d
,
2and
ð8Þ
"
#
hy2d ð1PÞ2 ð1PÞ
yd ð1PÞ
TCPðyd Þ ¼ w
hþ K ¼
þK:
2b
2
b
a
ð9Þ
Taking expected values, adding Eqs. (8) and (9) and then
dividing by the expected cycle time from Eq. (3) yields, by the
ðK þ nd Ks Þ
bþ
ETCUd ðyd ,nd Þ ¼
ð1mÞyd
hyd
2
h
i
ð1mÞ2 þ s2 ð1amÞb þ amnbd
ð1mÞ
ð10Þ
:
ð11Þ
For a fixed number of shipments, nd, the first-order optimality
conditions (which are sufficient since ETCUd(.) is convex in yd)
imply that the optimal production quantity is
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðK þ nd Ks Þb
,
yd ¼
y1 h
n
ð12Þ
where
m b
:
nd a
The optimal value for nd that minimizes the expected total cost
is found by minimizing the following:
ETCUd ðynd ðnd Þ,nd Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffi
g1 ðnd Þ=ð1mÞ,
ð13Þ
where g1 ðnd Þ ¼ 2ðK þnd Ks Þbhy1 . Then, applying the first-order
optimality conditions to g1 ðnd Þ equation yields the optimal
number of imperfect quality items shipments within a production
Overall Inventory
Level
–β
t1=yd/α
:
Upon simplification
y1 ¼ ð1mÞ2 þ s2 1m
TCIðyd ,nd Þ ¼ nd Ks þh
i
1a m þ amnd þ K þ nd Ks
t2=(z-Pyd)/β
Perfect Quality
Inventory
w
–β
(1-P)α-β
Time
T=(yd/β)(1-P)
Imperfect Quality
Inventory
Pyd/nd
αP
yd/(αnd)
Fig. 2. Augmented model showing one production cycle with 3 imperfect shipments.
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A. Yassine et al. / Int. J. Production Economics 135 (2012) 345–352
cycle:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðb=aÞmðK=Ks Þ
n
nd ¼
:
ð1mÞ2 þ s2 ð1mÞðb=aÞ
349
Section 4 as follows:
hyc
b
b
Kb
ECPUðyc Þ ¼
1 m 2
:
þ m2 þ s2 þ
2ð1mÞ
a
a
yc ð1mÞ
ð14Þ
From the lower part of Fig. 3, we can calculate the holding cost
of imperfect quality items over a shipping cycle composed on nc
production cycles as follows:
8
2
nX
nX
<nX
c 1
c 2
c 1
yc
y2
b
yc
Pi þ
CIh ðyc ,nc Þ ¼ h
Pi þ c 1Pi yc Pi
ð1Pj Þ
:
2
a
b
a
b
i¼1
i¼1
j ¼ iþ1
The convexity of g1 ðnd Þ can easily be shown as ð@2 g1 ðnd ÞÞ=
ð@nd Þ2 40:
It is worth nothing that the number of shipments, nnd , is
independent of the holding cost, h. This is due to a cancelation
effect which can be described as follows. A high h yields a low
production quantity, ynd , in order to reduce the holding cost for
both perfect and imperfect quality items, which in turn decreases
nnd . On the other hand, a high h increases nnd in order to reduce the
holding cost of imperfect quality items. These two opposing
phenomena results in canceling the effect of h on nnd , and thus
we see nnd in Eq.(14) independent of h.
9
=
y2c
y2c
þ
Pj þ
Pnc :
;
2a
2a
j¼1
nX
c 1
In contrast to the situation in the previous section, it might be
advantageous to carry imperfect quality items over multiple production cycles. This is could be due to economies of scale related to
shipping cost. For example, when there may not be enough
imperfect quality items in a single production cycle, to fill a truck
load; therefore, these items are carried over to the next cycle. In
addition to how much to produce, the manufacturer needs to
decide when to ship imperfect quality items, in order to minimize
total expect cost. In this case nc is defined to be the number of
production cycles between two consecutive shipments of imperfect
quality items. The inventory levels of this model are shown in Fig. 3,
where we defined t1i and t2i as the production and consumption
phases within a production cycle i. We also define Pi as the fraction
of imperfect items in cycle i; note that Pi, i¼1, 2, y, are
independent and identically distributed random variables. The
upper part of the figure shows the accumulation of total inventory
along the timeline corresponding to the base model. The lower part
of the figure shows how the inventory for the imperfect quality
items accumulates over multiple production cycles.
