International Review of Business Research Papers
Vol. 9. No. 3. March 2013 Issue. Pp. 127 – 145
The Joint Settings of Quality Investment, Commission Fee,
and Order Quantity under the Consignment Policy
Chung-Ho Chen*
Chen and Liu (2008) proposed an optimal consignment policy
considering a fixed fee and a per-unit commission. However, their
model assumed the perfect product and didn’t consider the product
quality. The objective of this study is to present a modified Chen and
Liu’s (2008) model by considering quality investment for increasing
both manufacturer’s and retailer’s profits. Taguchi’s (1986) quadratic
quality loss function is adopted for evaluating the product quality. The
optimal parameters of manufacturer’s quality investment and
commission fee and retailer’s economic order quantity are
simultaneously determined by maximizing the expected profit of supply
chain system. A comparative study between the modified model
with/without quality investment is provided for illustration. Numerical
results show that the modified Chen and Liu’s (2008) model with
quality investment has a larger order quantity, a smaller per unit
commission, a larger fixed commission fee , a larger manufacturer’s
expected profit, a larger retailer’s expected profit, and a larger
expected profit of the supply-chain system than those of one without
quality investment.
1. Introduction
In a supply chain system, the trade-off problem between manufacturer and retailer to
obtain the maximum expected total profit of society always receive considerable
attention. The manufacturer considers the sale, manufacturing, inventory, shipment,
and management of products for obtaining the maximum expected profit. Meanwhile,
the retailer considers the order quantity, the holding cost, and the goodwill loss for
obtaining the maximum expected profit. Recently, Glock (2012) reviews the integrated
inventory problems with stochastic demand and stochastic lead time, order/setup cost
reduction and lead time reduction, product quality, product deterioration and decay,
and learning. Many researchers have addressed the important work for supply chain
system with cooperation between the retailer and the manufacturer.
Chen and Liu (2007) presented the optimum profit model between the producers and
the purchasers for the supply chain system with pure procurement policy from the
regular supplier and with mixed procurement policy from the regular supplier and the
spot market. Chen and Liu (2008) further proposed an optimal consignment policy
considering a fixed fee and a per-unit commission. Their model determines a higher
manufacturer’s profit than the traditional production system and coordinates the
retailer to obtain a large supply chain profit. However, their model assumed that all of
the manufacturer’s products are perfect. In fact, it’s difficult to provide 100% perfect
product in the manufacturer’s process. Hence, the product quality loss problem should
be considered in their model. Chen and Huang (2010) addressed the problem that the
retailers purchase their products from options and online spot markets for hedging the
risk of demand uncertainty. For Chen and Liu’s (2008) model, they considered the
simple manufacturing cost and neglected the
*Dr. Chung-Ho Chen, Department of Management and Information Technology, Southern Taiwan
University of Science and Technology, Taiwan. Email : chench@mail.stust.edu.tw
Chen
used cost of customers in traditional production system. Hence, Chen (2010)
proposed a modified Chen and Liu’s (2008) model with imperfect product for
determining the optimum product and process parameters. Recently, Yu et al. (2012)
have presented an extensional work for Chen and Liu’s (2008) model with multiretailer and generalized demand distributions.
The on-line 100% rectifying inspection can be used as a short-term method for
controlling the product quality shipped to the customers. However, quality investment
is an available method for improving the process parameters in the long-term. For
example, one can buy a new machine for manufacturing the product and address the
continuous education training for personnel. Hong et al. (1993), Ganeshan et al.
(2001), Chen and Tsou (2003), and Tsou (2006) have presented the declining
exponential reduction of process mean and standard deviation as the function of
quality investment. Abdul-kader et al. (2010) further adopted Chen and Tsou’s (2003)
quality investment function for determining the optimum quality investment and
corresponding improved process mean and standard deviation.
Quality investment is a long-term method for improving product quality, decreasing the
product’s cost, and promoting the enterprise’s profit. The objective of this study is to
present a modified Chen and Liu’s (2008) model by considering quality investment for
increasing both manufacturer’s and retailer’s profits. The author proposes a modified
Chen and Liu’s (2008) model with imperfect product and quality investment under the
consignment policy. The motivation behind this work stems the fact that neglect of
product quality loss within the specification limits could overestimate the expected total
profit of the system. Long-term quality improvement can promote the service level of
product and the customer’s satisfaction. In this study, Taguchi’s (1986) quadratic
quality loss function is adopted for evaluating the product quality and represents the
real-world quality loss situations. The optimal parameters of manufacturer’s quality
investment and commission fee and retailer’s economic order quantity are
simultaneously determined by maximizing the expected profit of supply chain system.
A comparative study between the modified model with/without quality investment is
provided for illustration. In the next section, the notations is given. Then a brief review
of modified Chen and Liu’s (2008) model with traditional production system and
modified one with imperfect product under the consignment policy are reported. Then
the modification of Chen and Liu’s (2008) model is developed by using symmetric
quadratic quality loss function and quality investment concept. The numerical example
and sensitivity analysis of parameters are subsequently presented for illustration.
Some final concluding remarks are then provided based on the analysis.
2. Literature Review
For comparing the difference of profit between the modified Chen and Liu’s (2008)
model with imperfect product and quality investment under consignment policy and
traditional production system, we firstly need to review the literature about Chen’s
(2010) research in this section. Then we can propose the modified model with
imperfect product and quality investment in next section.
2.1 Modified Chen and Liu’s Traditional Production System Model
Chen (2010) proposed a modified Chen and Liu’s (2008) model with traditional
production system by assuming that the demand of customer (X) and the quality
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Chen
characteristic of product (Y) are independent. The manufacturing cost per unit
includes the constant and variable production costs, where the variable production
cost per unit is proportional to the value of quality characteristic of product. The 100%
inspection of product is executed before shipping to the customer. The nonconformance of product is scrapped and sold at a lower price. Chen (2010) also
considered the used cost of product for the retailer by adopting Taguchi’s (1986)
quadratic quality loss function in his modified model. From Chen (2010), the retailer’s
expected profit is
 
