Journal of Applied Science and Engineering, Vol. 15, No. 1, pp. 41-48 (2012)
41
Optimal Determination of Purchaser’s Order Quantity
and Producer’s Process Mean
Chung-Ho Chen1* and Chao-Yu Chou2
1
Department of Management and Information Technology, Southern Taiwan University,
Tainan, Taiwan 710, R.O.C.
2
Department of Finance, National Taichung University of Science and Technology,
Taichung, Taiwan, R.O.C.
Abstract
Determination of optimum mean of the process characteristic is an important theme in quality
improvement. Meanwhile, economic selection of order quantity is a key factor in inventory management
since an inappropriate order quantity may result in unreasonably high inventory or stockout cost. In the
present paper, we modify Chen and Liu’s model [8] to simultaneously determine the optimal
purchaser’s order quantity and producer’s process mean by incorporating an asymmetric loss function.
The solution procedure for this modified model is developed and numerical examples are given for
illustration. Based on the sensitivity analyses, the common parameters having significant effects on the
expected total profit of the system are the selling price per unit to the customer (R) and the mean
demand of customer (mx). Therefore, accurate estimation of R and mx in order to obtain the optimal
control on the order quantity and the maximum expected total profit of the system would be highly
desired.
Key Words: Order Quantity, Process Mean, Asymmetric Quadratic Quality Loss Function, Uniform
Distribution, Normal Distribution
1. Introduction
Quality improvement of processes and/or products
is always significant for enhancing customers’ satisfaction. One of important themes in quality improvement
is the determination of optimum mean of the process
characteristic, which usually has a major effect on the
defective fraction of process, the inspection cost, the reprocessing cost, the scrap cost, and the expected total
profit or cost associated with the process and products
[1]. Recently, there are considerable attentions paid to
this theme, e.g., Chen [2], Chen and Lai [3,4], and Chen
and Khoo [5,6]. On the other hand, among various logistics costs in a company, inventory cost generally
amounts to almost half of the company’s total distribution dollar expense [7]. The economic selection of or*Corresponding author. E-mail: chench@mail.stut.edu.tw
der quantity is a key factor in inventory management
since an inappropriate order quantity may result in unreasonably high inventory or stockout cost. The economic determinations of order quantity and process mean
look like two different problems for inventory management and quality improvement, respectively. However,
for a modern supply chain system, the producer needs
to determine the optimum process mean and the purchaser needs to consider the optimum order quantity.
That is, a producer’s decision on optimum process mean
and a purchaser’s decision on optimum order quantity
could be simultaneously obtained such that the expected
total profit of the system, including both the purchaser
and the supplier, may be maximized.
For previous years, several researchers focused their
studies on modeling the profit of the producer and the
purchaser in the supply chain system. For example, Chen
and Liu [8] presented an optimum profit model between
42
Chung-Ho Chen and Chao-Yu Chou
the producer and the purchaser for the supply chain system with pure procurement policy from the regular supplier and with mixed procurement policy from both the
regular supplier and the spot market. In 2008, Chen and
Liu [9] further proposed an optimal consignment policy
considering a fixed fee and a per-unit commission,
which leads to a higher manufacturer’s profit than the
traditional production system and coordinates the retailer to obtain a higher supply chain profit. Li and Liu
[10] considered the problem associated with the retailer
for determining his optimal order quantity and the manufacturer for determining his optimal reserve capacity.
Their model is able to increase profit for both sides of
the supply chain. Darwish [11] proposed the optimum
process mean setting for a single producer and a single
purchaser in the filling industry, where the process
mean, shipment quantity of product, and numbers of
shipment are three decision variables to be determined.
Chen and Huang [12] addressed that retailers purchase
their products from options and online spot markets for
hedging the risk of demand uncertainty. Arshinder and
Deshmukh [13] presented a review of supply chain coordination and then proposed its research directions for
further study.
