Optimal Economic Production Quantity and Inspection Plan that
Considers Inspection Time and Allows for Defective Rework, Minimal
Repair, and Inspection Errors
Ya-Hui Lin1, Wen-Ying Wang2, Cheng-Yi Lin3 and Yan-Chun Chen4
Department of Industrial Engineering and Management,
Asia-Pacific Institute of Creativity, Taiwan1
Department of Business Administration, Tungnan University, Taiwan2
Department of Industrial Management, Tungnan University, Taiwan3,4
[email protected]
[email protected]
[email protected]
[email protected]
Corresponding Author: [email protected]
Abstract
The paper proposes the integrated production, inspection, preventive maintenance,
minimal repair, and inventory problem, and find the optimal inspection interval, inspection
frequency, and economic production quantity yielding the maximum unit expected profit in
an imperfect production process where inspection time, rework, minimal repair and
inspection errors exist. In the deterioration of production systems, we investigate the
effectiveness of imperfect preventive maintenance, and use numerical analysis to explore the
effect of inspection time, rework, minimal repair and inspection errors on profit.
Keywords: Imperfect rework, preventive maintenance, minimal repair, inspection errors,
inspection time
International Conference on Innovation and Management, Palau, January 27-30, 2016.
Ya-Hui Lin, Wen-Ying Wang, Cheng-Yi Lin and Yan-Chun Chen
1. Introduction
Conventional economic production quantity (EPQ) models assume that all production
system outputs comprise conforming items. Many studies that use the EPQ model adjust the
assumptions and restrictions used. For example, Tsou et al. (2012) propose the EPQ model
for items with continuous quality characteristic and rework. They develop the optimal lot size
that minimizes total cost. Giri and Sharma (2014) propose the lot sizing and unequal-sized
shipment policy for an integrated production-inventory system. They determine the optimal
production and shipment policy.
Preventive maintenance (PM) can improve the reliability of a system. In many PM
modes, it is usually presumed that, subsequent to the PM, a system will be as good as new
every time. However, the reality is that the failure pattern will change accordingly. Ben-Daya
(2002) propose an integration mode that combined EPQ and varying degrees of PM, and
considered the best inspection cycle, inspection times and production quantity of
distributions. Darwish and Ben-Daya (2007) explored the influence of inspection errors and
PM effects on a production inventory system. Therefore, Wang et al. (2009) brought the
thesis closer to actual manufacturing conditions by incorporating inspection time, advocating
the optimal production quantity and an inspection strategy that considered inspection time
and allowed for minimal repairs. Chen (2013) considers the integrated problem with
production, PM, inspection, and inventory for an imperfect production process. The paper
determined the optimal inspection interval, inspection frequency, and production quantity.
In an actual production system, a non-conforming part can be turned into a
non-defective part through reworking. Chiu et al. (2007) advocated an optimal production
quantity strategy that allows for defective rework and a proportion of defective resulting in
rejects during the process of reworking. This study thus proposes, by combining the concepts
of Chui et al. (2007) and Chen (2013), an expected unit profit maximization strategy that
considers inspection time and allows for defective rework, and minimal repairs and
inspection errors integrated mode.
2. Mathematics Mode
To facilitate comparison, this study used the same symbols as Chen (2013):
D
: demand rate in units per unit of time
P
Pr
: production rate in units per unit of time, where P  D
: production rate in units per unit of time, with the non-conforming items
reworked
: setup costs per production cycle
: storage cost per product, per unit time
: cost of each inspection
: unit cost of non-conforming item reworking
S
Ch
CI
Cr
International Conference on Innovation and Management, Palau, January 27-30, 2016.