E  rPS
{
1
2 x


 U  y
 
   y



 L  y
  

 y


2
2

1 
1

  QW  H  
 Q   x   x 
x 
2
  

R  H   Q 2    x  1  x 
2


  1 Q 2    x  1  x  
 2 x 
2

 



 Loss y  f  y dy  (R  W  S )  Q   
U
L
 U  y
  

   y
 U  y

   y

 L  y
  

 
y



 L  y
  

 
y


x


  S  1

2 x

 1
 
 
x

1
 1
 x  Q 
2
 x
1


 (  x   x ) 2  Q 2  
2


U
1


   x   x  Q Q   Loss y  f  y dy
L
2


(1)
where
U

L
Loss( y ) f  y dy
 U  y
 k{ (  y  y 0 ) 2   y2  
   y




 L  y 
U  y
   y  2 y 0  U  
  y [(  y  2 y 0  L) 
 

 
y
y






 L  y
  

 
y



]}


(2)
The manufacturer’s profit may be given by
W  b  cY  i, L  Y  U
S p  b  cY  i, Y  L or Y  U
 mPS  
(3)
129
Chen
Then, the manufacturer’s expected profit is
 
E  mPS

 U  y 
 L   y 
  

b  c y  i  S p 1  
 

  

y
y




}
 Q  {W 
 U  y 
 L  y 
  


 

  
y
y




(4)
*
From Chen (2010), the optimum order quantity Q for the retailer and the optimum
*
selling price W for the manufacturer are as follows:
A  R  H  S B x  1 A  R  S  H  2W B x
2
Q* 
A  (R  H  S )B
(5)
1
 x ( S  R  H  2t ) B  A   x R  H  S   B  A
W*  2
2 B x
(6)
where
 U  y
A  k{ (  y  y 0 ) 2   y2  
   y


 L  y
  y [(  y  2 y 0 ) 
 
y

U  y
B  
 
y


 L  y
  

 
y



 L  y
  

 
y







U  y
   y  2 y 0  U  

 
y



]}


(7)




(8)

U   y 
 Ly
  
b  c y  i  S p 1  
 

 

y
y



t
U   y 
 Ly 
  


 

 

y
y








(9)
The trade-off model between Eqs. (1) and (4) needs to obtain the optimal order
quantity ( Q * ) , the optimal wholesale price ( W * ) , and the optimal process quality level
(  *y ) with the maximum expected profits for the retailer and the manufacturer.
Substituting Eqs. (5) and (6) into Eq. (4) leads to only one unknown decision variable
130
Chen
 y to be determined. Let L   y  U . The direct search method may be applied for
obtaining the optimal  *y with the maximum manufacturer’s expected profit for the
given order quantity and wholesale price. By substituting this optimal  *y into Eqs. (5)
and (6), the optimal solution with the maximum expected profit for the retailer and the
manufacturer can be obtained.
2.2 Modified Chen and Liu’s Consignment Policy with Imperfect Product
To simplify mathematical manipulations, some assumptions are made in the modified
Chen and Liu’s (2008) model as follows:
1. The demand of customer (X) and the quality characteristic of product (Y) are
independent.
2. The variable production cost per unit is proportional to the value of quality
characteristic of product.
3. Imperfect products occur in the production system and the 100% inspection of
product
is executed before shipping to the customer.
4. The non-conformance of product is scrapped and sold at a lower price.
5. The used cost of product is measured by adopting Taguchi’s quadratic quality loss
function.
6. The manufacturer guarantees that the retailer will be at least as well-off as in
the traditional production system.
By applying Taguchi’s (1986) quadratic quality loss function, the retailer’s profit may
be expressed by


