Tamkang Journal of Science and Engineering, Vol. 12, No. 2, pp. 109-112 (2009)
109
The Modified Economic Manufacturing Quantity
Model for Product with Quality Loss Function
Chung-Ho Chen
Department of Management and Information Technology, Southern Taiwan University,
Yung-Kang, Taiwan 710, R.O.C.
Abstract
Traditional economic manufacturing quantity (EMQ) model addressed that the perfect
production for product. However, there possibly exists the defective product in the manufacturing
process. Hence, it is necessary to include the quality cost in the EMQ model. In this paper, we propose
a modified EMQ model with quality loss and inventory cost. By solving the modified EMQ model, we
can obtain the optimum production run length and process mean by considering the minimum
expected total cost. Taguchi’s symmetric quadratic quality loss function will be adopted for evaluating
the product quality.
Key Words: Economic Manufacturing Quantity, Process Mean, Taguchi’s Symmetric Quadratic
Quality Loss Function
1. Introduction
Traditional economic manufacturing quantity (EMQ)
model is assumed implicitly that items produced with
perfect quality. However, product quality is not always
perfect and is usually a function of the production process. Recently, there are some works addressing the economic models integrating production, maintenance and
quality, e. g., Lin [1], Rahim and Ohta [2,3], Hariga and
Al-Fawzan [4], Rahim and Al-Hajailan [5], Chen [6],
and Chen and Lai [7].
Chen and Chung [8] presented the quality selection
problem to the imperfect production system for obtaining the optimum production run length and target level.
Rahim and Tuffaha [9] further proposed the modified
Chen and Chung’s [8] model with quality loss and sampling inspection. Chen [10] considered the process mean
of in-control is not equal to the target value for modified
Rahim and Tuffaha’s [9] model.
In this paper, we further propose a modified EMQ
model with inventory cost and quality loss. The symmetric quadratic quality loss function is adopted for
*Corresponding author. E-mail: [email protected]
evaluating the product quality. The optimum production
run length and process mean can be determined under
the minimum expected total cost. The iterative-type heuristic solution procedure will be provided for illustration.
2. Theory Review – Economic Manufacturing
Quantity Model
The expected inventory cost of EMQ model includes
the set-up cost and the holding cost is as follows: (Silver
and Peterson [11])
(1)
where ETC is the expected inventory cost per unit time;
d is the demand rate (units/unit time); S is the set-up
cost for each production run ($/set up); p is the production rate (units/unit time); B is the inventory holding
cost ($/unit/unit time); T is the production run length in
each production cycle (time).
One sets the first derivative of ETC for T equal to
zero, and solves for economic manufacturing quantity, Q
(= pT). We have the optimum economic manufacturing
110
Chung-Ho Chen
quantity and production run length are QE* =
2dSp
B ( p - d)
2dS
, respectively.
p( p - d ) B
and TE* =
Hence, the expected total cost per unit time for modified EMQ model with inventory cost and quality loss
is as follows:
3. Modified EMQ Model
According to Rahim and Tuffaha [9], there are some
assumptions made in the modified EMQ model:
(1) The quality characteristic is normally distributed
with unknown process mean m and known process
variance s2.
(2) When the production cycle starts, the process is in
control state. Once the shift has occurred, the process will remain in an out-of-control state until it is
discovered by inspection and followed by some restoration work. Otherwise, the out-of-control state
will continue until the end of the production run.
(3) The symmetric quadratic quality loss function is
adopted for evaluating the product quality.
(4) The elapse time until the occurrence of the assignable cause assumed to be exponentially distributed
with a mean of 1/l.
(5) The process mean of in-control process is m which
maybe not be equal to the target value. The process
mean of out-of-control process is m + ds.
(3)
It is difficult to show that the Hessian matrix of Eq.
(3) is positive definite. To minimize the expected total
cost per unit time, partially differentiate Eq. (3) with respect to m and T equal to zero:
(4)
(5)
From Eq. (4), we get an explicit expression of m in
terms of other variables:
From Rahim and Tuffaha [9], the expected quality
loss per item for each production cycle T is
(2)
(6)
Substituting Eq. (6) into Eq. (5), we obtain
where g0 is expected quality loss per item of in-control
¥
g0 =
process,
ò k ( y - m)
2
(7)
f ( y) dy = k [(m - m) 2 + s 2 ],
-¥
y~N (m, s ), f (y) is the probability density function of y; g1
is the expected quality loss per item of out-of-control process; m is the target value; k is the quality loss coefficient;
2
g1 =
¥
ò k ( y¢ - m)
2
f ( y¢ ) dy¢ = k [(m + ds - m) 2 + s 2 ], y¢
-¥
~N (m+ds, s2), f (y¢) is the probability density function
of y¢; d is the magnitude of mean shift; l is the parameter of exponential distribution.
We can adopt numerical analysis method, e. g., Newton’s method, for solving the optimum T* of Eq. (7).
Then the optimum m* can be obtained by Eq. (6).
4. Numerical Examples
Suppose that the production rate is p = 40 items per
unit time. The demand rate is d = 30 items per unit time.
