American Journal of Oper ations Research, 2011, 1, 84-99
doi:10.4236/ajor.2011.13012 Published Online September 2011 (http://www.SciRP.org/journal/ajor)
Copyright © 2011 SciRes. AJOR
Two-Stage Ordering Policy under Buyer’s
Minimum-Commitment Quantity Contract
Hsi-Mei Hsu, Zi-Yin Chen
Department of Industrial Engineering & Management, National Chaio Tung University,
Hsinchu, Chinese Taipei
E-mail: hsimei@cc.nctu.edu.tw, yahuilin2@ydu.edu.tw
Received July 1, 2011; revised July 22, 2011; accepted August 16, 2011
Abstract
In this paper we consider a two-stage ordering problem with a buyer’s minimum commitment quantity con-
tract. Under the contract the buyer is required to give a minimum-commitment quantity. Then the manufac-
turer has the obligations to supply the minimum-commitment quantity and to provide a shortage compensa-
tion policy to the buyer. We formulate a dynamic optimization model to determine the manufacturer’s two
stage order quantities for maximizing the expected profit. The conditions for the existence of the optimal so-
lution are defined. And we also develop a procedure to solve the problem. Numerical examples are given to
illustrate the proposed solution procedure and sensitivity analyses are performed to find managerial insights.
Keywords: Two Stages Ordering, Commitment, Bayesian Information Updating
1. Introduction
In this paper we study a two-stage component ordering
problem with a buyer minimum-commitment quantity
contract. Under the contract, the buyer is required to
commit a minimum order quantity 1
and for returning
the buyer’s commitment the manufacturer has the obli-
gations to supply the minimum-commit quantity 1
and
to give shortage compensation if the manufacturing sup-
ply level is under

1
1
where
is a shortage
compensation range coefficient 01
. Because of
the presence of the long lead time of key components,
the manufacturer has two opportunities to place his order
to supplier before the buyer’s demand realized.
The buyer’s real demand X is uncertain following a
normally distribution (0
N
,2
0
) where 0
is uncertain
having a normal distribution N(
,2
1
). When the
manufacturer makes his first order quantity (1) decision
at stage 1, the unit cost of key component at stage 1
is known but the unit cost at the stage 2 is uncer-
tain. The possible values of and their corresponding
probabilities are known, denoted as
q
(1)
c
(2)
(2)
C
C

(2)(2) (2)(2)
12
,,,
n
Ccc c and respe-
ctively. After receiving the buyer’s minimum-commit
quantity 1

12
,,,
n
Ppp p
, the manufacturer uses 1
as an estimator
of
and places his first order quantity (1) to his sup-
plier. At stage 2 the marketing department provides an
observation 2
q
of X. The posterior distribution of X is
defined by the observations of 1
and 2
. Then manu-
facturer places his second order quantity 2 if necessary.
The time events of key component procurement process
are shown in Figure 1.
q
We assume that the outputs are mainly limited by the
available amounts of the key component, and the pro-
duction cycle times are very short that can be neglected.
After receiving the key components, the manufacturer
produces the products immediately. Products are deliv-
ered to the buyer at the end of period (immediately after
second stage). Due to the demand uncertainty, the manu-
facturer is difficult to determine two stage order quanti-
ties.
In this paper we develop a two-stage dynamic optimi-
zation model to decide the order quantity of a key com-
ponent under a buyer’s minimum-commitment quantity
contract. The model is formulated to maximize a manu-
facturer’s profit. The following costs are considered in
the model. 1) Key component unit cost: the unit cost of
key component at stage 1 is and the unit cost at
stage 2 is . 2) Holding cost: two kinds of inventories
are considered. One is buyer responsible inventory,
which only exists in the case of buyer’s real demand (x)
below the minimum guaranteed quantity 1
(1)
c
(2)
C
. In this case,
customers only take away real demand x, and the re-
maining products (1
x
) are buyer responsible inventory
H.-M. HSU ET AL.
85
Figure 1. Time events of key component procurement process.
and will be paid in the near future. The unit holding costs
of buyer responsible inventory are the interest and in-
surance. The other is manufacturer responsible inventory.
The unit holding cost of manufacturer responsible in-
ventory is the interest, insurance and obsolete costs. The
holding costs of buyer responsible inventory and manu-
facturer responsible inventory are 1h and 2h
c respec-
tively and 21hh
. 3) Shortage cost: two kinds of
shortage cost are considered according to whether or not
to pay shortage compensation. If the manufacturing out-
put level is below
c
cc
1