X  hr (Q  X )  A1  X  Loss(Y ),  x  x  X  Q, L  Y  U


2
 rCP  
(10)
Q  S ( X  Q)  A  Q  Loss(Y ), Q  X     x , L  Y  U
r
1
x

2

Then the retailer’s expected profit can be obtained as follows:
 
E  rCP
2
 2 

1   
  hr   Q    x   x    Qhr 1  Q   x  1  x 
{
2 x
2   
x 
2 


1
 sr 
1
1


 (  x   x ) 2  Q 2   (  s r )
2 x 
2
x

1
1


  x   x  QQ  A1 }  B
2


2
A  2 
1  
1

 A

Q    x   x    Q    x   x  Q  
2 x 
2  
2

 x

(11)
131
Chen
where
 U  y
A  k{ (  y  y 0 ) 2   y2  

   y


U  y
  y  2 y 0  U  
 
y

U  y
B  
 
y


 L  y
  

 
y



 L  y
  

 
y



 L  y
   y [(  y  2 y 0  L) 

 
y







]}


(12)




(13)
CP
Letting the partial derivative of Eq. (11) with respect to Q be zero, i.e., E ( r )  0 ,
Q
results in the optimal order quantity for the retailer as follows:
Qr*  (  x 
x
2
)
[(  s r ) B  A] x
[(  s r  hr ) B  A]
(14)
The profit of the supply chain system can be given by
 tCP
x

 RX  (b  cy  i )Q  ht (Q  X )  X  Loss(Y ),  x  2  X  Q, L  Y  U

x

, L Y U
  RQ  (b  cy  i )Q  st ( X  Q)  Q  Loss(Y ), Q  X   x 
2




( S p  b  cy  i )Q,  x  x  X   x  x , Y  L or Y  U

2
2

(15)
Therefore, the supply chain system’s expected profit can be obtained as follows:
 
E  tCP
2
 2 

1   
R  ht   Q    x   x    Qht 1  Q   x  1  x 
{
2 x
2   
x 
2 


1
 st 
1
1


 (  x   x ) 2  Q 2   ( R  st )
2 x 
2
x

1
1


  x   x  QQ}  B
2


2
1
A  2 
1  

 A
 S p (1  B)Q  (b  c y  i)Q

Q    x   x    Q    x   x  Q  
2
2 x 
2  


 x
(16)
132
Chen
CP
Letting the partial derivative of Eq. (16) with respect to Q be zero, i.e., E ( t )  0 ,
Q
leads to the optimal order quantity for the supplier chain system as follows:
Qt*  (  x 
x
2
)
[( R  s r ) B  A  Z 11 ] x
[( R  st  ht ) B  A]
(17)
where
Z11  S p (1  B)  b  c y  i
(18)
Setting Qr*  Qt* gives the optimal per-unit commission as follows:
 *  s r 
[( R  st ) B  A  Z 11 ]hr B  A(ht B  Z 11 )
(ht B  Z 11 ) B
(19)
The profit of the manufacturer can be expressed by:
 mCP
( R   ) X  (b  cy  i)Q  hm (Q  X )  A1 , X  Q, L  Y  U

 ( R   )Q  (b  cy  i)Q  s m ( X  Q)  A1 , X  Q, L  Y  U

( S  b  cy  i)Q,    x  X     x , Y  L or Y  U
p
x
x

2
2

(20)
Thus, the manufacturer’s expected profit can be obtained as follows
 
E  mCP
{
1
2 x
 sm 

 
1
  Qhm
x
  
R    hm   Q 2    x  1  x 


2
2
1
1


 (  x   x ) 2  Q 2   ( R    s m )
2 x 
2
x

1
1 

 Q   x   x 
2 

1


  x   x  QQ  A1 }  B
2


 S p (1  B)Q  (b  c y  i)Q
(21)
There are four decision variables in the modified Chen and Liu’s (2008) model with
consignment policy and imperfect quality, including order quantity (Q), per-unit
commission ( ) , fixed commission ( A1 ) , and process mean of product (  y ) . The
mathematical model for obtaining the optimal values of these decision variables is as
follows:
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Chen
 