Optimal Economic Production Quantity and Inspection Plan that Considers Inspection Time and Allows for
Defective Rework, Minimal Repair, and Inspection Errors
Cd
C mr
Pu
: cost of production of each scrapped, non-conforming item
k
: minimal repair cost per unit
: selling price per unit
: probability of the process in the type II out-of-control state when it is in the
out-of-control state
: non-conforming rates when the process is in the Type I out-of-control state
: non-conforming rates when the process is in the Type II out-of-control state
: the frequency of inspection during each production cycle
hj
: arrival interval of the jth inspection
tj
: jth PM time point, where t j  i 1 hi  ( j  1)s

dI
d II
j
: the conditional probability that the process shifts to the out-of-control state
during the time interval ( t j 1 , t j ), given that the process is in the in-control state
at time t j 1 ; p j  F (b j )  F (a j 1 ) F (a j 1 )

: probability of the type I inspection error, that is the probability of judging the
system to be out of control while it is in control

: probability of the type II inspection error, that is the probability of judging the
system to be in control while it is out of control
When considering a system for manufacturing a product, there are two production
modes in the manufacturing process to be noted, namely control mode and non-control mode.
Supposing the control mode is on when production starts, the manufacturing process will
pj
switch to the non-control mode. Moreover, the current mode of the manufacturing process
can be learned through inspection. Preventative maintenance can be executed if it is in
control mode and will bring down the failure rate. Moreover, the age of a system under
maintenance is relevant to the degree of maintenance. There can be two circumstances if the
manufacturing process is in non-control mode. The first circumstance is a light deviation of
the manufacturing process, which can be returned to control-mode through minimal repair. In
other words, the manufacturing process is switched from a non-control mode to a
control-mode and the failure rate remains the same.
The second circumstance is a significant deviation, which cannot be returned to a
control-mode with minimal repair. The only solution is to stop production and repair and
switch in order to restore the mode to its original condition. Supposing a proportion of
defectives are produced during the non-control mode, the proportion of the defectives can be
reworked into conformations, but a proportion cannot and will become defective.
Consequently, the manufacturing process switch conforms to the generalized distribution at
an increasing hazard rate. The inspection happens at time t j ， j  1,2,...., k , the inspection
time is a fixed value s , so the preventative maintenance happens at time
t j  s , j  1,2,...., k  1 , and the time of the preventative maintenance is negligible. The end of
International Conference on Innovation and Management, Palau, January 27-30, 2016.
Ya-Hui Lin, Wen-Ying Wang, Cheng-Yi Lin and Yan-Chun Chen
a production cycle happens when the system is in the non-control mode under the second
circumstance or after k times of inspection; the accumulated hazard rate of every inspection
cycle is the same. The expected production time of a cycle is:
k 1
j 1
k 1
j 1
i 1
i 1
E (T )  {( h j  s)[(1  pi )(1   )  pi (1     )]}  hk  [(1  pi )(1   )  pi (1     )].
(1)
I(t)
Pr-D
-D
-D
P-D
t
h1
s
h2
Tr
s
t1
tk
T
CT
Figure 1. Inventory cycle.
When the manufacturing process is in non-control mode, defectives will be produced.
There are two non-control modes. Hypothesis N 1j is the quantity of the expected defective
number from the j th section of the first non-control mode. Thus the quantity of the expected
defective number from the j th section of the first non-control mode is:
E ( N 1j )  
b j s
a j 1
d I P(b j  s  t )
(1   ) f (t )[ F (t )]
dt.
[ F (a j 1 )]1
And the quantity of the expected defective number from the j
second non-control mode is:
E ( N 2j )

b j s
a j 1
d II P(b j  s  t )
f (t )[F (t )] 1
[ F (a j 1 )]
(2)
th
time interval of the
dt.
So the quantity of the expected defective number of a cycle is:
International Conference on Innovation and Management, Palau, January 27-30, 2016.
(3)
Optimal Economic Production Quantity and Inspection Plan that Considers Inspection Time and Allows for
Defective Rework, Minimal Repair, and Inspection Errors
k
E ( N )  [(1  p j )  p j (1   )][(1   ) E ( N 1j )  E ( N 2j )]
j 1
j 1
 [(1  pi )(1   )  pi (1     ).
(4)
So the rework time of the expected defective number of a cycle is:
E(Tr )  (1  d1 ) E( N ) Pr .
(5)
i 1
The anticipated inventory time is:
E (CT ) 
1
{P[ E (T )  (k  1) s ]  E ( N )  (1  d1 )(1  d 2 ) E ( N )}.