In Chen and Liu’s model [8], they only considered
the order quantity but did not account for the effect of
product quality on the demand quantity of the end customers. Taguchi [14] refined the quality of product and
presented the quadratic quality loss function for reducing total losses to the society. Taguchi’s loss function,
which explicitly considers the quality loss due to process
variability, can be used for the designs of control charts,
sampling plans, process mean, manufacturing tolerances
and specification limits [15-19]. In the present paper, we
modify Chen and Liu’s model by incorporating Taguchi’s quadratic loss function for determination of the
producer’s optimum process mean and the purchaser’s
optimum order quantity such that the expected total
profit of the supply chain system may be maximized. The
advantage of this modified model is to obtain the joint
control on purchaser’s order quantity and producer’s
process quality at the same time. The motivation behind
this work stems from the fact that neglect of quality loss
within the specification limits could overestimate the
expected total profit of the system. The asymmetric loss
function is employed in the present paper to well re-
present the real-world quality loss situations. In the next
section, a brief review of Chen and Liu’s model [8] is
given. Then the modification of Chen and Liu’s model is
developed using an asymmetric loss function. Two numerical examples are subsequently presented for illustration. Some final concluding remarks are then provided
based on the analysis.
2. A Brief Review of Chen and Liu’s Model
Chen and Liu’s pure procurement model [8] is actually based on the standard news-vendor model without
spot markets. They consider a single period supplierbuyer relationship in which a regular supplier produces
short life-cycle products and a buyer orders products
from the regular supplier and then sells to the end customer. Some assumptions in their model are as follows:
1. A buyer purchases a finished product from a regular
supplier and resells it at a price, R, to the customer.
2. The regular supplier produces each unit at a cost, C.
3. The regular supplier and the buyer enter into a contract at a wholesale price, W.
4. The regular supplier sets the wholesale price to maximize his expected profit, while offering the buyer a
specific order quantity, Q.
5. When realized demand exceeds procurement quantity, the unmet demand is lost; therefore, demand uncertainty exposes the buyer to the risk associated with
mismatches between the procurement quantity and
demand.
6. The procurement lead time is long relative to the selling season, such that the buyer cannot observe demand before placing an order.
7. The consumer demand, X, is uniformly distributed,
i.e., X ~ U[mx - (sx / 2), mx + (sx / 2)], where mx is the
mean of X and sx is the variability of X.
Let S be the salvage value per unit of product. From
Chen and Liu [8], the buyer’s profit, denoted by pRPS , is
given by
(1)
Then, the buyer’s expected profit, E ( pRPS ), may be expressed by
Optimal Determination of Purchaser’s Order Quantity and Producer’s Process Mean
(2)
where f (t) is the density function of T.
According to Chen and Liu [8], the optimum wholesale price and order quantity are as follows:
(3)
and
(4)
Consequently, the buyer’s expected profit and the supplier’s expected profit, denoted by E ( pSPS ), can be respectively obtained through
(5)
43
distributed representing the process characteristic, i.e.,
Y ~ U [m y - 3s y , m y + 3s y ], where my is the unknown
mean of Y to be determined for maximization of profit
and sy is the known standard deviation of Y, Loss(Y) is
the quality loss per unit of product, and mathematically,
where k1 is the quality loss coefficient for y < m, k2 is
the quality loss coefficient for y ³ m, m is the target
value of process characteristic, U is the upper specification limit of process characteristic which may be expressed by U = my + bsy, L is the lower specification
limit of process characteristic and can be expressed by L
= my - asy, where both a and b are constants, the random
variable X represents the customers’ demand quantity
and is assumed to be normally distributed, i.e., X ~
N (m x , s 2x ), where mx is the known mean of X and sx is
the known standard deviation of X. It can be shown that
the purchaser’s expected profit may be expressed by
and
(8)
(6)
3. Development of the Modified Chen and
Liu’s Model
Let customers’ demand quantity be normally distributed with a large variability and the quality characteristic of product have a finite value and be uniformly distributed. Assume that these two variables (i.e., demand
quantity and product characteristic) are independent. In
our modified model, an asymmetric loss function is incorporated to account for the quality loss of the conformance
products. Hence, the purchaser’s profit can be given by
where f (x, y) is the joint probability density function of
X and Y. Note that f (x, y) = f (x) f (y) since X and Y are independent. Eq. (8) can be rewritten as
(7)
(9)
where the random variable Y is assumed to be uniformly
where
44
Chung-Ho Chen and Chao-Yu Chou
Then the expected total profit of the system, including
both purchaser and supplier, is
(14)
(10)
For a given Q, we can obtain the optimum process
mean m *y . Differentiating Eq. (14) with respect to m and
setting equal to 0 gives
where F(×) is the cumulative distribution function of
the standard normal random variable, and f(×) is the
probability density function of the standard normal random variable.