1
, then there are two shortage
types may occurred. The one includes general shortage
cost and the compensation cost. The other is only the
general shortage cost.
The buyer’s minimum commitment and demand fore-
cast updating in this paper belong to the category of
minimum purchase commitment contract [1] and inven-
tory management with demand forecast updates respec-
tively [2]. Durango-Cohen and Yano [3] pointed out that
increasing the level of commitment and information
sharing will lead to the cost down of entire supply chain.
Nowadays minimum commitment quantity contracts are
commonly used in electronic industry. Anupindi and
Bassok [1] classified the contract of quantity commit-
ments and flexibility as three types. The first type is the
total minimum quantity commitment contract. The sup-
ply contract with total minimum quantity commitment is
that a buyer gives his supplier a minimum ordering
quantity commitment, and the supplier offers the buyer a
discount price in return for the buyer’s commitment [4].
The second type is the total minimum dollar volume
commitment contract. This contract is similar to the total
minimum quantity commitment contract, but a buyer
commits to a minimum business on the basis of dollar
volume [5], and the supplier offers discounts based on
the commitment of dollar volume. The third type is the
periodical commitment with flexibility contract. Under
such a contract, a buyer receives discounts for commit-
ting to purchase in advance, and the buyer is allowed to
update his order amount in the rolling horizon basis. The
rolling horizon flexibility (RHF) contract [6-8] is one
kind of the third type. The RHF contract means the buyer
has a “limited” flexibility to update his advance order
after he commits to purchase certain quantity.
Gallego and Ozer [9] and Sethi et al. [2] classified the
inventory information with demand updating problems
as three types. The first type is the Bayesian analysis.
This approach learns about further demand from the past
history [10]. Dvoretzky et al. [11] first analyzed Bayesian
models in the inventory problem. In this type, specific
classes of demand distribution were discussed, such as
exponential family of distribution [12], gamma family
[13,14], negative binomial distribution [15], uniform-
Pareto distribution [16] and normal distribution [17,18].
The second type is time-series models used in updating
demand forecast, where they assume a correlation exists
in the demand realization and construct the demand as a
time-series model [10,19]. The third type is concerned
with forecast revisions, such as Markovian forecast revi-
sions model [20-22], single-period, two-stage ordering
problem with demand forecast updating [23-26] and
multiple period ordering problem with demand forecast
updating [27,28]. A more comprehensive discussion can
be found in [2].
Our study differs from the previous papers because we
consider a shortage compensation policy for reducing
demand uncertainty. Under the buyer’s minimum-com-
mitment quantity contract and shortage compensation
policy, two kinds of inventory and shortage costs are
respectively formulated in the two-stage dynamic opti-
mization model to decide the optimal order quantities.
In this paper, we define the conditions for the exis-
tence of the optimal solution and also develop a proce-
dure to determine the two-stage optimal order quantities.
The rest of this paper is organized as follows: Section 2
states the assumptions and notations for the proposed
model; the two-stage dynamic optimization model and
the solution procedure are proposed. Section 2 illustrates
Copyright © 2011 SciRes. AJOR
86 H.-M. HSU ET AL.
some numerical examples, and sensitivity analyses of the
major parameters of the model are performed. Finally,
Section 4 concludes this article.
2. Problem Formulation
2.1. Notations
1
: buyer’s minimum-commitment quantity. 10
: shortage compensation range coefficient 01
.
[1
,

1
1
] is the shortage compensation range.
(1
c): unit ordering cost of key component at stage 1.
(2)
C: unit ordering cost of key component at stage 2 is a
random variable, and the corre-
sponding probability .
(2)(2) (2)(2)
12
,,,
n
Ccc c
{,, ,}
n
Ppp p
(1)
pc
12
: product unit selling price. . p
2
: the demand observation at stage 2.
1h
c
c: unit holding cost of buyer responsible inventory.
2h: unit holding cost of manufacturer responsible in-
ventory.
1
s
c: unit shortage compensation cost;
10
s
c
2
s
c: unit general shortage cost;
12ss
0cc
X
: buyer’s real demand, realization is denoted by
x
.
(1)
X
: (1)
X
is a random variable to forecast the buyer’s
demand at stage 1, (1)
1
X
X
.
(2)
X
: )(2
X
is a random variable to forecast the buyer’s
demand at stage 2, (2)
12
,XX
.
1(.)
f
: probability density function (pdf) of (1)
X
,

2 2
10 1
,XN
(1)

 
2(.)

f
: probability density function (pdf) of (2)
X
,


(2)22 222 2
12 01010 1
22222
010 01
,
XN
 



1(.)
F
: cumulative density function (cdf) of. (1)
X
2(.)F: cumulative density function (cdf) of (2)
X
.
(.)
: standard normal probability density function.
(.): the cumulative distribution function for standard
normal distribution.
1(.)
(.)
: inverse function of . (.)
: the standard linear loss function:
()()d ()
a
axa
 
x.
Decision variables:
1
q
q: order quantity at stage 1.
10q
0q
2: order quantity at stage 2.
2
Intermediate variables:
1: the decision space defined by 1
112
(1 )qq


(1 )qq .
2
: the decision space defined by 112
.
11
q: optimal order quantity in 1
at stage 1
21
q: optimal order quantity in 1
at stage 2
12
q: optimal order quantity in 2
at stage 1
22
q
: optimal order quantity in at stage 2
2
1
q
: optimal order quantity at stage 1
2
q
: optimal order quantity at stage 2
2.2. Problem Assumptions and Formulation
The mathematical model is formulated to determine the
two stage ordering quantities of the key component for
maximizing profit. The buyer’s real demand X is uncer-
tain to be assumed following a normal distribution with
an uncertain mean 0
and a given variance 2
0
, where
0
follows N(
,2
1
) with an unknown
and a given
variance 2
1
. At stage 1, after receiving the buyer’s
minimum commitment quantity 1
, the manufacturer
uses 1
as an estimator of
. The posterior distribu-
tion of X after receiving 1
at stage 1 is denoted as
(1)
1
X
X
where
(1)2 2
110
,XXN
1

  (1)
At stage 2, the marketing department collects informa-
tion 2
about buyer’s real demand. We call it as an
observation of X. The posterior distribution of X at stage
2 is denoted as (2)
12
,XX
where
(2) 2
1222
,,XX Nk

(2)

222222
2 1201 0101
k
 

(3)