Max E  mCP
(22)
subject to E  rCP  E ( rPS )
 
(23)
where
 
E  rPS
2
 2 

1   
R  H   Q1    x   x    Q1 W1  H   1  Q1   x  1  x 
{
2 x
2   
x 
2 


1
 U  y
 
   y

 L  y
  

 
y



  1
 2
x

2
 2 
1  
Q1    x   x  }  A
2  


1

 1
 ( R  W1  S )  Q1    x   x  Q1  
2

 x
 U  y
  
   y
 U  y

   y

 L  y
  

 
y



 L  y
  

 
y



  S  1

2 x

 1
 
 
x

1


   x   x  Q1 Q1  A
2


 U  y
A  k{ (  y  y 0 ) 2   y2  
   y


 L  y
  y [(  y  2 y 0  L) 
 
y

Q1 
1


 (  x   x ) 2  Q12  
2



 L  y
  

 
y


(24)





U  y
   y  2 y 0  U  

 
y



]}


(25)
A  R  H  S B x  1 A  R  S  H  2W1 B x
2
A  (R  H  S )B
(26)
1
 x ( S  R  H  2t ) B  A   x R  H  S   B  A
2
W1 
2 B x
(27)
U  y
B  
 
y


 L  y
  

 
y






(28)
134
Chen

U  y 
 L  y
  
b  c y  i  S p 1  
 

 

y
y



t
U  y 
 L  y 
  


 

 

y
y








(29)
 
E1  rCP
 
1 
1 
 Qt   x   x 
  Qt hr
2 x
2   
x 
2 


1 
1
1 
1


 sr 
 (  x   x ) 2  Qt2   (  s r )
  x   x  Qt Qt }  B
2 x 
2
x 
2


{

1

  hr   Qt2    x  1  x 
2
A  2 
1
 
Q




 t  x
x    Qt
2 x 
2

 
Qt  (  x 
x
2
)
2
1

 A
   x   x  Qt  
2

 x
(30)
[( R  s r ) B  A  Z 11 ] x
[( R  st  ht ) B  A]
(31)
Z11  S p (1  B)  b  c y  i
(32)
[( R  st ) B  A  Z 11 ]hr B  A(ht B  Z 11 )
(ht B  Z 11 ) B
(33)
CP
CP
E ( r )  E1  r   A1 B
(34)
  s r 
 
A1  [ E ( rPS )  E1  rCP ] / B
(35)
The following solution procedure may be applied for obtaining the optimal solution of
Eqs. (22) and (23):
Step 1. Substituting Eqs. (31), (33), and (35) into Eq. (22) results in the only one
unknown decision variable,  y , in the model.
Step 2. Let L   y  U . One may use the direct search method for obtaining the
optimal  *y with the maximum manufacturer’s expected profit for the
manufacturer.
135
Chen
Step 3. Substitute this optimal  *y from Step2 into Eqs. (31), (33), and (35). Then we
can obtain the optimal values of order quantity (Q * ) , per-unit commission ( * ) ,
and fixed commission ( A1* ) .
Step 4. Substituting the combination of decision variables ( Q * ,  * , A1* ,  *y ) into Eqs.
(11), (24), and (16) leads to the optimal expected profit of retailer for
consignment policy, the optimal expected profit of retailer for traditional
production system, and the optimal expected profit of supply chain system for
consignment policy, respectively.
3. Methodology---The Modified Chen and Liu’s Consignment Policy
with Imperfect Product and Quality Investment
Consider the quality investment can improve the known process mean and standard
deviation of product quality characteristic and decrease the bias and variability of
product. For example, we can address the quality train for person, use the new
material and manufacturing equipment for process. It can improve product output
quality, increase the manufacturer’s profit, and promote total profit of supply chain. We
adopt Chen and Tsou’s (2003) quality investment function which has the declining
exponential reduction of process mean and standard deviation as the function of
quality investment. Hence, the optimum profit model is as follows:
 
Max E  mCP
(36)
subject to E  rCP  E ( rPS )
 