D
(6)
The production cost of every product is fixed and thus negligible. As for the expected
storage costs, hypothesis the calculation of the expected cost of ownership considered the unit
product’s cost of ownership * quantity in unit time, all unit products’ costs of ownership in
unit time are the same C h . Thus the expected storage cost is:
E ( HC )  Ch 
CT
0
I (t )dt  Ch E ( H ),
(7)
E (H ) is the anticipated inventory. The solution for E (H ) is as follows:
1 k
E ( H )  {[h j ( I j 1  I j  Ds )  s(2I j  Ds )]
2 j 1
j 1
 [(1  pi )(1   )  pi (1     )]
i 1
k 1
j 1
j 1
i 1
  [(1  p j )  p j (1   )][(1  pi )(1   )  pi (1     )]
 {[ 2 I j  2(1  d1 ) E ( N )  ( Pr  D) E (Tr )]E (T )  r
[ I j  (1  d1 ) E ( N )  ( Pr  D) E (Tr )]2
D
}
k 1
  [(1  p j )(1   )  p j (1     ) ] {2[I k  2(1  d1 ) E ( N )
j 1
 ( Pr  D) E (Tr )]E (Tr ) 
[ I k  (1  d1 ) E ( N )  ( Pr  D) E (Tr )] 2
}}.
D
(8)
j
I j is the inventory level at time t j  s , I j   ( P  D)hi  D  s  , j  1,2,  , k , and
i 1
I 0  0 , if I j  0 , then hypothesis it was 0. The parameter  is a degradation factor, which
impacts the effect of PM activities on the “used age” of the process. Let C mpm be the cost of
PM during the maximum state, C apm is the cost of PM actually implemented, and rk is the
imperfect coefficient at the kth PM, then
International Conference on Innovation and Management, Palau, January 27-30, 2016.
Ya-Hui Lin, Wen-Ying Wang, Cheng-Yi Lin and Yan-Chun Chen
rk   k 1
C apm
C mpm
(9)
.
Let bk be the actual age of the system before the kth PM and a k be the actual age of
the system after the kth PM. Therefore
ak  (1  rk )bk .
(10)
In time t j , the effective age of the system is:
b1  h1,
b j  a j 1  h j , j  2,3,...., k .
(11)
Since PM will result in changes in the age of a system. The expected cost of PM and the
repair of a cycle is:
k 1
j
E ( PM )  C apm [(1  pi )(1   )  pi (1     )]
j 1 i 1
k 1
j 1
j 1
i 1
 C mr  (1   )[(1  p j )  p j (1   )][(1  pi )(1   )  pi (1     )].
(12)
The inspection cost is:
k 1
j
E ( IC )  C I {1  [(1  pi )(1   )  pi (1     )]}.
(13)
j 1 i 1
As the rejection rate of a defective that cannot be reworked is d 1 , the rework cost of
defective is:
E( RW )  Cr  (1  d1 ) E( N ).
(14)
For a defective that cannot be reworked, the reject rate d 2 during the rework process is:
E ( DC )  Cd  [d1  (1  d1 )d 2 ]E ( N ).
(15)
Where r0 and r1 are constant and the restoration delay cost for the jth interval is:
E ( RC j )  
bj
a j 1
R(b j  t )
f (t )[ F (t )] 1
[ F (a j 1 )]
bj
f (t )[ F (t )] 1
a j 1
[ F (a j 1 )]
  [r0  r1 (b j  t )]
 (r0  r1b j )[1  (
F (b j )
F (a j 1 )
dt.
bj
f (t )[ F (t )] 1
a j 1
[ F (a j 1 )]
) ]  r1  t

dt.
dt.
Therefore, the expected the restoration delay cost is:
k
j 1
j 1
i 1
E ( RC )    [(1  p j )  p j (1   )][(1  pi )(1   )  pi (1     )]
International Conference on Innovation and Management, Palau, January 27-30, 2016.
(16)
Optimal Economic Production Quantity and Inspection Plan that Considers Inspection Time and Allows for
Defective Rework, Minimal Repair, and Inspection Errors
 {( r0  r1b j )[1  (
F (b j )
F (a j 1 )
) ]  r1 
bj
a j 1
t
f (t )[ F (t )] 1
[ F (a j 1 )]
(17)
dt}.