222222
201001

 (4)
Because the manufacturer has the obligation to pro-
vide the minimum-commitment quantity 1
to the buyer,
the total order quantities 12
must be larger than 1
qq
.
Now we will formulate the expected profit function and
use a backward dynamic programming to determine the
optimal 1
q
and 2
q
.
We formulate the problem as a dynamic programming
(DP) problem. For the DP formulation, the ordering
times are given as the stages, stage 1 and stage 2. Deci-
sion variable for stage n (n = 1, 2) is the ordering quan-
tity n. The profit at the current stage depends upon the
current decision and the ordering quantity in the
preceding stage
q
n
q
1n
q
. We set states for each stage n as
1n
q
.
With the backward solving procedure, first we should
determine the optimal order quantity 2 at stage 2 in a
given state 2
q
1
s
q
. We denote the expected profit func-
tion at stage 2 as
(2)2 1
Eqq
. The state at stage 1 is
known as 10s
. The expected profit function at stage 1
is denoted as
1
q
(1)
E
, where

(1)
(1)11(2)2 1
EqEcqEqq




(5)
Copyright © 2011 SciRes. AJOR
H.-M. HSU ET AL.
Copyright © 2011 SciRes. AJOR
87
The optimal expected profit of the manufacturer is de-
termined as follows:
lows:
1) Expected products sales



1
1
(1) 1(1) 1
0
(1)
1(2) 21
0
max
max
q
q
Eq Eq
EcqEqq

 






(6)

122
min, d
X
p xqqfxx 
(9)
2) Ordering costs when
(2) (2)
Cc
where (2)
2
cq (10)
2
(2)21(2)21
0
max
q
Eqq Eqq

 

(7) 3) Expected holding costs:
The unit holding cost of buyer responsible inventory is
1h and holding cost of manufacturer responsible in-
ventory is . The holding costs can be expressed as
follows:
c
2h
c
The items considered in
(2)2 1
Eqq
are expected
product sales, ordering costs, expected holding costs and
expected shortage costs.

(2)2 1Expected revenues
Expected costs
Expected product sales
Ordering costsExpected holding costs
Expected shortage costs
Eqq




(8)

112 1211
2121 1
for
for
0othe
hh
h
cxcqq x
cqqx xqq

 
2
rwise
 
(11)
The expected holding costs can be formulated as fol-
lows:
The relevant items are formulated respectively as fol-
 

1121212
max,0 max,0d
hh
X
cxcqqxf


xx (12)
4) Expected shortage costs:
The decision space of total order quantity 21
qq
can
be divided into two subspaces 1 and , 1
2
: total
order quantity (2
) is less than
1
qq
1
1
2
qq
, as shown
in Figure 2(a); 2: total order quantity (1) is lar-
ger than
1
1
, as shown in Figure 2(b). In 1
, two
kinds of shortage types may occur: 1) the shortage oc-
curred between and
1
qq2
1
1
belongs to short-
age compensation range, its unit shortage costs are the
shortage compensation cost and general shortage cost; 2)
the others does not belong to shortage compensation
range, its unit shortage cost is general shortage cost. The
unit shortage costs of shortage compensation and general
shortage are 1
s
c and 2
s
c respectively, 12
s
s. In 2
cc
,
the shortage compensation cost does not occur. Two
cases of shortage can be expressed as follows:

 


112 12
11112 211
for 1,
expected shortage costs in 1 + 1for 1,
0ot
s
ss
cxqqqq x
cqqcx
1
herwise.
x
 
 

 


(13)
 
212 112
2
for 1,
expected shortage costs in
0 otherwise.
s
cxqqqq x

 



(14)
Expected shortage cost can be combined as follows:
 


 

11122121
maxmin1,,0maxmax, 1,0d
ss
cxqqcxqq
 
 

2
fxx (15)
(2)2 1
Eqq
depend on the domain of the total order
quantity 21
qq
, i.e.,
112112 1
2.3. Optimal Solution
,1qqqq

  
and
,1q q
11
qq
212 1
To solve the DP problem we first provide the optimal de-
cisions for stage 2 under a given state 21
s
q. As men-
tioned above the expected shortage costs in
, each domain cor-
responds to a shortage cost equation respectively (Figures
(a) and (b)). Therefore
(2)2 1
Eqq
is formulated 2
H.-M. HSU ET AL.
Copyright © 2011 SciRes. AJOR
88
(a)
(b)
Figure 2. (a) in : total order quantity q1 + q2 is less than
1
1
1
; (b) in 2
: total order quantity q1 + q2 is larger than
1
1
.
for each domain as follows:


 
  






112 1
12 1
112
112
1
1
12 1
(2)211 22122112
,
212122122
1
(2)
1122212 2
1
d()dd
dd
d1d
qq
h
qq qq
qq
hh
ss
qq
Eqqpfxxpxfxxpqqfxxc xfx
cqqfxxcqqxfxx
cxqqfxxcxfxxcq
 




  





 
 
 


dx
(16)
and


  
 

112 1
12 2
112
112
1
12
(2)211 22122112
,
21212212 2
(2)
2122 2
d()d d
dd
d
qq
h
qq qq
qq
hh
sqq
Eqqpfxxpxfxxpqqfxxcxfx
cqqfxxcqqxfxx
cxqqfxxcq
 

  




 

 

dx
(17)
We will show


12 1
(2)2 1,qq
Eqq 


and


12 2
(2)2 1,
qq
Eqq 


are both concave functions.
Then we can determine the optimal in the interval
for the two cases respectively.
2
q
[0, )
Proposition 1. If and
hold for, then
12
0
sh
pc c22
0
sh
pc c
2[0, )q


12 1
1,
qq
q
(2)2
Eq
and


2

12
(2)2 1,
qq
Eqq
are concave functions, i.e.,


12 1
,0
qq
2
(2)2 1
2
2
dEq q
dq

 and


12 2
2
(2)2 1,
2
2
0
qq
dEq q
dq



for .
2[0, )q
Proof. See appendix A.
Proposition 2. Maximizing


12 1
(2)2 1,qq
Eqq


and

12 2
(2)2 1,qq
Eqq


with respect to , we can
2
q
get the optimal order quantity 2 denoted as 21
q q
and
22
q
for the two domains respectively as follows:

 


 


221
202 11
221
12021
11
21
221
202 11
11
221
202 11
max 0,
if 1
max0,1
if 1
max0,
if
ktq
kt
q
q
kt
q
kt

1
 

 
 
 
 

 
 
,
where


(2)
1121 2
1
sh
tpcFc pcc

 1
s
(18)
H.-M. HSU ET AL.
89


 


 
11
221
12022
22 221
202 21
221
202 21
max0,1
if 1
max 0,
if 1
q
kt
qktq
kt

1
 

 

 
 
 
,
where


(2)
22 2
sh
tpcc pcc 2
s
(19)
Proof: See appendix B.
Then we provide the optimal solution for stage 1. At
stage 1, the profit functions


12 112 1
(1)
11(2)211
(, ),
qq qq
qcqEqq




and


12 212 2
(1)
11(2)221
(, ),
qq qq
qcqEqq




correspond to 1
and 2
respectively. Due to 2
and being uncertain, the sample space of is
k
)(2)
C(2
C
(2)) (2(2)
12 ,
n
c
,p
(2
Cc
12
{,Pp
)
,,c
, }
n
p with respect to the probability
, and the distribution of is
2
k
422
110 1
,N
  
. Let

12 1
12 1
(1)1 (, )
(1)
1(2) 211
(,)
qq
qq
Eq
EcqE qq





 

and

12 2
12 2
(1)1 (, )
(1)
1(2) 221
(, )
qq
qq
Eq
EcqE qq





 

be the expectation of
12 1
1(, )
qq
q
and
12 2
1(, )
qq
q
respectively, that are formulated as fol-
lows:
 





1
113 1
2
12 1
12 1
12
12 1
113 1
()
(1)1( 2)2122
,
(, )1
(1)
(2)21221
,
()
0, d
0,d
nqdd t
iK
qq
qq i
K
qq
qdd t
EqpEqqfkk
Eqqfkkc


 



 

 



q
(20)
 





11
3 2
2
12 2
12 2
12
12 2
113 2
()
(1)2 1(2)2122
,
(, )1
(1)
(2)212 21
,
()
0, d
0,d
qd
d t
iK
qq
qq i
K
qq
qdd t
EqpEqq fkk
Eqq fkkc


 



 

 



1
n
q
(21)
where
















12 1
22 22
(2)211 20 2120 2
,
22 1
21 021
22 22
12021202
22
12112021
(2)2 2
1211202 2
(2)
0,
1
11
11
hh
qq
hs
ss
hh s
hh s
h
Eqq pcck
pc ct
cc k
pc cck
ccc ckk
cc


 
 
 




 
 





 










2222 1
2112020 21
22 (2)
1112021
11
11 ,
s
s
ck
ckcq

 





 


t
(22)



1
(2)
121 2
11
s
hs
tpcFc pcc


  

Copyright © 2011 SciRes. AJOR
90 H.-M. HSU ET AL.















12 1
22 22
(2)2121021 202
,
22 22
12 021202
22 22
12021202
22
1211202
22
2112021
1
0,
1
11
11
11
hs
qq
hh
ss
hh s
hs
s
Eqq pccqk
pc ck
cc k
pc cck
cck q
c
 

 
1

 


 

 
 








22
1202112
,
h
kc


 


k
(23)













12 2
22 22
(2)211 20 2120 2
,
22 1
22 022121
(2)(2)2 21(2)
1222 0221
(2)
22 22
0,
,
hh
qq
hs hh
hh h
shs
Eqq pcck
pcctpc c
ccckcctcq
tpcc pcc

 



 

 
 
 
(24)







12 2
22 22
(2)212 20 2120 2
,
22 22
12 021202
1211221
0,
.
hs
qq
hh
hhh h
Eqq pccqk
pc ck
pccckcq

 





 
(25)
Proof: See appendix C.
We will show and


12 1
(1)1 ,qq
Eq 




12 2
(1)1 ,qq
Eq 

 are both concave functions. Then
we can search for optimal ordering quantity for each
domain, expressed as (,
11
q
21
q
) and (, ) respec-
tively.
12
q
22
q
Proposition 3. If 12
0
sh
pcc
 and
22
0
sh
pc c
hold for1[0, )q
, then and


12 1
(1)1 ,qq
Eq 



12
1,qq
Eq 2
(1)


are concave functions of .
1
q
where




12 1
2221
(1)1 ,102 12
124
2
1
11
n
qq
ish
i
Eq qt
ppccB
qB




 









B
(26)




12 2
2221
(1)1 ,102 22
224
2
1
11
n
qq
ish
i
Eq qt
ppccB
qB




 









B
(27)


422
120
144222
11020
,B
 




4222
111 1020
242222
11010
,
q
B
 

 
 
2


42222 22
1111020
342222
11020
,
q
B


 



2
23 1
22 2
101
4222
12 0
.
2π
B
BB
B
Be

 
Proposition 4. If (2)(1) 0cc
where
(2) (2)
1ii
i,
n
cpn
c
1, ,i
0
, then at stage 1 the optimal
order quantity 11
q
and .
12 0q
Proof: See appendix D.
Proposition 5. There exists an optimal ordering quan-
tity for each domain respectively, and the optimal order-
ing quantity (11
q
, 21
q
) and (, ) can be determined
12
q
22
q
Copyright © 2011 SciRes. AJOR
H.-M. HSU ET AL.
91
by the following procedure:
Step 1: At stage 1, we find , such that
11
q
12
q