(37)
where
 y2  T2  (02 - T2 )exp(-I )
(38)
 y2   T2  ( 02 -  T2 )exp(-I )
(39)
The constraint of the above mathematical model, Eq. (37), is based on the larger
retailer’s profit under consignment policy than that of traditional production system. It
assures the retailer’s benefit when the retailer agrees to adopt the consignment policy.
The objective function of this model, Eq. (36), is to obtain the maximum
manufacturer’s profit by process improvement through quality investment. It is
anticipated that a good long-term quality policy will promote the customer’s
satisfaction on product/service and manufacturer’s profit. The trade-off model needs
to obtain the optimal order quantity ( Q * ) ,per-unit commission ( * ) , fixed
commission ( A1* ) , and the optimal quality investment ( I * ) with the maximum expected
profit for the manufacturer under the constraint Eq. (37) satisfied. Substituting Eqs.
(24)-(35) and (38)-(39) into Eq. (36), there are one unknown decision variable, I . One
can adopt the direct search method for obtaining the optimal I * with the
136
Chen
corresponding improved process mean (  *y ) and standard deviation (  *y ) under the
maximum manufacturer’s expected profit. Substitute these optimal  *y ,  *y , and
I * values into Eqs. (31), (33), and (35). Then we can obtain the optimal values of order
quantity (Q * ) , per-unit commission ( * ) , and fixed commission ( A1* ) . Substituting the
combination of decision variables ( Q * ,  * , A1* , I * ) into Eqs. (11), (24), and (16) leads
to the optimal expected profit of retailer for consignment policy, the optimal expected
profit of retailer for traditional production system, and the optimal expected profit of
supply chain system with consignment policy, imperfect quality,
and quality
investment, respectively.
4. Result---Numerical Example and Sensitivity Analysis
The market demand is assumed to have a uniform distribution with mean 100 and
variability 200. The numerical example employs financial parameters of R = 30, S
= 0.1, H = 0.36, Sp = 0.1, sr = 0.1, sm = 0.1, st = 0.2, ht = 0.36, hm = 0.18, hr = 0.18,
  0.5,  T  0,  0  4.5, T  5,
b = 0.1, i = 0.02, c = 1 and quality parameters 0
  0.01,   0.05, y0 = 5, k = 4, L = 3, U = 7. By applying the aforementioned
solution procedure for solving Eqs. (36) and (37), we can obtain the optimal
process mean I *  108.01 , which results in the optimal order quantity Q *  163 , the
*
per unit commission  *  2.027 and the fixed commission fee A1  416.24 with the
manufacturer’s expected profit E  mCP   1329.43 , the retailer’s expected profit for
traditional system E  rPS  469.27, the retailer’s expected profit for consignment
 
 
policy E 
 469.27, and the expected profit of the supply chain system
CP
E  t  1798.71 . If there is no quality improvement for production process, the
optimal solution of modified Chen and Liu’s (2008) model is: the order quantity
Q *  161 , the per unit commission  *  4.123 and the fixed commission fee
A1*  374.99 with the manufacturer’s expected profit E  mCP   1206.17, the retailer’s
 
CP
r
 
expected profit for traditional system E  rPS  392.91, the retailer’s expected profit
for consignment policy E  rCP  392.91, and the expected profit of the supply chain
system E  tCP   1599.07 . In this numerical example, it shows that the modified
model with quality investment has larger manufacturer’s expected profit, retailer’s
expected profit, and expected profit of the supply chain system than those of
modified one without quality investment.
 
Tables 1-2 list  20% magnitude change for parameter values and show their effects
on the order quantity, process mean, per unit commission, fixed commission fee,
manufacturer’s expected profit, retailer’s expected profit, and expected profit of the
supply chain system for modified Chen and Liu’s (2008) model with/without quality
investment. A parameter is considered as a major effect on the expected profit if the
change magnitudes of manufacturer’s, retailer’s and supply chain system’s expected
profits are greater than 10% . From Tables 1-2, we may have the following
observations:
1. As the selling price per unit, R, increases, the order quantity, the per unit
commission, and the fixed commission fee increase. The selling price per unit has a
137
Chen
major effect on the manufacturer’s expected profit, retailer’s expected profit, and
expected profit of the supply-chain system for the modified Chen and Liu’s (2008)
model with/without quality investment. By the quality improvement, the manufacturer
can sell the product at a higher price and all of the membership in supply chain
system obtain a larger expected profit.
138
Chen
Table 1: The Effect of Parameters of Optimal Solution for Modified Chen and
Liu’s (2008) Model with Quality Investment
Q
I

A1
E  mCP
 
E  rCP
E  tCP
24
30
36
153
163
169
104.06
108.01
110.38
1.880
2.027
2.195
290.70
416.24
543.35
899.655
1329.43
1766.09
1224.41
1798.71
2381.66
x
Q
I

A1
 
324.77
469.27
615.57
E  mCP
E  rCP
E  tCP
80
100
120
142
163
183
87.20
108.01
126.70
2.307
2.027
1.811
287.17
416.24
502.50
999.14
1329.43
1707.48
1324.62
1798.71
2277.65
x
Q
I