The Modified Economic Manufacturing Quantity Model for Product with Quality Loss Function
The holding cost is B = 0.1 per item per unit time and the
set-up cost is S = 50 per production run. The quality characteristic is normally distributed with unknown mean
m and known standard deviation s = 1. Let the magnitude
of mean shift d = 0.5, the quality loss coefficient k = 5,
the target value m = 10, and the parameter of exponential
distribution l = 0.05.
By solving Eq. (7), we obtain the optimum production run length T* = 5.94. Substituting T* into Eq. (6),
we have m* = 9.933. Hence, we get ETC 1* = 163658
. .
Tables 1-9 list the effect of parameters on the optimum solution for modified EMQ model when the parameter varies about between -20% and +20%. From Tables 1-9, we have the following conclusions:
1. As the l increases, the m* increases, T* decreases,
and ETC 1* slightly increases.
2. As the d increases, the m* and T* decrease but
ETC 1* slightly increases.
3. As the s increases, the m* and T* decrease but
Table 1. The effect of l on the optimal solution for
modified EMQ model
l
0.03
0.04
0.05
0.06
0.07
m*
T*
ETC1*
9.955
9.943
9.933
9.922
9.912
6.44
6.15
5.94
5.79
5.69
162.135
162.946
163.658
164.289
164.853
Table 2. The effect of d on the optimal solution for
modified EMQ model
d
0.3
0.4
0.5
0.6
0.7
m*
T*
ETC1*
9.951
9.940
9.933
9.928
9.925
7.44
6.69
5.94
5.25
4.65
160.620
162.011
163.658
165.490
167.455
111
Table 4. The effect of p on the optimal solution for
modified EMQ model
p
m*
T*
ETC1*
32
36
40
44
48
9.896
9.920
9.933
9.941
9.948
9.68
7.20
5.94
5.11
4.51
161.972
162.992
163.658
164.138
164.504
Table 5. The effect of d on the optimal solution for
modified EMQ model
d
m*
T*
ETC1*
24
27
30
33
36
9.944
9.939
9.933
9.925
9.916
4.81
5.34
5.94
6.66
7.58
133.049
148.416
163.658
178.769
193.733
Table 6. The effect of B on the optimal solution for
modified EMQ model
B
0.1
0.2
0.3
0.4
0.5
m*
T*
ETC1*
9.933
9.944
9.951
9.956
9.960
5.94
4.81
4.17
3.74
3.42
163.658
166.312
168.544
170.514
172.301
Table 7. The effect of k on the optimal solution for
modified EMQ model
k
m*
T*
ETC1*
3
4
5
6
7
9.924
9.929
9.933
9.936
9.939
6.79
6.33
5.94
5.60
5.30
101.819
132.761
163.658
194.512
225.331
Table 3. The effect of s on the optimal solution for
modified EMQ model
Table 8. The effect of m on the optimal solution for
modified EMQ model
s
0.8
0.9
1.0
1.1
1.2
m*
T*
ETC1*
m
m*
T*
ETC1*
9.940
9.936
9.933
9.930
9.928
6.69
6.31
5.94
5.58
5.25
108.011
134.307
163.658
196.054
231.490
8
9
10
11
12
07.933
08.933
09.933
10.933
11.933
5.94
5.94
5.94
5.94
5.94
163.658
163.658
163.658
163.658
163.658
112
Chung-Ho Chen
Table 9. The effect of S on the optimal solution for
modified EMQ model
S
m*
T*
ETC1*
40
45
50
55
60
9.940
9.936
9.933
9.929
9.926
5.22
5.59
5.94
6.28
6.61
162.313
163.001
163.658
164.271
164.854
ETC 1* largely increases.
4. As the p increases, the m* increases, T* decreases,
and ETC 1* slightly increases.
5. As the d increases, the m* decreases, T* increases,
and ETC 1* largely increases.
6. As the B increases, the m* increases, T* decreases,
and ETC 1* slightly increases.
7. As the k increases, the m* increases, T* decreases,
and ETC 1* largely increases.
8. As the m increases, the T* and ETC 1* are the same
constants but m* increases.
9. As the S increases, the m* decreases, T* increases,
and ETC 1* slightly increases.
5. Conclusion
From the above numerical example, we have the following three conclusions: (1) The target value has a major effect
on the optimal process mean, but other parameters only has a
slight effect on the optimal process mean; (2) The target
value has no effect on the optimal production run length, but
other parameters have a major effect on the optimal production run length; (3) The process standard deviation, the demand rate, and the quality loss coefficient have a major effect
on the expected total cost per unit time for modified EMQ
model. We need to emphasis the reasonable estimation of parameters in order to obtain the optimal solution.
In this paper, we have presented a modified EMQ
model based on the minimum expected total cost per unit
time. The solution procedure of proposed method is easy,
clear, and efficient. Further study should address that
more cost parameters, e.g., inspection cost, manufacturing cost, and defective cost, used in the modified model.
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References
[1] Lin, C. Y., “Optimization of Maintenance, Production
Manuscript Received: Nov. 6, 2007
Accepted: Sep. 9, 2008