12 111
(1) 11
,0
qq q
Eq q




and


12 212
(1) 11
,0
qq q
Eq q




,
where












12 1
1
113 1
2
2
(2) 221422
(1)1111021110 1
,1
()
(2) (1)22
12120222
(2)
11 121
d
1,
n
is
qq i
qdd t
sh K
sxh s
Eqqppcc qt
cc pccqkfkk
tpcFc pcc
 









 



 


 

 

(28)











12 2
1
113 2
2
(2) 221422
(1)1121022110 1
,1
()
(2) (1)22
2212 0222
(2)
22 22
d
n
is
qq i
qdd t
sh K
shs
Eqqppcc qt
cc pccqkfkk
tpccpcc













 


 
 
(29)
then and can be derived as follows:
11
q
12
q
1111 1
min, 1qq

(30)
and
12 12
qq
(31)
Step 2: At stage 2, and can be derived by
Proposition 2 as follows:
21
q
22
q


 



 



221
202 111
221
12021
111
21
221
202 11
111
221
202 11
max0,
if 1
max0,1
if 1
max0,
if
ktq
kt
q
q
kt
q
kt

1
 

 
 

 
 

 
 
,
where




(2)
1121 2
1
s
tpcFc pcc

 1hs
. (32)



 

 
112
221
120221
22 221
202 212
221
202 21
max 0,1
if 1
max 0,
if 1
q
kt
qktq
kt

 

 


 
 
 
,
where

(2)
22 2sh
tpcc pcc 
Step 3:
After determining optimal ordering quantity (11
q
, 21
q
)
and (12
q
, 22
q
) for each domain, the optimal order quan-
tity 1
q
and 2
q
can be derived as follows:





111
12
1(1)11
(1)11 12
max ,
qq
q
qArgE qqq
Eq qq



11





(34)
211 11
2
221 12
, if
, if
qqq
qqqq


(35)
In next section we will demonstrate the proposed pro-
cedure with some given numerical examples and the sen-
sitivity analysis.
3. Computational Study
3.1. Numerical Examples
Three examples are presented to demonstrate the pro-
posed solution procedure. The relevant parameters are
shown in Table 1. The optimum 1, 2 and the cor-
responding expected profit for each example are also
shown in Table 1.
qq
Example 1. Suppose the buyer demand follows a nor-
mal distribution with the standard deviation terms 03
,
15
, and buyer’s minimum-commitment quantity (1
)
is 30 and
is 0.1, that is, shortage compensation range
is (30, 33). The demand observation at the second stage
(2
) is 33, and the other relevant parameters are given as
follows: product unit selling price p is 100, the unit cost
of key component at stage 1, = $30, there is a 70%
(1)
c
2s
.
(33)
Copyright © 2011 SciRes. AJOR
92 H.-M. HSU ET AL.
Table 1. Examples.
Example 1 Example 2 Example 3
0
3 3 3
1
5 5 5
p
100 100 100
1
c 30 30 30

(2)
11
0.7cp 40 40 40

(2)
22
0.3cp 20 20 20
1
30 30 30
0.1 0.4 0.1
2
33 33 38
1h
c 10 10 10
2h
c 15 15 15
1
s
c 15 15 15
2
s
c 10 10 10
Fitted domain 2
1
2
Optimal Solution
**
12
,qq
1
q
= 27.1216
2 = 5.8784 q
()
(2) 40c
2 = 7.3876 q
()
(2) 20c
1
q
= 27.2491
2
q = 5.7159
()
(2) 40c
2
q = 7.3787
()
(2) 20c
1
q
= 27.1216
2 = 9.3574 q
((2) 40c
)
2 = 11.0641q
((2)20c
)
Optimal Profit 2084.91. 2079.22 2220.19
chance that the ordering cost at stage 2 will 40 and a
30% that the ordering cost at stage 2 is 20 ((2)
140c
,
1, , 2), per unit holding cost of
buyer responsible inventory (1h) = $10, per unit holding
cost of manufacturer responsible inventory (2h) = $15,
per unit shortage compensation cost (
0.7p(2)
220c0.3pcc
1
s
c) = $15 and per
unit general shortage cost (2
s
c) = $10.
With the proposed solution procedure, in step 1 we
find that and , then
11 27.3127q12 27.1216q

11 1
min27.3127, 13327.3127q




and . In step 2, if at stage 2,
12 27.1216q(2)
140c
 


221
12021
1
30 32.4876
133
kt



 
and
 


221
202 2
1
30 32.8025
133
kt



 
,
then
21 max0,32.487627.31275.1749q
and
22 max0, 3327.12165.8784q.
If (2)
220c
at stage 2,



221
12021
1 3334.0793kt
 
 
and



221
12022
1 3334.5092kt
 
 ,
then
21 max0,3327.31275.6873q
and
22 max0, 34.509227.12167.3876q.
In step 3, we get







111
12
1
1(1)11
(1)11 12
max ,
max2081.85, 2084.9127.1216,
qq
q
q
qArgE qqq
Eq qq
Arg



11






2 (if 5.8784q(2)
14
0
c
) or 2 (if 7.3876q(2)
220c
)
and optimal expected profit is 2084.91.
Example 2. In example 1 the value of
is changed
from 0.1 to 0.4 while other parameters remain unchanged.
With the proposed solution procedure, in step 1 we find
that 11
q
= 27.2491 and 12
q
= 27.1216, then
11 1
min27.2491, 14227.2491q




and 12
q
= 27.1216. In step 2, if at stage 2,
(2)
140c
 


221
12021
1
30 32.965
142
kt



 
and
 


221
202 2
1
30 32.8025
142
kt



 
,
then
21 max0,32.96527.24915.7159q
and
22 max 0, 4227.121614.8784q.
If (2)
220c
at stage 2,
Copyright © 2011 SciRes. AJOR
H.-M. HSU ET AL.
93