Consider the situation that conformance product is
sold to the primary market and the non-conformance
product is scrapped and sold to the secondary market.
Then the supplier’s profit is given by
(15)
(11)
where W is the selling price per unit of the conformance
product, Sp is the discounted price per unit of the nonconformance product that is scrapped, f is the constant
fixed production cost, z is the variable production cost
per unit of product, and i is the inspection cost per unit of
product. According to the definition of expectation, the
supplier’s expected profit per unit may be expressed by
Since the supplier needs to produce
Let A = [m x F(
F(
Q -mx
Q -mx
) - s xf (
)] and B = Q[1 sx
sx
Q -mx
)] . If the second derivative of Eq. (14) is negsx
ative, that is,
(12)
then m *y is optimal for a given Q. In general, we have A >
Q
items
P (L £ y £ U )
0 and B > 0. Therefore, the necessary condition to obtain the optimal solution is [k1(L - m) + k2(U - m)] > 0,
k U -m
that is, 1 <
. By substituting L = my - asy and U =
k2 m - L
in order to satisfy the purchaser’s order quantity, hence,
the supplier’s expected profit would be
my + bsy into Eq. (15), we can solve Eq. (15) to obtain
the optimum process mean for a given Q. Eq. (15) can
be rewritten as
(13)
(16)
Optimal Determination of Purchaser’s Order Quantity and Producer’s Process Mean
45
4. Numerical Examples and Sensitivity Analysis
Eq. (16) has an optimum process mean m *y =
2
-b1 + b1 - 4a 1 c1
2a 1
as b12 - 4a1c1 > 0. Because b12 -
4a1c1 = 4[ k 1 k 2 ( a + b) 2 s 2y -
2 3Qz
( k 2 - k 1 )] ,
( a + b)( A + B )
we need the following necessary condition for obtaining the optimal solution, that is,
The general procedure for obtaining the optimal solution of the aforementioned modified model in Eq. (14)
is summarized as follows:
Step 1. Set the minimum order quantity Qmin = mx - 5sx
and the maximum order quantity Qmax = mx + 5sx.
Step 2. Let Qmin £ Q £ Qmax. For a given Q, the optimum
process mean may be obtained by
under the necessary conditions
k1 U - m
and
<
k2 m - L
Step 3. Select the maximum expected total profit of the
system from Step 2 as the best policy. The corresponding combination of parameters (m *y , Q*)
having the maximum ETP is the optimum solution.
The optimal solution of the modified Chen and Liu’s
model [8] depends on the several cost and profit parameters. The influences of these parameters need to be
illustrated by conducting the sensitivity analysis of parameters in the next section.
4.1 Example 1
Assume that the parameters in the system are as follows: a = 0.8, b = 0.6, k1 = 5, k2 = 10, R = 25, S = 5, W =
10, mx = 100, Sp = 5, sx = 20, m = 11, sy = 0.9, i = 0.2, f =
2, and z = 0.5. By solving Eq. (14), we have the optimal
order quantity Q* = 118 and the optimal process mean
with the expected profit of the purchaser
m *y =10941
.
R
E ( pPS ) = 771.66, the expected profit of the supplier
E ( pSPS ) = 139.86, and the expected total profit of the society ETP = 911.52 for the our model.
Table 1 lists ±50% range for parameter values and
presents their effects on the order quantity, the process
mean, the expected profit of the purchaser, the expected
profit of the supplier, and the expected total profit of the
system for the modified Chen and Liu’s model. If the
change percentage of the expected profit is greater than
20%, then the corresponding parameter is considered as
a major effect on the expected profit. From Table 1, the
following results can be observed:
1. The mean demand of the customer (mx), the variable
production cost per unit (z), and the target value (m)
have significant effects on the purchaser’s order quantity.
2. The quality loss coefficients (k1 and k2), the coefficient of the lower specification limit (a), the coefficient of the upper specification limit (b), and the target
value (m) have significant effects on the process mean.