Therefore, the problem is to determine the production quantity, yc, and the number of production cycles, nc, between two
consecutive shipments of imperfect quality items that minimize
the total expected cost. We start by deriving cost expressions for
perfect and imperfect quality items separately. The expected cost
for perfect quality items per unit time is computed similar to
þ
ðnc 1Þðnc 2Þy2c 2
ðnc 1Þy2c
y2
m þ c m:
ðm mÞ þ
2b
a
2a
Upon simplification
hy2c
b
nc 1
E½CIh ðyc ,nc Þ ¼
nc nc mð1mÞ þ m2s2
:
nc
2b
a
The expected shipping cycle (SC) duration is
"
#
nc
X
yc
nc yc
ð1pi Þ ¼
ð1mÞ:
EðSCÞ ¼ E
i¼1
b
b
t11= yc/α
P1yc
ð18Þ
ð19Þ
Therefore, the expected total cost is obtained by combining
Eqs. (15) and (20), which yields the following:
1
ðKs =nc þKÞb hyc
b
ð1mÞ2 ð12mÞ
þ
ETCUc ðyc ,nc Þ ¼
ð1mÞ
yc
2
a
–β
(1-P2)α-β
t12=(z-P1yc)/β
t21= yc/α
τ22=(z-P2yc)/β
Time
T2=(yc/β)(1-P2)
T1=(yc/β)(1-P1)
Imperfect Quality
Inventory
ð17Þ
Then, the expected holding cost of imperfect quality items per
unit time becomes
E½CIh ðyc ,nc Þ
hyc
b
nc 1
¼
nc mð1mÞ þ m2s2
E½CIhu ðyc ,nc Þ ¼
:
EðSCÞ
nc
2ð1mÞ
a
ð20Þ
Perfect Quality
Inventory
(1-P1)α-β
ð16Þ
In (16), the first term is the inventory from cycle i produced in
cycle i, the second term is the inventory from production cycle i
carried through production cycles iþ1,y, nc 1 of a shipping
cycle, the third term is the inventory from cycle i carried to
production cycle nc and the last term is the inventory of production cycle nc.
Taking the expected value of (14) gives the following:
1 1
ðn 1Þy2c 2
E½CIh ðyc ,nc Þ ¼ ðnc 1Þy2c
m c
ðm þ s2 Þ
b 2a
b
5. Consolidating shipments of imperfect items across
multiple production cycles
w
ð15Þ
αP2
αP1
Shipping cycle
Fig. 3. Augmented model showing how perfect and imperfect quality items accumulate over multiple production cycles.
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þ ðnc 1Þðmm2 Þ þ
2
1 s2 :
nc
ð21Þ
Taking the derivative of Eq. (21) to solve for optimal production quantity yields:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u u2 K þ Ks b
t
nc
n
,
yc ¼
y2 h
ð22Þ
where
b
2
y2 ¼ ð1mÞ2 ð12mÞ þ ðnc 1Þðmm2 Þ þ
1 s2 :
a
nc
The optimal value of nc that minimizes the expected total cost
is found by minimizing the following:
~ Uc ðnc Þ ¼ ETCUc ðnc ,yn ðnc ÞÞ ¼
ETC
c
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2g2 ðnc Þ=ð1mÞ,
ð23Þ
~ Uc ðnc Þ is
where g2 ðnc Þ ¼ ðK þ ðKs =nc ÞÞbhy2 . It can be shown that ETC
unimodal in nc assuming that nc is a continuous variable. The
proof is shown in appendix. The continuous value of nc, n~c , that
minimizes ETCUðnc Þ can be easily found by a simple singlevariable numerical search using the following cubic equation:
n3c þ
gnc þ p ¼ 0,
ð24Þ
where
h
g¼
p¼
i 2m2 þ 3m þ ba ð12mÞ þ s2 1 KKs 2s2
ðmm2 Þ
,
ð24aÞ
4 KKs s2
:
ðmm2 Þ
ð24bÞ
The optimal value of nc is the integer value closest to n~c ,
~ Uc ðnc Þ value. That is:
whichever results in a lower ETC
~ Uc ðnc ÞÞ,
nnc ¼ arg min ðETC
bn~c c, dn~c e
ð25Þ
where bxc is the largest integer rx and dxe is the smallest
integer Zx.
