American Journal of Computational Mathematics, 2011, 1, 202-207
doi:10.4236/ajcm.2011.13023 Published Online September 2011 (http://www.SciRP.org/journal/ajcm)
Copyright © 2011 SciRes. AJCM
Computational Optimization of Manufacturing Batch
Size and Shipment for an Integrated EPQ
Model with Scrap
Yuan-Shyi Peter Chiu1, Hong-Dar Lin1, Ming-Hon Hwang2*, Nong Pan 3
1Department of Industrial Engineering & Management, Chaoyang University of Technology, Taichung, Chinese Taipei
2Department of Marketing and Logistics Management, Chaoyang University of Technology, Taichung, Chinese Taipei
3Department of Business Administration, Chaoyang University of Technology, Taichung, Chinese Taipei
E-mail: *hwangmh@cyut.edu.tw
Received March 12, 201 1; revised April 2, 2011; accepted April 28, 2011
Abstract
This paper employs mathematical modeling and algebraic approach to derive the optimal manufacturing
batch size and number of shipment for a vendor-buyer integrated economic production quantity (EPQ) model
with scrap. Unlike the conventional method by using differential calculus to determine replenishment lot size
and optimal number of shipments for such an integrated system, this paper proposes a straightforward alge-
braic approach to replace the use of calculus on the total cost function for solving the optimal production-
shipment policies. A simpler form for computing long-run average cost for such a vendor-buyer integrated
EPQ problem is also provided.
Keywords: Computational Optimization, Manufacturing Batch Size, Shipments, EPQ Model, Random Scrap
Rate, Algebraic Approach
1. Introduction
The economic production quantity (EPQ) mod el was first
introduced by Taft [1] to assist practitioners in produc-
tion and inventory control field to determine the economic
replenishment batch size that minimizes total produc-
tion-inventory costs. Classic economic production quan-
tity model assumes a continuous inv entory issuing policy
for satisfying customer’s demand. However, in real
world vendor- buyer system, multiple or periodic deliv-
eries of finished products are often adopted. Therefore,
“how many shipments should a manufacturing lot be
broken down to?” becomes another critical issue that
practitioners must address in order to minimize overall
production-inventory-delivery costs.
Studies related to various aspects of supply chain op-
timization have been extensively carried out (see for ex-
ample [2-9]) in past decades. Goyal [2] examined an
integrated single supplier-single customer problem. He
proposed a method that is typically applicable to those
inventory problems where a product is procured by a
single customer from a single supplier. Example was
provided to demonstrate his proposed model. Schwarz et
al. [3] considered the system fill-rate of a one-warehouse
N-identical retailer distribution system as a function of
warehouse and retailer safety stock. They used an ap-
proximation model from a prior study to maximize sys-
tem fill-rate subject to a constraint on system safety stock.
As results, properties of fill-rate policy lines are sug-
gested. They may be used to provide managerial insight
into system optimization and as the basis for heuristics.
Lu [4] studied a one-vendor multi-buyer integrated in-
ventory model with the objective of minimizing vendor’s
total annual cost subject to the maximum costs that the
buyers may be prepared to incur. Lu’s model required to
know buyer’s annual demand and previous order fre-
quency. As a result, an optimal solution for the one-
vendor one-buyer case was obtained and a heuristic ap-
proach for the one-vendor multi-buyer case was also
provided. Sarker and Khan [5] considered a production
system that procures raw materials from suppliers in a lot
and processes them into finished products which are then
delivered to outside buyers at fixed points in time. A
general cost model was formulated considering both raw
materials and finished products. Using this model, a sim-
ple procedure was developed to determine the optimal
Y.-S. P. CHIU ET AL.
203
ordering policy for raw materials as well as the manu-
facturing batch size, so that the overall costs for such a
supply chain system can be minimized. Chiu et al. [9]
incorporated a multi-delivery policy and quality assur-
ance into an imperfect economic production quantity
(EPQ) model with scrap and rework. They assumed that
the random defective items produced are partially re-
pairable and are reworked in each cycle when regular
production ends, and the finished items can only be de-
livered to customers if the whole lot is quality assured at
the end of rework. Fixed quantity multiple installments
of the finished batch are delivered to customers at a fixed
interval of time. The expected integrated cost function
per unit time was derived. A closed-form optimal batch
size solution to the problem was obtained.
Imperfect quality items produced in real world manu-
facturing environments is another inevitable and impor-
tant issue that practitioners in the production manage-
ment filed must deal with. In the past decades, many
studies have been carried out to address the issue of de-
fective items in the production lines (see for example
[10-14]). The nonconforming items sometimes can be
repaired through rework, hence overall production costs
can be significantly reduced [15-20]. Yu and Bricker [15]
presented an informative application of Markov Chain
analysis to a multistage manuf acturing problem. Jamal et
al. [16] studied the optimal manufacturing batch size
with rework process at a single-stage production system.
Cases of rework being completed within the same pro-
duction cycle, and rework being done after N cycles are
examined. They developed mathematical models for
each case and derived total system costs and optimal
batch sizes accordingly. Chiu et al. [19] proposed a nu-
merical method for expediting scrap-or-rework decision
making in EPQ model with failure in repair.
Algebraic approach for determining economic order
quantity (EOQ) model with backlogging was introduced
by Grubbström and Erdem [21]. They proposed algebraic
derivations to solve the optimal order quantity without
reference to the first-order or second-order differentia-
tions. Variou s aspects of supp ly chain optimization stud-
ies have employed the same or similar methodologies
[22,23]. This paper uses mathematical modeling to de-
rive the long-run average cost function for the proposed
vendor-buyer integrated EPQ model with scrap; then
employs such a straightforward algebraic derivation to
determine the optimal production-shipment policies for
the proposed EPQ model.
2. The Proposed Model and Mathematical
Modeling
The proposed economic production quantity model as-
sumes there is an x portion of defective items produced
randomly at a production rate d during regular produc-
tion time. All produced items are screened and inspection
cost per item is included in the unit production cost C.
All nonconforming items are assumed to be scrap and
will be discarded at the end of production. Under regular
supply (not allowing shortages), the constant production
rate P must be larger than the sum of demand rate λ and
production rate of scrap items d. That is: (
P
d λ) > 0.
The production rate of scrap items d can be expressed as
d = Px.
A multi-delivery policy is consid ered in this study and
it is also assumed that the finished items can only be de-
livered to customers if the whole lot is quality assured at
the end of production process. Fixed-quantity n install-
ments of finished batch are delivered to customers at a
fixed interval of time during the production downtime t2
(see Figure 1). Additional notation is listed in Nomen-
clature in Appendix.
TC(Q, n), the total production-inventory-delivery costs
per cycle consists of 1) setup cost; 2) variable production
costs; 3) variable scrap disposal costs; 4) fixed delivery
cost; 5) variable delivery costs; 6) variable holding costs
at the supplier side for all items produced (defective and
perfect quality items) in t1 and all items waiting to be
delivered in t2; and 7) holding cost for finished goods
stocked at customer’s end. Therefore, TC(Q, n) is