So the expected total cost of each cycle is:
E (TC )  S  E ( HC )  E ( PM )  E ( IC )  E ( RW )  E ( DC )  E ( RC ).
(18)
The expected total income of each cycle is:
E (TR)  Pu  {P[ E (T )  (k  1)s]  [d1  (1  d1 )d 2 ]E ( N )}.
(19)
The expected profit in unit time is:
E (TR)  E (TC )
EU ( ) 
.
E (CT )
(20)
3. Optimal Solution
The solution for the model is as follows:
(1) Give the fixed C apm value according to the different degrees of preventative
maintenance, and then find the optimum C apm value.
(2) When k  1 ，calculate the expected profit values for different unit time h1 , and find
the maximum expected profit EU1 ( ) .
(3) When k  2,3,...., k max ( k max is the maximum value of the inspection times) ，
EU 2 ( ), EU 3 ( ),...., EU kmax ( ) is found.
(4) EU ( )  Max{EU j ( ), j  1,2,...., k max } the optimal values h1* and k * are thus
gained.
4. Numerical Analysis
A numerical example was used to explore this model, and the parameters are as
follows:   5 ,   2.5 , D  500 , P  1000 , Pr  750 , C h  $0.5 , S  $150 , C d  $20 ,
C mpm  $30 , Cr  $5 , Pu  $10 , C I  $10 , r0  $10 , r1  0.5 ,   0.99 , d 2  0.1 ,
d I  0.2 , d II  0.4 .
C apm
C mpm
Table 1. Effect of PM level on anticipated profita.
C apm
C apm
C apm
 0 .0
 0.25
 0 .5
 0.75
C mpm
C mpm
C mpm
4639
4683
4711
EU ( )
a
k  3, h1  0.2500, s  0.05,  0.01,   0.01, d1  0.8,  0.5.
C apm
C mpm
4731
 1 .0
4743
Table 1 shows the effects of various PM levels. The results show that expected profit
increases with the PM level. Figure 2 indicates that the rejection rate of defectives that cannot
be reworked has a significant influence on expected unit profit, and that the lower the
International Conference on Innovation and Management, Palau, January 27-30, 2016.
Ya-Hui Lin, Wen-Ying Wang, Cheng-Yi Lin and Yan-Chun Chen
rejection rate, the higher the expected unit profit is. In other words, the higher the rate of
defective rework, the higher the expected unit profit is.
EU ( )
4760
d1  0.0
4740
d 1  0 .5
d1  1.0
4720
0.11
0.16
0.21
0.26
0.31
0.36
0.41
0.46
h1
Figure 2. Effect of different reworking scrapping rates on unit expected profit.
( k  3, s  0.05,   0.01,   0.01,  0.5. )
5. Conclusions
The major contribution of this study is that it integrates production, inspection, minimal
repairs, preventative maintenance and inventory, and proposes an expected unit profit
maximization strategy that considers inspection time and allows for defective rework,
minimal repairs and inspection errors. In today’s trend toward green technology and an
environmentally protected world, it is indeed a good strategy to recycle defective products in
that it reduces resource wastage and increases a corporation profits. On the other hand,
manufacturing system deterioration does exist in the manufacturing industry. Therefore, by
understanding the correlation between production, inspection, minimal repairs, preventative
maintenance and inventory, management can deliver more efficient job control and quality
assurance in order to increase a company’s competitiveness. This study explores the
respective influence of inspection time, defective rework, minimal repairs, and inspection
errors on expected unit profit. The results show that all four factors have a significant
influence on expected unit profit.
Acknowledgements
This research was supported by the National Science Council of the Republic of China
(NSC 100-2410-H-236 -001).
International Conference on Innovation and Management, Palau, January 27-30, 2016.
Optimal Economic Production Quantity and Inspection Plan that Considers Inspection Time and Allows for
Defective Rework, Minimal Repair, and Inspection Errors
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International Conference on Innovation and Management, Palau, January 27-30, 2016.