Finally, the optimal production quantity is found from Eq. (22)
as ync ¼ ync ðnnc Þ.
Similar to the disaggregation case in Section 3, it is also worth
nothing that nnc is independent of the holding cost, h, but due to
the opposite of the logic discussed in disaggregation. The cancelation effect in this case can be described as follows. A high h yields
a low order quantity, ync , which in turn results in a large nnc in
order to reduce the holding cost for both perfect and imperfect
quality items. On the other hand, a high h implies that we prefer
to ship fast, which means that nnc tends to be smaller. These two
opposing phenomena results in canceling the effect of h on nnc , and
thus we see Eq. (24) independent of h.
6. Numerical results
In this section, we develop numerical results which illustrate
the application of the proposed models. Consider a situation with
the following parameters: production rate, a ¼100,000 units/year,
demand rate, b ¼50,000 units/year, ordering cost, K¼$100/order,
shipping cost, Ks ¼$50/shipment, and holding cost, h¼$5/unit/year.
For the fraction of imperfect quality item, P, we consider two
scenarios where P is uniformly distributed on (0, b) with b¼0.04
and 0.15. With P U(a, b), then m ¼ (aþb)/2 and s2 ¼(ba)2/12.
We first start by solving the base model by calculating the optimal
base order quantity ðynb Þ and corresponding expected total cost, as
discussed in Section 3. Then, we solve both the disaggregation and
Table 2
Effect of changing K.
Key: bold entries: P U(0, 0.04); non bold entries: P U(0, 0.15)
Base model
Case #
K
Recommendation
ynb
Consolidation
ETCUbn
Disaggregation
nnc
ync
ETCUcn
nnd
ynd
ETCUdn
1
70
Aggregate
Aggregate
2235
2356
5479
5507
4
2
1750
1951
4823
5264
–
–
–
–
–
–
2
50
Aggregate
Aggregate
2040
2150
5002
5027
5
2
1466
1734
4178
4677
–
–
–
–
–
–
3
130
Aggregate
Aggregate
2737
2885
6712
6745
3
2
2376
2492
6298
6723
–
–
–
–
–
–
4
200
Aggregate
Base
3226
3400
7908
7949
2
–
3000
–
7652
–
–
–
–
–
–
–
5
400
Aggregate
Base
4328
4562
10,610
10,665
2
–
4123
–
10,517
–
–
–
–
–
–
–
6
600
Base
Base
5201
5482
12,752
12,817
–
–
–
–
–
–
–
–
–
–
–
–
7
800
Base
Base
5928
6264
14,583
14,657
–
–
–
–
–
–
–
–
–
–
–
–
8
3000
Base
Disaggregate
11,267
11,876
27,623
27,764
–
–
–
–
–
–
–
2
–
11,876
–
27,377
9
5000
Disaggregate
Disaggregate
14,488
15,282
35,544
35,726
–
–
–
–
–
–
2
3
14,646
15,898
35,534
35,020
10
10,000
Disaggregate
Disaggregate
20,452
21,558
50,142
50,399
–
–
–
–
–
–
2
4
20,610
22,461
50,041
49,095
11
15,000
Disaggregate
Disaggregate
25,028
26,381
61,360
61,674
–
–
–
–
–
–
3
5
25,287
27,528
61,136
59,891
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A. Yassine et al. / Int. J. Production Economics 135 (2012) 345–352
consolidation cases as discussed in Sections 4 and 5, respectively. The
costs across all three models are compared and the model with the
least cost is selected. (For consistency, the shipping cost, Ks, is added
to the setup cost, K, when solving the base model.) For the base case,
the base model yields an optimal order quantity ynb ¼ 2499 units and
a corresponding expected cost of $6003. The disaggregation model for
the base case, gives an optimal number of shipments of nnd ¼ 1,
yielding a similar expected cost as the base model. Next, we solved
for the consolidation model, which yielded an optimal number of
shipments of nnc ¼ 4, and an optimal order quantity ync of 2043, and a
corresponding optimal cost of $5507. In this case, it is recommended
that the consolidation model is adopted with a savings of about 8%
over the base model. The above analysis was repeated for various
values of K and Ks in order to assess their impact on the optimal
solution and draw useful insights.