A1
 
325.48
469.27
570.18
E  mCP
E  rCP
E  tCP
160
200
240
150
163
175
109.99
108.01
108.00
2.003
2.027
2.027
408.78
416.24
378.72
1423.50
1329.43
1280.85
1888.35
1798.71
1709.70
k
Q
I

A1
 
464.86
469.27
428.85
E  mCP
E  rCP
E  tCP
3.2
4.0
4.8
163
163
163
90.27
108.01
12.69
1.955
2.027
2.054
417.73
416.24
415.68
1350.65
1329.43
1311.45
1821.09
1798.71
1780.28
b
Q
I

A1
 
470.44
469.27
468.83
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
163
163
163
107.99
108.01
107.90
2.031
2.027
2.025
416.71
416.24
415.74
1331.87
1329.43
1327.00
1801.95
1798.71
1795.42
c
Q
I

A1
 
470.08
469.27
468.42
E  mCP
E  rCP
E  tCP
0.8
1.0
1.2
170
163
156
109.39
108.01
106.37
2.225
2.027
1.900
437.26
416.24
390.24
1453.84
1329.43
1210.17
511.16
469.27
429.02
1965.00
1798.71
1639.19
R
 
 
 
 
 
 
 
 
 
 
 
 
i
.016
.020
.024
Q
I

A1
E  mCP
 
E  rCP
E  tCP
163
163
163
108.01
108.01
108.03
2.028
2.027
2.026
416.34
416.24
416.15
1329.92
1329.43
1328.94
1799.35
1798.71
1798.06
Sp
Q
I

A1
 
469.43
469.27
469.12
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
163
163
163
108.03
108.01
108.03
2.027
2.027
2.027
416.25
416.24
416.25
1329.43
1329.43
1329.43
1798.71
1798.71
1798.71
sr
Q
I

A1
 
469.28
469.27
469.28
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
163
163
163
108.06
108.01
108.05
2.047
2.027
2.007
414.25
416.24
418.25
1329.50
1329.43
1329.36
1798.79
1798.71
1798.65
sm
Q
I

A1
 
469.29
469.27
469.29
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
163
163
163
108.01
108.01
108.21
2.027
2.027
2.025
416.24
416.24
416.30
1329.50
1329.43
1329.36
1798.78
1798.71
1798.70
st
Q
I

A1
 
469.27
469.27
469.34
E  mCP
E  rCP
E  tCP
.16
.20
.24
163
163
163
108.17
108.01
108.17
2.024
2.027
2.027
416.41
416.24
416.16
1329.43
1329.43
1329.43
469.33
469.27
469.33
1798.76
1798.71
1798.76
 
 
 
 
 
 
 
 
 
 
139
Chen
Table 1 (Continued)
hr
Q
I

A1
E  mCP
 
E  rCP
E  tCP
.144
.180
.216
163
163
163
108.06
108.01
107.83
1.871
2.027
2.185
428.92
416.24
403.53
1331.81
1329.43
1327.06
1801.10
1798.71
1796.27
hm
Q
I

A1
 
469.29
469.27
469.21
E  mCP
E  rCP
E  tCP
.144
.180
.216
163
163
163
108.21
108.01
108.06
2.025
2.027
2.027
416.30
416.24
416.25
1331.81
1329.43
1327.06
1801.15
1798.71
1796.35
ht
Q
I

A1
 
469.34
469.27
469.29
E  mCP
E  rCP
E  tCP
.288
.360
.432
163
163
162
107.87
108.01
108.01
2.039
2.027
2.017
415.20
416.24
417.22
1329.42
1329.43
1329.42
1798.65
1798.71
1798.69
S
Q
I

A1
 
469.32
469.27
469.27
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
163
163
163
108.01
108.01
108.01
2.027
2.027
2.027
417.43
416.24
415.06
1327.92
1329.43
1330.95
1798.71
1798.71
1798.71
H
Q
I

A1
 
470.79
469.27
467.76
E  mCP
E  rCP
E  tCP
.288
.360
.432
163
163
163
108.01
108.01
108.01
2.027
2.027
2.027
417.43
416.24
415.06
1328.24
1329.43
1330.62
1798.71
1798.71
1798.71
1
Q
I

A1
 
470.46
469.27
468.09
E  mCP
E  rCP
E  tCP
.008
.010
.012
163
163
163
108.01
108.01
108.01
2.027
2.027
2.027
416.24
416.24
416.24
1329.43
1329.43
1329.43
1798.71
1798.71
1798.71
1
Q
I

