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Apr. 2007, Volume 6, No.4 (Serial No.46)
Chinese Business Review, ISSN1537-1506, USA
Supply chain coordination with CCPP for a short-life-cycle product
LIU Bin, ZHANG Rong
(College of Information and Management Science, Henan Agricultural University, Zhengzhou 450002, China)
Abstract: Supply chain coordination for short-life-cycle products with price-dependent demand is a focus in
present operation management and management science fields. This paper considers the supply chain coordination
problem for a single-cycle and two-selling-stage short-life-cycle product. Firstly, the inventory model of whole
system is depicted with a stochastic dynamics programming. Then, a new combined contract CCPP (combined
contract with pending prices) is developed to coordinate the decentralized system, and its validity for channel
coordination is proved. Finally, the analysis on system performances is simulated with some examples.
Key words: supply chain coordination; short-life-cycle product; combined contract with pending prices
(CCPP)
1. Introduction
In the research on supply chain coordination, the prices of products are endogenetic variables for the setting,
and they are not in the range of retailer’s decision. In fact, this is an assumption for the perfect competitive market,
and selling prices of products are determined by supply-demand situation. However, in practices, due to limited
retailers in the similar products market, the products supplied by different retailers are always different in
performances, qualities, appearances, brands and after-selling services, especially for short-life-cycle products. So,
these products will come into different markets, and manufacturers and retailers are able to set the selling prices.
At the same time, for customers, though the functions of the same kind of products are very similar, customers can
select their objectives, and the selling prices of products will be one of important factors affected customers. To
sum up, retailers consider not only the selling prices of similar products, but also its competitive ability in relative
products. It is known that firms can get steady profits and the faithfulness from end customers with suitable
pricing. Namely, the selling prices of products will become one of key factors to affect the system performance of
supply chain, and how to set the optimal selling prices, which can bring the steady profits for channel agents and
the optimal performance for the whole system, will be a focus issue in operation management and management
science fields.
This paper will investigate the channel coordination problem for a short-life-cycle product under
price-dependent demand. This kind of product, such as mobile phones, notebook PC, is characterized with the
higher uncertainty in demand, the stronger style, and the dropping in selling prices due to the emergence of the
substitutable products. In fact, with the rapid growth in technology and the improvement of people’s living
This paper was partly supported by Natural Science Foundation of China (No. 70321001), Nature Science Foundation of Henan
Province Education Committee (No. 2006120004), the Ph.D. Science Research Foundation of Henan Agricultural University (No.
30700300), and the Post-doctor Research Foundation of China (No. 20060390280).
LIU Bin, Ph.D., associate professor of Department of Management Science, College of Information and Management Science of
Henan Agricultural University; research fields: supply chain management, grey systems.
ZHANG Rong, Ph.D. candidate of Department of Management Science, College of Information and Management Science of
Henan Agricultural University; research fields: supply chain management, grey systems.
1
Supply chain coordination with CCPP for a short-life-cycle product
standard, people have higher requirement for the varieties and specifications of commodities, and wish to get
better services and quality for productions. So, more and more merchandise can be characterized with
short-life-cycle, and their selling prices always drop with the selling periods like the mobile. Similar research was
considered by literature (LIU Bin, CHEN Jian & LIU Si-feng, 2006), but it only investigates the channel
coordination under price-independent demand. In relative literature on inventory coordination, Whitin (1955)
depicted the uncertain demand depended the selling prices for the first time, and considered the jointed decision
on ordering and pricing. Then, Mills (1959) extended Whitin’s model, and explained that the mean of the
uncertain demand was the function on selling prices. Recently, Petruzzi et al. (1999) reviewed the literatures on
the jointed decision, and depicted the jointed decision on ordering and pricing with the stochastic additive demand
and multiple demands. Literature considered the optimal inventory question under the single-cycle and two selling
periods, which was like our question, but they only dealt with the uncertain demand with independent selling price.