Table 2 shows the impact of changing K on the optimal
solution. All other model parameters are held fixed at their base
values given above. In this table, we show the base model solution
(optimal order quantity, ynb and corresponding expected cost,
ETCUbn ), and the solution for either the consolidation or the
disaggregation models when either of these two models is
recommended. The recommendation column represents the optimal model in each scenario.
Carefully inspecting the results in Table 2 reveal the following
insights:
Consolidation occurs at low values of K (e.g., Cases 1–3) and
saves significantly on cost relative to the base case. That is, in
Cases 1 and 2, the cost savings are, respectively, 12% and
16.5%; significant indeed.
Disaggregation typically occurs at very large K values and
saves little cost over the base model, (e.g., Cases 8–11 with
P U(0, 0.04)). However, at large P values, this is not the case.
That is, in Cases 10 and 11 with P U(0.0.15), the savings are
respectively 2.6% and 2.9%, which may be considered significant in some cases.
Relative to the base model, the order quantity decreases in
consolidation (e.g., ync oynb in Cases 1–3) and increases in
351
disaggregation (e.g., ynd 4ynb in Cases 9–11). This is due to the
effect of holding cost as it increases in consolidation and
decreases in disaggregation.
Table 3 shows the impact of changing the shipping cost Ks on
the optimal solution. All other model parameters are held fixed at
their base values given above. Similar to Table 2, in this table, we
show the base model solution, the solution for the consolidation
and the disaggregation cases. Carefully inspecting the results in
Table 3 shows that:
Consolidation occurs at high Ks (e.g., Cases 7–9) and saves
significantly on cost relative to the base model. That is, in
Cases 8 and 9, the savings are 14.6% and 16.5%, respectively.
Disaggregation occurs at very low Ks values (e.g., Cases 5 and
6) and saves little cost over the base model. However, at large
P, this is not always the case (e.g., in Cases 5 and 6 with P U(0,
0.15), the savings are 2% and 2.6%, respectively).
Similar to Table 2, relative to the base model, the order quantity
decreases in consolidation and increases in disaggregation.
7. Summary and conclusion
In this paper we have relaxed an assumption in a standard EPQ
model relating to the possibility of producing imperfect quality
items (production lines with imperfect yields). A 100% screening
process (which is assumed to be faster than the demand rate) is
used to identify all imperfect quality items, which are then sold as
a single batch at the end of a production cycle. Similar to a
standard EPQ model, the objective of our model is to find the
economic production quantity that minimizes the total cost per
unit time, which results from a tradeoff between production setup
costs and inventory holding costs. We have derived a closed-form
expression for the new economic production quantity.
Then, we questioned whether there would be an economic
benefit from either disaggregating the shipment of imperfect
Table 3
Effect of changing Ks.