3. The selling price per unit to the customer (R), the standard deviation of the quality characteristic (sy), the
mean demand of the customers (mx), and the wholesale
price per unit paid by the buyer to the regular supplier
(W) have significant effects on the expected profit of
the purchaser.
4. The mean demand of the customer (mx), the coefficient
of the lower specification limit (a), the coefficient of
the upper specification limit (b), the variable production cost per unit (z), the wholesale price per unit paid
by the buyer to the regular supplier (W), the target
value (m), the discounted price per unit for the nonconformance product scrapped (Sp), and the fixed production cost (f) have significant effects on the expected profit of the supplier.
5. The selling price per unit for the end customer (R), the
mean demand of the customer (mx), the variable pro-
46
Chung-Ho Chen and Chao-Yu Chou
Table 1. Effects of parameters on the optimal solution for
the modified Chen and Liu’s model
S
Q
my
E (p RPS )
E (pSPS )
ETP
2.5
4
6
7.5
111
114
121
113
10.941
10.941
10.940
10.938
750.62
763.62
782.89
796.84
131.46
135.06
143.46
157.93
882.09
898.68
926.35
954.77
sy
0.45
0.72
1.08
1.35
Q
119
118
117
114
my
10.927
10.940
10.940
10.935
E (p RPS )
909.54
838.72
690.93
544.08
E (pSPS )
142.19
139.93
138.79
135.59
ETP
1051.73
978.65
829.72
679.67
sx
10
16
24
30
Q
109
114
121
126
my
10.943
10.942
10.940
10.938
E (p RPS )
815.38
789.65
754.94
729.16
E (pSPS )
129.00
135.03
143.49
149.54
ETP
944.38
924.68
898.43
878.70
mx
50
80
120
150
Q
068
097
138
168
E (p RPS )
my
10.936 342.12
10.940 601.04
10.941 943.47
10.942 1201.190
E (pSPS )
080.84
115.04
163.48
198.93
ETP
422.95
716.07
1106.96
1400.11
k1
2.5
4
6
7.5
Q
119
118
117
116
my
10.752
10.879
10.992
11.055
E (p RPS )
836.07
795.00
752.29
726.44
E (pSPS )
157.09
145.07
134.39
128.01
ETP
993.16
940.07
886.68
854.45
k2
6
8
12
15
Q
117
117
118
117
my
11.070
10.997
10.895
10.841
E (p RPS )
810.72
789.78
757.43
740.99
E (pSPS )
127.83
133.95
143.69
147.02
ETP
938.55
923.73
901.12
888.02
a
0.4
0.64
0.96
1.2
Q
112
116
118
118
my
10.657
10.830
11.049
11.210
E (p RPS )
618.15
720.75
809.70
835.56
E (pSPS )
0.-6.40
090.74
177.80
222.23
ETP
611.75
811.49
987.50
1057.79
b
0.3
0.48
0.72
0.9
Q
112
116
119
120
my
11.199
11.045
10.835
10.676
E (p RPS )
664.80
735.34
800.10
825.73
E (pSPS )
.-10.56
086.53
185.14
241.72
ETP
654.74
821.88
985.24
1067.46
W
5
8
12
15
Q
110
114
121
130
E (p RPS )
E (pSPS )
ETP
my
10.942 1161.150 -419.73- 741.41
.-92.94 842.08
10.941 935.02
10.940 597.73
385.46 983.19
10.938 291.80
804.31 1096.10
Table 1. Continued
m
Q
my
E (p RPS )
E (pSPS )
ETP
8
8.8
13.2
16.5
145
131
109
099
07.935
08.738
13.142
16.444
700.24
744.05
775.84
759.91
483.17
361.37
.-42.21
-271.79-
1183.42
1105.42
733.62
488.12
f
1
1.6
2.4
3
Q
129
121
114
110
my
10.938
10.940
10.941
10.942
E (p RPS )
749.44
767.13
775.42
776.15
E (pSPS )
337.38
212.61
069.92
.-26.88
ETP
1086.82
979.74
845.33
749.27
duction cost per unit (z), and the target value (m) have
significant effects on the expected total profit of the
system.