1
112
222
,
1
1
22
2
ST
TCQ n
K
CQ CxQnKCQx
Hdt n
ht Ht
n
hHtTH t
n
 













(1)
Figure 2 shows supplier’s inventory holding during
delivery time t2. The variable holding costs for finished
products kept by the supplier in delivery time t2 are
Figure 1. On-hand inventory of perfect quality items in the
proposed EPQ model with scrap and a multiple shipment
policy.
Copyright © 2011 SciRes. AJCM
204 Y.-S. P. CHIU ET AL.
1) When n = 1, total holding cost in delivery time = 0.
2) When n = 2, total holding costs in delivery time be-
come (see Figure 2)
22
2
1
22 2
t
H
hh






 Ht
(2)
3) When n = 3, total holding costs in delivery time are
22 2
2
21
33
tt
HH
hh



  




Ht
(3)
4) When n = 4, total holding costs in delivery time be-
come
222 2
2
444 4
ttt
HHH
hh
 


 





Ht
(4)
Therefore, the following general term for total holding
costs during t2 can be obtained (as shown in Equation (1)
above):
1
22
22
1
11(1)
22
n
i
nn n
hiHt hHt hHt
n
nn


 
 
 
 
 

2
1
(5)
Taking randomness of scrap rate into consideration
and employing the expected values of it, and with further
derivations, the long-run average costs per unit time for
the proposed EPQ model, E[TCU(Q, n)] can be derived
as follows (refer to a similar derivation procedure in [9]):
















1
2
,
,
111
1
1
22
21
11
1
2
S
T
ETCUQn
ETCQn
ET
CEx
KnK
CC
Ex QEx Ex
hQE x
hQ nhQ
nP
PEx
hQ n
Ex
nnP




 


 



 

 

 

(6)
3. Deriving Optimal Production-Shipment
Policies without Derivatives
This study employs algebraic approach to derive the op-
timal production-shipment policies, instead of using dif-
ferential calculus on E[TCU(Q, n)] with the need o f pr ov-
ing its optimality [21-23]. In Equation (6), both Q and n
are decision variables, by rearranging terms in Equation
(6) as the constants, Q–1, Q, nQ–1, and Qn–1, one has
 


1
12 3
1
45
,
ETCUQnQ Q
nQn Q
 


 


Figure 2. On-hand inventory of the finished items kept by
supplier during t2 in the proposed EPQ model.
where β1, β2, β3, β4, and β5 den ote the following :



111
S
T
CEx
CC
Ex Ex
 
(8)




22
1
22
21
hEx
hhh
P
PEx

 

(9)


31
K
Ex
 (10)


1
41
K
Ex
 (11)

52
1
22
Ex hh
P

 