Key: bold entries: P U(0, 0.04), non bold entries: P U(0, 0.15)
Base model
#
Ks
Consolidation
Disaggregation
Recommendation
ynb
ETCbn
nnc
ync
ETCcn
nnd
ynd
ETCdn
1
30
Aggregate
Base
2326
2452
5702
5732
3
–
2058
–
5454
–
–
–
–
–
–
–
2
10
Aggregate
Base
2140
2255
5246
5273
2
–
2050
–
5228
–
–
–
–
–
–
–
3
5
Base
Base
2090
3204
5126
5151
–
–
–
–
–
–
–
–
–
–
–
–
4
3
Base
Disaggregate
2070
2182
5077
5102
–
–
–
–
–
–
–
2
–
2264
–
5063
5
1
Disaggregate
Disaggregate
2050
2161
5027
5052
–
–
–
–
–
–
2
3
2071
2248
5026
4953
6
0.5
Disaggregate
Disaggregate
2045
2156
5014
5040
–
–
–
–
–
–
2
4
2061
2246
5000
4909
7
70
Aggregate
Aggregate
2660
2804
6521
6555
4
2
2088
2326
5743
6275
–
–
–
–
–
–
8
87
Aggregate
Aggregate
2790
2941
6840
6875
5
2
2050
2398
5844
6469
–
–
–
–
–
–
9
100
Aggregate
Aggregate
2885
3041
7073
7110
5
2
2073
2452
5908
6614
–
–
–
–
–
–
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352
A. Yassine et al. / Int. J. Production Economics 135 (2012) 345–352
quality items, or aggregating them across multiple production
cycles. For both scenarios, we have derived a closed-form solution
for the economic production quantity and the optimal batching
policy, and show that our desegregation/consolidation schemes
can lead to significant cost savings over the base model. Under
specific production parameters (e.g., low production setup cost
and/or high shipment cost), consolidating the shipment of imperfect quality items resulted in 16.5% savings compared to the base
model. Alternatively, when the setup cost is high and/or shipment
cost is low, disaggregation had a modest cost savings of about 3%
over the base model.
Finally, future work may include applying similar aggregation/
disaggregation schemes to the EPQ system with other models of
random supply (e.g. Rosenblatt and Lee, 1986).
Appendix
The following supporting lemma is needed to prove the
~ Uc ðnc Þ:
desired result on the structure of ETC
Lemma 1. If a cubic polynomial of the form q(x)¼x3 þax þc, where
a a0 and ca 0, has three real roots, then not all three may be
positive or negative.
Proof. By contradiction, assume that q(x) has three nonnegative
roots, x1, x2, and x3, then q(x) can be written as:
qðxÞ ¼ ðxx1 Þðxx2 Þðxx3 Þ ¼ x3 ðx1 þ x2 þ x3 Þx2
þ ðx1 x2 þ x1 x3 þ x2 x3 Þx þ x1 x2 x3 :
Therefore, it must be true that x1 þx1 þx3 ¼0, which is not
possible if x1, x2, and x3 are all nonnegative. ’
~ Uc ðnc Þ is unimodal in nc for nc A ð0,1Þ:
Theorem 1. The function ETC
~ Uc ðnc Þ is equivalent to minimizProof. Note that minimizing ETC
ing g2(nc), where
Ks
b
2
g2 ðnc Þ ¼ K þ
1 s2 :
ð1mÞ2 ð12mÞ þðnc 1Þðmm2 Þ þ
nc
a
nc
First, note that limnc -0 g2 ðnc Þ ¼ 1 and limnc -1 g2 ðnc Þ ¼ 1. This
proves that g2(n2) admits at least one local minimum for
nc A ð0,1Þ: Due to Lemma 1 the equation ð@g2 ðnc ÞÞ=@nc ¼ 0
@g2 ðnc Þ
Ks
b
2
¼ 2 ð1mÞ2 ð12mÞ þðnc 1Þðmm2 Þ þ
1 s2
@nc
nc
a
nc
2
Ks
2
s
mm2 2 ¼ 0,
þ Kþ
nc
nc
has at most two positive roots, which solve the cubic Eq. (24).
If this equation has exactly one root, then the result follows. If it
has exactly two positive roots, nc,1 and nc,2 , with nc,1 rnc,2 , then,
due to limnc -0 þ g2 ðnc Þ ¼ 1 ¼ limnc -1 g2 ðnc Þ, not both of them can
be minimizing or maximizing points. Then, one of them will be a
minimizing point. Due to limnc -0 þ g2 ðnc Þ ¼ 1, the smallest one,
nc,1 , will be a minimizing point and the other, nc,2 (the larger), will
be a maximizing point. In this case, due to limnc -1 g2 ðnc Þ ¼ 1, the
function g2 ðnc Þ must have another minimizing point, say nc,3 ,
falling to the right of nc,2 . In this case, ð@g2 ðnc ÞÞ=@nc ¼ 0will have
three positive roots, which is a contradiction. ’
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