4.2 Example 2
Consider the situation that the mean and standard deviation of the demand of customers are the same as those
of Example 1 in Chen and Liu’s model [8]. Assume that
the parameters are as follows: R = 25, S = 5, mx = 100, sx
= 40 3, and C = 2. By solving Eqs. (3)-(6), we have the
optimal order quantity Q* = 73 and the optimal wholesale price W * = 22.93 with the expected profit of the purchaser E ( pRPS ) =142.44, the expected profit of the supplier E ( pSPS ) =151805,
. and the expected total profit of
the system ETP = 1660.49 for the Chen and Liu’s model
[8].
Table 2 lists ±50% range for parameter values and
presents the effects on the order quantity, the wholesale
price, the expected profit of the purchaser, the expected
profit of the supplier, and the expected total profit of the
system for the Chen and Liu’s model [8]. If the change
percentage of the expected profit is greater than 20%,
then the corresponding parameter would be a major effect on the expected profit. From Table 2, the following
conclusions may be obtained:
1. The mean demand of customer (mx) has a significant
effect on the purchaser’s order quantity.
2. The standard deviation of demand of customer (sx)
and the selling price per unit to the customer (R) have
significant effects on the wholesale price.
3. The standard deviation of the demand of customer
(sx) and the mean demand of customer (mx) have significant effects on the expected profit of the purchaser.
Optimal Determination of Purchaser’s Order Quantity and Producer’s Process Mean
Table 2. Effects of parameters on the optimal solution for
Chen and Liu’s model
S
Q
W
E (p RPS )
E (pSPS )
ETP
2.5
4
6
7.5
68
71
75
78
24.11
23.41
22.46
21.75
059.19
108.41
177.62
232.17
1505.68
1511.65
1526.74
1544.96
1564.87
1620.06
1704.36
1777.93
sx
Q
W
E (p RPS )
E (pSPS )
ETP
20 3
32 3
61
37.37
-890.08- 2166.56
1276.48
68
26.54
-108.19- 1669.20
1561.01
48 3
60 3
77
20.53
302.86
1427.04
1729.90
84
18.12
453.27
1350.66
1803.94
mx
50
80
120
150
Q
48
63
83
98
W
15.72
20.05
25.82
30.15
E (p RPS )
291.84
267.15
.-68.87
-548.22-
E (pSPS )
0651.78
1128.24
1965.59
2745.16
ETP
0943.62
1395.39
1896.72
2196.93
C
1
1.6
2.4
3
Q
74
73
72
71
W
22.43
22.73
23.13
23.43
E (p RPS )
179.13
157.01
128.01
106.62
E (pSPS )
1591.43
1547.19
1489.18
1446.40
ETP
1770.56
1704.21
1617.19
1553.01
R
12.5
20
30
37.5
Q
81
74
71
71
W
10.79
18.08
27.79
35.08
E (p RPS )
125.46
134.35
151.05
164.42
E (pSPS )
0713.36
1193.57
1843.56
2332.73
ETP
0838.81
1327.92
1994.61
2497.15
4. The standard deviation of the demand of customer
(sx), the mean demand of the customer (mx), and the
selling price per unit to customer (R) have significant
effects on the expected profit of the supplier.
5. The standard deviation of the demand of customer
(sx), the mean demand of customer (mx) and the selling
price per unit to customer (R) have significant effects
on the expected total profit of the system.
5. Conclusion
In the present paper, we modify Chen and Liu’s
model [8] by involving an asymmetric quality loss function. By assuming that the demand quantity of the customer is normally distributed, the quality characteristic
of the process is uniformly distributed, and both random
variables are independent, the optimum process mean of
47
product and the order quantity can be simultaneously
determined in this modified model by maximizing the
expected total profit of the supply chain system, including both the purchaser and the producer. Based on the
sensitivity analyses, the common parameters having significant effects on the expected total profit of the system
are the selling price per unit to the customer (R) and the
mean demand of customer (mx). Therefore, accurate estimation of R and mx in order to obtain the optimal control on the order quantity and the maximum expected
total profit of the system would be highly desired.
The extension of this method to the modified Chen
and Liu’s model [8] with rectifying sampling inspection
plan or mixed procurement policy may be left for further
study.
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Manuscript Received: Dec. 8, 2010
Accepted: Nov. 28, 2011