(12)
With further rearrangements, Equation (7) becomes

123
45
12
2
11
,
ETCUQnQ Q
nQ nQ



 






(13)




 




 
12 323
4545
23 45
22
`
22
11
11
,
2
2
22
ETCUQn
QQ Q
n QnQnQ
QQ nQnQ
 

 





 


1
1






(14)




12
45
23 45
2
1
2
11
,
22
ETCUQn QQ
nQ nQ


 


 






3
(15)
1
(7)
Copyright © 2011 SciRes. AJCM
Y.-S. P. CHIU ET AL.
Copyright © 2011 SciRes. AJCM
205
** 123 4
,22ETCUQ n
It is noted that if the following square terms (Equa-
tions (17) and (18)) equal zero, then Equation (15) will
be minimized:
5
 


 (22)
4. Demonstrative Example

2
123
0QQ

 
(16)
Consider a product can be produced at an annual rate of
60,000 units and this item has experienced a flat demand
rate of 3,400 units per year. Assume that during produc-
tion process a random scrap rate which follows a uniform
distribution over the interval [0, 0.3]. In additions, the
following values of related variables are considered:


2
11
45
0nQ nQ




(17)
or
*3
2
Q
(18)
C = $100 per item,
and
CS = $20, disposal cost per scrap ite m,
*
5
4
*n
Q (19) h = $20 per item per year,
h2 = $80 per item kept at the customer’s end per unit
time,
Substituting Equations (9) and (10) in Equation (18),
the optimal replenishment lot size Q* can be obt ains: K = $20,000 per production run,
K1 = $4350 per shipment, a fixed cost,





*
2
2
2
1
K
QhhExhh Ex
PP

 1
(20) CT = $0.1 per item delivered.
Substituting Eq uations (11) , (12), and (20) in Equation
(19), the optimal number of shipments is










2
2
1
11
*
11
Kh hExEx
P
nh
KhExhhEx
PP

 
 
2
From Equations (21), one obtains the optimal number
of delivery n* = 3. By plugging n* back into Equation (7)
and resolving the algebraic solution for Q* one finds the
optimal production batch size Q* = 2652. Calculating
Equation (22) one obtains the long-run average cost E ×
[TCU(Q*, n*)] = $512,047. Figure 3 shows the convexity
of the long-run integrated cost function E[TCU(Q, n* =
3)].
It is noted that n* should practically be an integer nu-
mber, but Equation (21) gives a real number. In order to
obtain the optimal integer value of n, one should com-
pute the E[TCU(Q, n)] for both integers that are adjacent
to real number n* respectively (for instance, in this ex-
ample Equation (21) gives n* = 3.1733, so both n = 3 and
n = 4 must be plugged in E[TCU(Q, n)]), and select the
one with minimum co st as our optimal n*.
(21)
One notes that Equation (21) is identical to what was
obtained by using the conventional differential calculus
method on E[TCU(Q, n)] [24]. Further, from Equation (7)
the optimal cost function E[TCU(Q*, n*)] is
Figure 3. Convexity of the long-run integrated cost function E[TCU(Q,n* = 3)].
Y.-S. P. CHIU ET AL.
Copyright © 2011 SciRes. AJCM
206
5. Conclusions
This paper derives the optimal manufacturing batch size
and number of shipment for a vendor-buyer integrated
EPQ model with scrap using mathematical modeling and
algebraic approach. It is confirmed the research results
from the proposed algebraic derivations are identical to
what were derived by the use of conventional differential
calculus. In additions, this study also reveals a simpler
computation formula (i.e. Equation (22)) for the long-run
average cost function for such a vendor-buyer integrated
EPQ problem. This straightforward algebraic approach
enables practitioners or students who with little or no
knowledge of calculus to learn or handle with ease the
real-life EPQ model.
6. Acknowledgements
The authors greatly appreciate the National Science Coun-
cil of Taiwan for supporting this research under grant
number: NSC 99-2221-E-324-017.
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Appendix
Nomenclature:
C = unit manufacturing cost,
CS = the disposal cost per scrap item,
h = unit holdi n g cost ,
K = setup cost per production run,
K1 = fixed delivery cost per shipment,
CT = unit delivery cost CT,
Q = production lot size, a decision variable, to be de-
termined for each cycle,
T = production cycle length,
n = number of fixed quantity installments of the fin-
ished batch to be delivered to customers, a decision vari-
able, to be determined for each cycle,
d = producti on rate of scrap items,
t1 = the production uptime for the proposed EPQ
model,
t2 = time required for delivering all finished products,
H = maximum leve l of o n-hand inve ntor y in un its when
regular producti on p rocess en ds,
tn = a fixed interval of time between each installment
of finished products delivered during production down-
time t2,
I(t) = on-hand inventory of perfect quality items at
time t,
TC(Q, n) = total production-inventory-delivery costs
per cycle for the proposed model,
E[TCU(Q, n)] = the long-run average costs per unit
time for the proposed model.
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