221
12021
1
30( )34.6278
142
kt




and


221
202 21
34.5092142kt
 
 
,
then


21 max0,34.627827.24917.3787q
and


22 max 0, 4227.121614.8784q
.
In step 3, we get







111
12
1
1(1)11
(1)1112
max ,
max2079.22,1846.8127.2491,
qq
q
q
qArgE qqq
Eq qq
Arg








11
2 (if ) or 2 (if 5.7159q(2)
140c7.3787q(2)
220c
)
and optimal expected profit is 2079.22.
Example 3. In example 1 the value of 2
is changed
from 33 to 38 while other parameters remain unchanged.
With the proposed solution procedure, in step 1 we find
that = 27.4702 and = 27.1216, then
11
q12
q

11 1
min27.4702, 14227.4702q



and = 27.1216. In step 2, if at stage 2,
12
q(2)
140c



221
12021
1 3335.7675kt
 
 
and



221
12022
1 3336.479kt
 
 ,
then


21 max0, 3327.47025.5298q
and


22 max0, 36.47927.12169.3574q
.
If at stage 2,
(2)
220c



221
12021
1 3337.3228kt
 
 
and



221
12022
1 3338.1857kt
 
 ,
then
21 max0,3327.47025.5298q
and
22 max0, 38.185727.121611.0641q.
In step 3, we get







111
12
1
1(1)11
(1)1112
max ,
max2131.75, 2220.1927.1216,
qq
q
q
qArgE qqq
Eq qq
Arg












11
2 (if 9.35 74q(2)
140c
) or 11.0641 (if (2)
220c
) and
optimal expected profit is 2220.19.
3.2. Sensitivity Analysis
2
is an observation of
22
10 1
,N

at stage 2. At
stage 1 we don’t know what value of 2
will be ob-
served. With Monte Carlo method we randomly generate
100 values of 2
from
22
10 1
,N

, denoted as ()
2
i
,
1, ,100.i
()
1
i
q With the proposed solution procedure, we
can find
with respect to ()
2
i
. Then we evaluate the
average expected profit value for using in 100 val-
ues of
()
1
i
q
()
2
i
. The steps of Monte Carlo method are stated
as follows:
Step 1: Randomly generate ()
2
i
, from
1, ,100i
22
10 1
,N

.
Step 2: With each ()
2
i
we can find by Proposi-
tion 5,
()
1
i
q
1, ,100i
respectively.
Step 3: Compute the expected profit values for using
()
1
i
q
in each ()
2
i
1, ,100i
, denoted as
(1)( )
1
i
q
, ,
)( )
1
i
q(100
(excluding 2
, parameters are fixed in
()
1
ii
qi
() , 1,,100). Let

*(1)() (2)()(100)()
()111100,
1, ,100.
ii i
iqq q
i
 

 