A1
 
469.27
469.27
469.27
E  mCP
E  rCP
E  tCP
.04
.05
.06
163
163
163
108.01
108.01
108.01
2.027
2.027
2.027
416.24
416.24
416.24
1329.43
1329.43
1329.43
469.27
469.27
469.27
1798.71
1798.71
1798.71
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. As the mean of the demand of customer,  x , increases, the order quantity
increases, the per unit commission decreases, and the fixed commission fee
increases. The mean of the demand of customer is a major effect to the
manufacturer’s expected profit, retailer’s expected profit, and expected profit of the
supply-chain system for the modified Chen and Liu’s (2008) model with/without
quality investment. Hence, the increase of mean demand of customer can promote
a larger expected profit for all of the membership in supply chain system
3. The mean of the demand of customer,  x , and the quality loss coefficient, k, have
the major effects on the quality investment, I. Hence, the mean demand of
customer and the cost of quality improvement are two important factors for
consideration in deciding the long-term quality investment policy.
4. The modified Chen and Liu’s (2008) model with quality investment has a larger
order quantity, a smaller per unit commission, a larger fixed commission fee
increase, a larger manufacturer’s expected profit, a larger retailer’s expected profit,
140
Chen
and a larger expected profit of the supply-chain system than those of one without
quality investment. Hence, the quality investment policy should be a correct method
for maintaining the good long-term relationship between the manufacturer and the
retailer
Table 2: The Effect of Parameters of Optimal Solution for Modified Chen and
Liu’s (2008) Model without Quality Investment
R
Q

A1
E  mCP
 
E  rCP
E  tCP
24
30
36
150
161
168
3.920
4.123
4.325
251.77
374.99
500.59
809.798
1206.17
1610.07
1070.65
1599.07
2137.57
x
Q

A1
 
260.86
392.91
527.50
E  mCP
E  rCP
E  tCP
80
100
120
141
161
181
4.123
4.123
4.123
261.25
374.99
448.27
924.09
1206.17
1525.74
1199.83
1599.07
1998.31
x
Q

A1
 
275.74
392.91
472.58
E  mCP
E  rCP
E  tCP
160
200
240
149
161
173
4.123
4.123
4.123
1289.19
1206.17
1162.20
1678.50
1599.07
1519.64
Q
A1
 
389.31
392.91
357.44
k

368.22
374.99
339.62
E  mCP
E  rCP
E  tCP
3.2
4.0
4.8
162
161
160
3.448
4.123
4.797
389.51
374.99
360.51
1252.89
1206.17
1159.53
1661.31
1599.07
1536.96
b
Q

A1
 
408.46
392.91
377.38
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
161
161
161
4.126
4.123
4.119
375.50
374.99
374.48
1208.58
1206.17
1203.76
1602.28
1599.07
1595.86
c
Q

A1
 
393.71
392.91
392.10
E  mCP
E  rCP
E  tCP
0.8
1.0
1.2
168
161
154
4.326
4.123
3.982
395.14
374.99
350.34
1317.01
1206.17
1100.16
1746.82
1599.07
1457.78
i
Q

A1
 
429.81
392.91
357.62
E  mCP
E  rCP
E  tCP
.016
.020
.024
161
161
161
4.123
4.123
4.122
1206.65
1206.17
1205.68
1599.71
1599.07
1598.43
Q
A1
 
393.07
392.91
392.75
Sp

375.09
374.99
374.89
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
161
161
161
4.122
4.123
4.123
374.95
374.99
375.03
1205.99
1206.17
1206.34
1598.84
1599.07
1599.31
sr
Q

A1
 
392.85
392.91
392.96
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
161
161
161
4.143
4.123
4.103
372.99
374.99
376.99
1206.24
1206.17
1206.09
1599.14
1599.07
1599.00
sm
Q

A1
 
392.91
392.91
392.91
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
161
161
161
4.123
4.123
4.123
374.99
374.99
374.99
1206.24
1206.17
1206.09
392.91
392.91
392.91
1599.14
1599.07
1599.00
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
141
Chen
Table 2 (Continued)
st
Q

A1
E  mCP
 
E  rCP
E  tCP
.16
.20
.24
161
161
161
4.121
4.123
4.124
375.12
374.99
374.86
1206.17
1206.17
1206.17
1599.07
1599.07
1599.07
hr
Q

A1
 
392.91
392.91
392.91
E  mCP
E  rCP
E  tCP
.144
.180
.216
161
161
161
3.976
4.123
4.269
386.76
374.99
363.22
1208.32
1206.17
1204.01
1601.22
1599.07
1596.92
hm
Q

A1
 
392.91
392.91
392.91
E  mCP
E  rCP
E  tCP
.144
.180
.216
161
161
161
4.123
4.123
4.123
374.99
374.99
374.99
1208.32
1206.17
1204.01
1601.22
1599.07
1596.92
ht
Q

