Research Journal of Applied Sciences, Engineering and Technology 4(22): 4755-4760, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: April 03, 2012
Accepted: April 25, 2012
Published: November 15, 2012
Effects of Breakdown, Backlog and Rework on Replenishment Lot Size
1
Tsu-Ming Yeh, 2Chia-Kuan Ting, 3Fan-Yun Pai and 4Jyh-Chau Yang
Department of Industrial Engineering and Technology Management, Da Yeh University,
Changhua 515, Taiwan
2
Department of Industrial Engineering and Management,
3
Department of Business Administration, National Changhua University of Education,
Changhua, Taiwan
4
Department of Business Administration, Chaoyang University of Technology,
Taichung 413, Taiwan
1
Abstract: This study studies the effects of machine breakdown, backlog and rework on the replenishment lot
size. In real-life manufacturing systems, random defective rate and breakdown of equipment are inevitable.
When backlogging is permitted during a production run, a random machine failure may take place either in
backlog filling stage or in inventory piling time; this study focuses on the former situation and considers all
defective items produced are repairable through a rework process. The objective of this study is to determine
the optimal run time that minimizes the long-run average production-inventory costs. The result can be directly
applied to the practical production planning and control field to assist practitioner in production management
cost reduction.
Keywords: Backlog, equipment failure, production, repair, replenishment lot size
INTRODUCTION
In manufacturing sector, the Economic Production
Quantity (EPQ) model is commonly adopted to assist
management in determining optimal replenishment lotsize that minimizes the long-run average production costs
(Taft, 1918). Regardless of its simplicity, the EPQ model
remains the basis for analyzing more complicated and
complex systems (Schneider, 1979; Yum and McDowell,
1987; Silver et al., 1998; Chiu, 2003; Chelbi and Rezg,
2006; Kreng and Tan, 2010; Wong, 2010; Hsieh et al.,
2010; Chen and Lu, 2011; Kreng and Tan, 2011;
Chiu et al., 2011a, b; Chen, 2011).
The classic economic production quantity model
assumes all items produced are of perfect quality.
However, in real-life manufacturing systems, due to
different unpredictable factors, the production of
nonconforming items seems inevitable. Studies have since
been conducted to address the imperfect quality issues
and its consequence quality control matters (Lee and
Rosenblatt, 1987; Bielecki and Kumar, 1988; Cheng,
1991; Makis and Fung, 1998; Boone et al., 2000; Giri and
Dohi, 2005; Banerjee and Sharma, 2010; Chiu et al.,
2010a, b; El Saadany and Jaber, 2010). Owing to the
unexpected excess demands, the stock-out situations may
arise occasionally. Shortages sometimes are permitted and
backordered; it will normally be satisfied in the next
replenishment order. Hence, the production-inventory
costs can be substantially decreased (Chiu and Chiu,
2006; Chiu et al., 2009).
Random breakdown of manufacturing equipment is
another troublesome reliability factor in the field. To be
able to effectively control and manage such a disruption
of service becomes a critical task to most production
planners. It is not surprising that this critical issue has
received extensive attentions from researchers in past
decades (Groenevelt et al., 1992; Kuhn, 1997; Chern and
Yang, 1999; Teunter and Flapper, 2003; Chiu, 2010; Chiu
et al., 2010c, d). This study incorporates issues of
equipment failure, rework and backlog into a
manufacturing system and studies their effects on the
optimal replenishment run time.
METHODOLOGY
Model description: The proposed manufacturing system