Step 4: optimal profit and optimal order quantity 1
q
can be derived as follows:
()
1
** *
1(1) (2)(100)
max,, ,
i
q
qArg
.
The relationships among optimal expected profit, short-
age compensation range coefficient (
), and buyer’s
minimum-commitment quantity (1
) are shown in Table
1 with the related parameters given in the illustrating
example. And the relationships among optimal expected
profit, shortage compensation range coefficient (
) and
shortage compensation cost 1
s
c, are shown in Table 2
and Table 3.
In Table 2 we find that: 1) given 1
, we define the
optimal expected profit turning point of decreasing as
Copyright © 2011 SciRes. AJOR
94 H.-M. HSU ET AL.
Table 2. Relationships among optimal expecte d profit, short-
age compensation coefficient and buyer’s minimum-com-
mitment quantity.
1
20 25 30 35 40
0 1836.02 2074.59 2345.81 2653.83 2990.32
0.1 1836.02 2074.59 2345.81 2653.83 2990.32
0.2 1836.02 2074.59 2345.81 2653.83 2990.32
0.3 1836.02 2074.59 2345.81 2653.83 2990.32
0.4 1836.02 2074.59 2345.81 2648.21 2985.97
0.5 1836.02 2067.19 2337.45 2645.42 2982.73
0.6 1832.17 2064.98 2335.56 2643.19 2980.85
0.7 1829.75 2064.86 2335.53 2643.11 2980.44
0.8 1829.53 2064.67 2335.47 2643.1 2980.41
0.9 1829.49 2064.65 2335.47 2643.1 2980.39
1 1829.48 2064.65 2335.45 2643.1 2980.39
Table 3. Relationships among optimal expecte d profit, short-
age compensation coefficient and unit shortage compensa-
tion cost (1) (given 1s
c
= 30).
1
s
c
15 30 45 60 75
0 2345.81 2345.81 2345.81 2345.81 2345.81
0.1 2345.81 2345.81 2345.81 2345.81 2345.81
0.2 2345.81 2345.81 2345.81 2345.81 2345.81
0.3 2345.81 2345.81 2345.81 2345.81 2345.81
0.4 2345.81 2345.81 2345.81 2345.81 2345.81
0.5 2337.45 2321.75 2312.85 2299.03 2270.26
0.6 2335.56 2315.03 2301.06 2283.1 2264.03
0.7 2335.53 2303.37 2290.98 2270.67 2243.73
0.8 2335.47 2302.57 2287.84 2266.5 2229.04
0.9 2335.47 2302.48 2286.71 2260.49 2228.85
1 2335.45 2302.47 2286.7 2260.49 2228.84
1
, that is, when 1
, the optimal expected profit is
kept unchanged, but when 1
, the optimal expected
profit decreases accordingly, i.e., 20
= 0.5, 25
= 30
=
0.4, 35
= 40
= 0.3. The larger 1
is, the smaller 1
is. 2) The marginal optimal expected profit is increased
as 1
increases. In Table 3, Table 4 and Figure 3 with a
given 1
we find that when 1
(30
= 0.4, 40
=
0.3), the optimal total order quantity 12
q is over q
1
1
which causes the shortage compensation cost
can not be occurred. Hence, if 40
then the manu-
facturer can give the buyer a larger shortage compensa-
tion value of 1
s
c and take buyer’s increasing the value
of 1
. For example, if the original shortage compensation
coefficient
= 0.2 and 1
= 30, then we find 30
= 0.4,
the manufacturer can give
= 0.3 and an arbitrarily
large value of compensation cost 1
s
c to the buyer and
take the buyer increasing the value of 1
from 30 to 40,
then the optimal expected profit in the case of
= 0.3
and 1
= 40 will be larger than it in the case of
= 0.2
and 1
= 30.
Base on the above description, the management in-
sights observed from Table 2 to Table 4 can be con-
cluded as follows:
1) The more the buyer’s minimum-commitment quan-
tity 1
is, the more the expected profit of manufacturer
is.
2) There are some ways to induce the buyer to in-
crease the value of 1
:
a) From Table 2, we know the upper bound of
( 1
)
which the manufacturer can give to the buyer to attract
the buyer increasing the minimum commitment value of
1
without losing the expected profit.
b) If the optimal total order quantity 12
is larger
than
q
q
1
1
, then the manufacturer can give attractive
values of shortage compensation cost 1
s
c (Tables 3 and
4) and shortage compensation coefficient (
) (Table 2 to
Table 4 to take buyer increasing the value of 1
.
The manufacturer can provide alternatives for the
buyer as Table 5 to induce the buyer to increase the
value of 1
.
Table 4. Relationships among optimal expecte d profit, short-
age compensation coefficient and unit shortage compensa-
tion cost (1) (given 1s
c
= 40).
1
s
c
15 30 45 60 75
0 2990.322990.322990.32 2990.32 2990.32
0.1 2990.322990.322990.32 2990.32 2990.32
0.2 2990.322990.322990.32 2990.32 2990.32
0.3 2990.322990.322990.32 2990.32 2990.32
0.4 2985.972963.112940.85 2934.03 2930.33
0.5 2982.732956.392932.06 2909.27 2887.73
0.6 2980.852952.732926.98 2903.1 2880.74
0.7 2980.442951.932925.84 2901.67 2879.04
0.8 2980.412951.842925.71 2901.5 2878.85
0.9 2980.392951.832925.7 2901.492878.84
1 2980.392951.832925.7 2901.492878.84
Copyright © 2011 SciRes. AJOR
H.-M. HSU ET AL.
Copyright © 2011 SciRes. AJOR
95
Figure 3. Relationships among the expected profit, shortage compensation cost (1), shortage compensation coefficient (
s
c
)
and unit shortage compe nsati on cost (1).
s
c
Table 5. Alternatives for the buyer.
Alternative
Buyer’s
minimum-commitment
quantity 1
Shortage
compensation
coefficient
Shortage
compensation
cost 1
s
c
1 120
0.1
115
s
c
2 130
0.2
130
s
c
3 140
0.3
145
s
c
4. Conclusions
This paper proposes a two-stage dynamic optimization
model for an ODM CMOS camera module manufacturer
to determine its optimal order quantities to maximize
optimal expected profit based on buyer’s minimum-
commitment quantity contract and shortage compensa-
tion policy. The manufacturer can update the distribution
of buyer’s demand by collecting the market information,
and this situation is common and realistic for entrepre-
neurs in industry. In this paper, two kinds of inventories
and shortage costs that are taken into consideration, the
conditions for the two-stage optimal order quantities are
derived, and the solution procedure is proposed. Nu-
merical examples are to be illustrated and some man-
agement insights are provided. The upper bound of
(1
) which the manufacturer can give to the buyer to
attract the buyer increasing the minimum commitment
value of 1
without losing the expected profit, the
manufacturer can use the upper bound of
( 1
) to re-
define an attractive values of shortage compensation co-
efficient (
) and shortage compensation cost (1
s
c) to in-
duce the buyer to increase the value of 1
that can im-
prove expected profit of the manufacturer. It would be of
interest to extend the model to allow for the manufac-
turer having third or above order opportunity before the
buyer’s real demand occurred.
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Copyright © 2011 SciRes. AJOR
H.-M. HSU ET AL.
97
Appendix A
Proof of Proposition 1:


 
 

12 1
1
12
1
12
1
12
(2)212
,
222
1
(2)
22 12
dd
dd
dd
qq
h
qq
qq
hs
qq
Eqq q
pfxxc fxx
cfxxcfxxc









After rearranging the above equation, we have



 

12 1
(2)2 12
,
(2)
2121212 1
dd
1
qq
hs s
Eqq q
pcc FqqpcFc




 
and


12 1
22
(2)2 12
,
dd
qq
Eqq 

 q


21212
.
hs
pccfq q 
If hold for , then
12
0
sh
pc c2[0, )q


12 1
22
(2)212
,
dd0
qq
Eqq q



 .
Therefore


12 1
(2)2 1,qq
Eqq 

 is concave for
. The proof of
2[0, )q


12
(2)2 1,qq
Eqq2


is the
same as


12 1
(2)2 1,qq
Eqq


.
Appendix B
Proof of Proposition 2: For, let
1





12
(2)2 12
,2
21212
(2)
12 1
10
qq
hs
s
Eqq q
pcc Fqqp
cF c


 