A1
 
392.91
392.91
392.91
E  mCP
E  rCP
E  tCP
.288
.360
.432
161
161
161
4.133
4.123
4.113
374.03
374.99
375.93
1206.15
1206.17
1206.15
1599.06
1599.07
1599.06
S
Q

A1
 
392.91
392.91
392.91
E  mCP
E  rCP
E  tCP
0.08
0.10
0.12
161
161
161
4.123
4.123
4.123
376.51
374.99
373.47
1204.76
1206.17
1207.57
1599.07
1599.07
1599.07
H
Q

A1
 
394.31
392.91
391.50
E  mCP
E  rCP
E  tCP
.288
.360
.432
161
161
161
4.123
4.123
4.123
376.15
374.99
373.84
1205.09
1206.17
1207.24
393.98
392.91
391.84
1599.07
1599.07
1599.07
 
 
 
 
 
 
 
 
 
 
 
 
5. Conclusions
The sensitivity analysis shows that the selling price per unit and the mean of the
demand of customer have a major effect on the manufacturer’s expected profit,
retailer’s expected profit, and expected profit of the supply-chain system. Hence, the
exact estimation on these parameters always requires to avoid any incorrect outcome.
The quality investment can promote the expected profit for all of the membership in
the supply chain system. The managerial implication of this paper is that the joint
optimization of quality investment, commission fee, and order quantity management
can improve the product/service quality shipped to the customer and increase the
expected profit of the supply chain system. Hence, the appropriate quality investment
policy can maintain the strong relationship for the manufacturer, retailer, and customer.
It can bring more benefit for them.
In the present paper, we present a modified Chen and Liu’s (2008) model with
consignment policy, imperfect quality, and quality investment, which may be
considered as a generation of the original Chen and Liu’s (2008) model. The order
quantity, per unit commission, fixed commission, and quality investment are
simultaneously determined in the modified model. The extension to the modified Chen
and Liu’s (2008) model with multi-retailers and multi-attribute quality characteristics
may be left for further study.
142
Chen
References
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143
Chen
Appendix
Notations
the per-unit commission paid by the manufacturer to the retailer

Q
the order quantity by the retailer
the per-unit carrying cost for the retailer
hr
the goodwill loss per unit of stock-out for the retailer
sr
X
the stochastic demand of customer having an uniform distribution,
1
1 

X ~ U  x   x ,  x   x 
2
2 

x
x
A1
R
c
ht
st
hm
sm
H
S
the mean demand of customer
the variability of customer demand
the fixed commission fee paid by the manufacturer to the retailer
the per-unit retailer price
the per-unit manufacturing cost/ the variable production cost per unit
the per-unit carrying cost in the supply chain system
the goodwill loss per unit of stock-out in the supply chain system
the per-unit carrying cost for the manufacturer
the goodwill loss per unit of stock-out for the manufacturer
the carrying cost per unit for the retailer in the traditional production system
the goodwill loss per unit of stock-out for the retailer in the traditional
production system
the known process mean of quality characteristic of product
0
0
T (
T
Y
y0 ) the target value of process mean for product quality characteristic
y
the improved mean of quality characteristic of product
y
the improved standard deviation of quality characteristic of product
the known process standard deviation of quality characteristic of product
the target value of process standard deviation for product quality characteristic

the quality characteristic of product, which is assumed that Y ~ N  y ,  y2

I
1
the quality improvement
the parameter of declining exponential function for the improved process
standard deviation
the parameter of declining exponential function for the improved process
1
mean.
L
the lower specification limit of product quality characteristic
U
the upper specification limit of product quality characteristic
Loss(Y ) the quality loss per unit, LossY   k Y  y0 2
k
the quality loss coefficient
the cumulative distribution function of standard normal random variable
()
the probability density function of standard normal random variable
 ()
W
the selling price per unit for the conformance product in the traditional
production system
Sp
the discounted price per unit for the non-conformance product scrapped in the
traditional production system
144
Chen
b
i
the constant production cost per unit
the inspection cost per unit
the manufacturer’s profit for consignment policy
 mCP
E  mCP  the manufacturer’s expected profit for consignment policy
 rCP
E  rCP 
 tCP
E  tCP 
 mPS
E  mPS 
 rPS
E  rPS 
the retailer’s profit for consignment policy
the retailer’s expected profit for consignment policy
the supply chain system’s profit for consignment policy
the supply chain system’s expected profit for consignment policy
the manufacturer’s profit in the traditional production system
the manufacturer’s expected profit in the traditional production system
the retailer’s profit in the traditional production system
the retailer’s expected profit in the traditional production system.
145