has an annual production rate P and the demand rate 8 per
year, where P is greater than 8. All items produced are
screened and the unit inspection cost is included in unit
manufacturing cost C. During the production process,
there is x portion of random defective items generated and
d is the production rate of defective items, where, d = Px.
Assuming (P-d-8)>0. When demand exceeds supply
occasionally, shortages are permitted and backordered,
they will be satisfied in next replenishment production
cycle. All nonconforming products are assumed to be
Corresponding Author: J.C. Yang, Department of Business Administration, Chaoyang University of Technology, Taichung 413,
Taiwan
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Res. J. Appl. Sci. Eng. Technol., 4(22): 4755-4760, 2012
repairable through a rework process. Furthermore,
according to Mean Time between Failures (MTBF)
analysis, a stochastic equipment failure may take place in
backlog filling time. When such a machine breakdown
occurs an abort/resume control policy is used. Under such
a policy, the production equipment is immediately under
repair, the repair time is assumed to be constant and the
interrupted lot will be resumed right after the machine is
fixed and put back to use. It is also assumed that during
the setup time, prior to the production uptime, the
working status of machine is fully checked and
confirmed. Hence, the chance of machine failure in a very
short period of time when production process begins is
relatively small.
It is also assumed that due to tight preventive
maintenance schedule, the probability of more than one
machine failure occurrences in a production cycle is very
small. However, if it does take place, safety stock will be
used to satisfy the demand during equipment repairing
time. Therefore, multiple equipment failures are assumed
to have insignificant effect on the proposed model.
The proposed manufacturing system has the
following cost parameters: the setup cost K, unit
production cost C, unit holding cost h, unit repair cost for
each defective item CR, unit holding cost per reworked
item h1, unit shortage backordering cost b and the cost for
repairing machine M. Other notation include:
T1 =
T
t
=
=
tr
=
Production run time to be determined by the
proposed study
Production cycle length
Production time before a random breakdown
occurs
Time required for repairing and restoring the
machine
tr’ =
Time required for producing sufficient stocks to
satisfy the demand during machine
t1 = Time for piling up stocks during the production
uptime in each cycle
t2 = Time needed to rework the defective items
t3 = Time required for depleting all available perfect
quality on-hand items
t4 = Shortage permitted time
t5 = Time required for filling the backorder quantity
B
Q = Production replenishment lot size for each cycle
B = The maximum backorder level allowed for each
cycle, to be determined by this study
H1 = Level of on-hand inventory when machine
breakdown occurs
H2 = Level of on-hand inventory when machine is
repaired and restored
H3 = Level of on-hand inventory when the remaining
regular production uptime ends
H4 = The maximum level of perfect quality inventory
when rework finishes
I (t) = On-hand inventory of perfect quality items in
time t
= Total production-inventory costs per
TC (T1, B)
cycle
= Total production-inventory costs per
TCU (T1, B)
unit time (e.g. annual)
E[TCU(T1, B)] = The expected total productioninventory costs per unit time
Modelling and system analysis: Applying the modeling
and analysis as that was given in Chiu et al. (2009), one
can obtain the total production-inventory cost per cycle
TC (T1, B) as follows:
TC(T1 , B)  C( PT1 )  K  M  C R ( PT1 ) x
H3  H4
H4
 H3