 

If
 
221
120211
1kt
,

 

221
2120 21 1
max 0,qk t


q
,
where


(2)
1121 2
1
sh
tpcFc pcc

 1
s
.
If
 
221
202 1
1kt
1

 
1
,
we take the upper bound of

1
1
 instead of

221
202 1
kt

 , If



211 1
max0, 1qq


221
202 11
tk
 , we take the lower bound of
11
instead of

221
202 1
kt

 ,
211 1
max 0,q
q
. The proof of 2 is the same as 1
.
Appendix C
Proof of Equations (20)-(25): From (18), when

221
210 21
kq t

 , we can know ; when
21 0q

221
210 21
kq t

 , . For given
21 0q

Eq
(2)
1,qc
and , we can discuss with two
2
k

12 1
(1)1 ,qq


conditions: 2, 2
0q0q
and expected profit at stage
2 can be expressed as follows:

221
221 2
when 0,o
qkq t


1


 









1221
221
22
12 11
1
1
221 22 2
202 1
221
202 1
2
1
(2)211
,
221 221
202 111220211
221
22011
0,d d
dd
d
o
kt
qq
hh
kt
kt
h
Eqq pfxxpxfxx
Pktf x xcxf x xcktf x x
cktxfxxc



 


 
 
 



 





d



















1
2
221
202 1
2
1
1
221
12021
(2) 2212222
21 202111202120
1
22 12222
21 02112021202
12112
d
1d
1
11
s
kt
shh
hs ss
hh s
xktfxx
cx fxxcktqpcck
pcctc ck
pc cck




 



 
 
 
 
2











22
021
(2)2 2
1211202 2
(2)2 2221
211202021
22 (2)
1112021
11
11
11
hhs
hs
s
ccc ckk
ccc kt
ckcq

 

 




 



 



 


Copyright © 2011 SciRes. AJOR
98 H.-M. HSU ET AL.
21
when 0,q



 








11 11
22 22
12 1
11
1
11
22 2
1
1
2
1
(2)21112 1
,
1
211211 1
22 22
21 21021202
1
0,d ddd
dd d
1d
q
h
qq q
q
hhs
q
shs
h
Eqqpfxx pxfxxpqfxxcxfxx
cqfxxcqxfxxcxqfxx
cx fxxpccqk
pc
 



  
  


 

 
 

 
 













22 222222
12 02120212021202
22 22
12112021211202
22
11202112
1
11 11
11
hss
hh sh s
sh
cqkcc k
pcc ckcckq
ckck

  

  





 


1
Therefore










1
113 1
2
12 1
12 1
12
12 1
113 1
(1)1(2)212 2
,
,1
(1)
(2)21221
,
0, d
0, d
nqddt
iK
qq
qq i
K
qq
qdd t
EqpEqq fkk
Eqq fkkcq


 



 

 



By the same discussions, can be expressed as follows:


12 2
(1)1 ,qq
Eq 



221
22102
when 0,qkqt


2


 






 


1221
202 2
22
12 11
1
221 22
202 2
221
1202 2
2 2
1
(2)2112 2
,
221
202 211
221 221
22 022122 022
0,d d
dd
dd
kt
X
qq
h
kt
kt
hh
Eqq pfxxpxfxx
pktfx xcxfx x
ck tfxxck txfx



 
 

 
 

 


 


 


 x









 



2
221
202 2
221(2) 221
2 2202220221
2222221
12 02120222 022
(2)2 21(2)
121122220221
d
s
kt
hhhs
hh hhh
cxk tfxxcktq
pc ckpcct
pc cc cckcctcq

 
 




 
 
22
when 0,q


 
   




11
22 2
12 2
11
1
11
2222
1
1
(2)2 111
,
112112121
22 2222 22
220212 0212021202
0,d dd
ddd
+
q
qq q
q
hhh s
q
hs hh
Eqq pfxxpxfxxpqfxx
cxfxxcqfxxcqxfxxcxqfxx
pccq kpcck
p

d
 
 





 
  

 

12112 21hhh h
cc ckcq

Therefore
Copyright © 2011 SciRes. AJOR
H.-M. HSU ET AL.
Copyright © 2011 SciRes. AJOR
99









1
113 2
2
12 2
12 2
12
12 2
113 2
()
(1)2 1(2)2122
,
,1
(1)
(2)212 21
,
0, d
0, d
nqdd t
iK
qq
qq i
K
qq
qdd t
EqpEqq fkk
Eqq fkkcq


 



 

 



Appendix D In 2
, we redefine


122
(1) 111
,
qq
Eq qG

 
 q
,
if (2)(1) 0cc
,
(2) (1)
10Gq cc
 as 1
and
q
(1)
12 0
h
Gqc c
  as 1
q, by the interme-
diate value theorem, there exists a such

12
q[,]
Proof of Proposition 4:
In 1, we redefine

12 1
(1) 111
,
qq
Eq qG

 
 q
.
If (2)(1) 0cc,

(2) (1)
10Gq cc as
1
and as 1, by the interme-
diate value theorem, there exists a such
q that

12 2
(1) 11
,0
qq
Eq q



 . If (2)(1) 0cc
,

(1)
11 0
h
Gqc c q
11
q[,
10Gq
as 1 and as 1,
hence 1
q

10Gq q
0q
when (2) (1)
cc0
and this completes
the proof.
]
that


12 1
(1) 11
,0
qq
Eq q

 

 . If (2)(1) 0cc,

10Gq
1
q
as 1 and as ,
hence when
q

10Gq1
q
0(2) (1)
cc0.