 dT1 
 h
t1  
t2 
t 3   h1 



 t 2 
2
2
 2

 2 

  B  H1 
 H2 
 H  H2 
B
t  1
tr  
t 5  t r  t1   t 4 
 b


2
2
2
2


(1)
  dt 

dt  dT1 
 t    dt  t r 
 h
t 5  t r  t  t1 

2
 2

This study employs the expected values of defective rate x and the renewal reward theorem in the productioninventory cost analysis to cope with the randomness of x. Further, because of the assumption of occurrence of a
stochastic equipment failure (with mean equals to $ per unit time), this study uses integration of TC (T1, B) to cope with
such a unexpected breakdown. Thus, the long-run expected cost per unit time E [TCU (T1, B)] can be computed as
follows:
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Res. J. Appl. Sci. Eng. Technol., 4(22): 4755-4760, 2012


E TCU T1 , B 
E
 TCT , B  f (t )dt   E TCT , B  (e  )dt 
T P /  (1  e  )
E  T  f (t )dt 
t5
t5
1
0
t
1
0
(2)
 t5
t5
1
0
With further derivations one can obtain E [TCU (T1, B)] as follows:


E TCU  T1 , B 
K  M 

B 2   T 1 P E  x  
  g 2 

2 P1
 

 hB  hg 

Bg
T1 P (1  e
  t5

  C  CR E  x  
T1 P
2
h1  h 

g hE x   b1  E  x 
T1


b  h
T1 P
2
1 x


E

1 x   / P 
hT1  P  hgB 
1

 E
 1 

2  
T1 P

1 x   / P 
(3)

1 x
x




  b. E 
  h  E1 x   / P 
1


x
P

/
) 




Let,
x
1 x
1






2
E1  E  x ; E 2   E  x  ; E 3  E 
; E4  E 
; E5  E 
 1  x   / P 
 1  x   / P 
 1  x   / P 
(4)
then Eq. (3) becomes:
 b  h  2
B2 
g
hE1  b1  E1  
g  

T
P
T1
21
 

hT1  P  T1 P
hgB
Bg
E3 
 1 
h1  hE2  T P E2  hB  hg 
bE3  hE5 
2  
 2 P1
T1 P 1  e  t5
1


E TCU  T1, B 
K  M 
T1 P

  C  CR 1    E1 


(5)

Proof of convexity and derivation of the optimal run time: In order to find the optimal replenishment run time, one
needs to prove convexity of E [TCU (T1, B)]. The Hessian matrix equations are employed in this section for the proof
(Rardin, 1998):
 2E


B  2
 E


T

1
TCU T , B
1

T
2
1

TCU T1 , B
 T1 B




 2 E TCU T1 , B 
 T1 B
  T1 
    0
 2 E TCU T1 , B   B 

 B2

(6)
A theorem is proposed in this study. Let T0 denote the following:


1 x

 0    2 K  M    b  h g 2 E 
  2 gP b  h E  x 
 1  x   / P  

(7)
Theorem 1: E [TCU (T1, B)] is convex if T0>2 gPb
Applying the Hessian matrix equations, one obtains:
T
1




  2 E TCU T B
1


T12
B  2
  E TCU T1 B

 T1B







 2 E TCU T1 B 

T1B

 2 E TCU T1 B 
B 2


 T1 
 
B
(8)
1 x
  


   2 K  M    b  h  g 2  E
 2 gP b  h E  x   b
T1 P  
1  x   / P 

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Res. J. Appl. Sci. Eng. Technol., 4(22): 4755-4760, 2012
Because the first part of the Right-Hand-Size (RHS) in Eq. (8) is greater than zero, so if the second part of RHS in
Eq. (8) is greater than zero then the Hessian matrix equations for E [TCU (T1, B)] is greater than 0. That is:
 

1 x


If    2 K  M    b  h  g 2  E 
   2 gP b  h E  x   b  0


/
1

x


P




(9)
then the minimum of E[TCU(T1, B)]exists.
or

 
1 x


If    2 K  M    b  h g 2  E 
   2 gP b  h E x   2 gPb .


1 x   / P 

 
(10)
then the minimum of E[TCU(T1, B)]exists.
From Eq. (7) and (10), proof of Theorem 1 is completed.
RESULTS AND DISCUSSION
In order to minimize the expected overall costs E [TCU (T1, B)], Eq. (10) must be satisfied. Now, to search for the
optimal uptime T1 and the optimal backorder level B, one can differentiate E [TCU (T1, B)] with respect to T1 and with
respect to B separately, then solve linear systems of Eq. (11) and (12) by setting these partial derivatives equal to zero:

 
 E TCU T1, B
 T1
 
K  M
T12 P

 b  h 2 2
P
h  h  E2 
 g   B 2 E3
2 P1 1
2T12


g
hgB
Bg
 2 hE1  b E1  1  2 E 4  2
T1 
T1 P
T1 P1  e  t



 E TCU T1 , B
B

bE  hE5   2h  P   
5
 3
B
hg
g
 b  h E 3  h 
E 
bE 3  hE5 
T1 P
T1 P 4 T1 P1  e  t5 
(11)
(12)
With extra derivations, the resulting optimal production lot size Q* and backorder level B* can be obtained as
follows:
Q* 


h 2 g 2 2 32
g 2 212
2hg 2 2 31
2 Pg
 2 K  M    g 2 2  2 


hE x  b1  E x
2 
 t


t
2



 2 1  e 5   2 1  e 5  2
2
h




2
h  h  E  x    h 1   

P1 1
P  2


gbE 3  hE 5 
hgE 4
 h  1 
B*  

   Q*
 b  h   E3 
b  h E 3  b  h E 3 1  e  t5 
(13)
(14)
where, T1, T2, T3 denote the following:
1 x
x




 h E

1
1

x


P
x
 / P 
/





1 x
x




 2  (b  h) E 
;3  E 
 1  x   / P 
 1  x   / P 
1  b  E 
(15)
Suppose the equipment failure factor does not exist, then machine repair cost and time are both zero, i.e., M = 0 and
g = 0; Eq. (13) and (14) become:
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Res. J. Appl. Sci. Eng. Technol., 4(22): 4755-4760, 2012
 Q* 
2 K

 
h 1   
h  h E x2 
p  p1 1





1
 h 
 Q*
B*  

 b  h  
1 x

 E

  1  x   / P  
h2


 1 x 

 b  h  E 
 1 x   

P
size Q* = 8096, the backorder level B* = 3310 and E
[TCU (T1*, B*)] = $10474.
Note that if the present research result is not
available, then one can probably use the most related
model from Chiu (2003) and obtains Q = 5,601 (or T1 =
0.4871) and B = 2320. Then by plugging T1 and B in Eq.
(5) one obtains E [TCU (Q, B)] = $10577. It is noted that
this costs $103 more than our optimal cost or 9.88% more
on total other related cost (i.e., E [TCU (Q, B)] excludes
$9430 (total manufacturing and rework costs)).
(16)
(17)
CONCLUSION
Further, if it is a perfect manufacturing system, i.e.,
the defective rate x = 0, then Eq. (16) and (17) become the
same equations as were given by classic EPQ model with
shortages permitted and backordered (Hillier and
Lieberman, 2001):
Q* 
2 K
b h


b

h 1  

P
 
 h 
B*  
1    Q *
P 
  b  h 
(18)
(19)
REFERENCES
NUMERICAL EXAMPLE WITH FURTHER
DISCUSSION
Suppose a manufacturing system has annual
production rate 11500 for a specific product and this item
has demand rate 4600 units per year. The production
equipment is subject to a stochastic failure which follows
a Poisson distribution with mean $ = 2 times per year.
Further based on the analysis of MTBF data, a machine
breakdown may randomly take place during backlog
filling time. Abort/Resume (AR) policy is used when such
a equipment failure occurs. Also, during the regular
production, x portion of defective items produced, where
x follows a uniform distribution over the interval [0, 0.2].
Other related parameters are given below.
K
C
h
b
CR
h1
g
In real world manufacturing environments, stochastic
machine failure and random defective items produced are
inevitable. This research specifically investigates the
effects of these factors on the optimal replenishment run
time. One notes that without an in-depth study of such a
realistic system, one will not be able to obtain the optimal
production run time. For future study, an interesting
direction may be to look into the effect of stochastic
demand on the model.
=
=
=
=
=
=
=
$450 for each production run
$2 per item
$0.6 per item per unit time
$0.2 per item backordered per unit time
$0.5 for each item reworked
$0.8 per item per unit time
0.018 years, time needed to repair and restore the
machine
M = $500 repair cost for each breakdown
To test for convexity of the cost function, from
computation of Eq. (10) and theorem 1, one obtains T0 =
3837 and 2 gPb = 83. Because T0>2 gPb, therefore E
[TCU (T1, B)] is a convex function. Then, by applying Eq.
(13), (14) and (5), one obtains the optimal production lot-
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Boone, T., R. Ganeshan, Y. Guo and J.K. Ord, 2000. The
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