In relative literature on channel coordination, Kandel (1996) analyzed the channel coordination issue with
price-dependent demand, and developed a return policy with fixed wholesale price to coordinate the supply chain
system; Emmons et al. (1999) also considered the supply chain coordination with return policy under the
price-dependent demand, but a demand density function, g ( x, r) f ( x / D(r)) / D(r) , where D(r ) is decreased with
the selling price r , is applied in his assume. It was shown that return policy could make improve the performance
of agents on supply chain and the whole system. HA (2001) developed three policies or contracts to coordinate the
supply chain channel under stochastic and elasticity demand in the view of suppliers, fixed ordering quantity,
two-part linear pricing and return policy with fixed wholesales price. YAO (2002) studied a jointed decision on
ordering and pricing for a newsboy problem, and extended this decision model by single decision-maker to the
channel system consisted of single supplier and single retailer. She considered the effect of three classic contracts,
such as return policy, quantity flexibility and penalty scheme. Then, she explained the reasons why with them the
channel coordination could get, and designed a profit sharing contract to optimize the expected profit of whole
channel. CHEN (2004) studied channel coordination problem of a short-life-cycle product consisted of several
universal components and several individuation components under Bayesian information updating and
price-dependent demand, and she designed a two-dual disk compensatory contract to coordinate the system
channel.
LIU, et al. (2006) considered supply chain for the same product as this paper, and designed a combined
contract to coordinate the system channel under only perfect competitive market. Furthermore, this paper will
investigate its supply chain coordination under price-dependent demand, and depict its marketing practice of
single-cycle and two selling periods with stochastic dynamics programming model. Then, a coordination
mechanism will be designed, and the necessary condition for system coordination will be found out. To sum up,
there are two contributions in this paper, one is that the research objective is different from the classic
newsboy-type product, and it is sold in single-cycle with two periods under price-dependent demand; and the
other is a new type contract, combined contract with pending prices will be developed.
2. Denotes for modeling and assumptions
2.1 Denotes for modeling
The notations are listed as follows:
c is the production cost per unit;
2
Supply chain coordination with CCPP for a short-life-cycle product
g i is the shortage cost or goodwill loss per unit for the unmet demand in the i -th selling period, i 1,2 ;
hi is the storage cost per unit for the unsold products in the i -th selling period, i 1,2 ;
v
is the salvage value per unit for the unsold product at the ending of the selling season;
b is the return price per unit at the end of selling cycle, decision variable;
m is the markdown money per unit (when marked down the wholesale price, supplier paid to retailer for the
unsold production in the first selling period), decision variable;
pi is the selling price per unit in the i -th selling period, i 1,2 , decision variable;
Q is the ordering/yielding quantity before the selling season, decision variable;
At the same time, for the superscripts and subscripts in this context, C is the centralized system; DC is the
decentralized system with coordination. Additionally, EPT is the expected profit of whole system, EPR is the
expected profit of the retailer, and EPS is the expected profit of the supplier.
2.2 Demand function
In present literatures, there are two kinds of demand functions to express the stochastic demand depended on
the selling price, additive and multiple. They are common in the price-quantity relation, and all combine the
determinate part with the uncertain part, which are applied widely in economics.
Only the additive demand is applied in this paper. We call D( p, x) y( p) x , where y( p) p as
x
the stochastic additive demand depended on the selling price;
is the random factor, that follows the cumulated
probability Function F (x) and its mean and variance are and , respectively; is the basic demand of
is the price elasticity, and it can be estimated by the selling
datum of relative products. We suppose that and are known by the dealer of this product.
Furthermore, we assume that the demand function in the first selling period is D1 ( p1 , x1 ) y1 ( p1 ) x1 ,
where y1 ( p1 ) 1 1 p1 and x1 ~ F1 () , whose mean and variance are 1 and 1 , respectively, and the
demand function in the second selling period is D2 ( p 2 , x 2 ) y 2 ( p 2 ) x 2 , where y 2 ( p2 ) 2 2 p2 and
this product, and it is relative to its popularity;
x2 ~ F2 () , whose mean and variance are 2 and 2 , respectively. At the same time, we assume that p2 < p1 , and
that for sample in modeling, the two stochastic factors are independent and 1 2 0 .
2.3 Assumptions
We assume that there are more demands and higher price-elasticity in the first selling period, and the
competitive strength of this product comes down due to the emergence of new substitutable product in the second
selling period, which implies that its basic demand decreases and the price-elasticity weakens. Namely,
1 2 , 1 2 0 ,
1 2
1 2
3. Jointed decision model on ordering and pricing for whole system
Under the centralized system, there is only one decision-maker, and all agents on the supply chain quest for
the optimal objective of the whole system. Though the centralized system is very rare in practice, we can view it
as the benchmark of the decentralized system, which can help us understand the meaning of the perfect
coordination. By the athwart reasoning method in dynamics programming, we first assume that there are
y
units
3
Supply chain coordination with CCPP for a short-life-cycle product
products unsold at the beginning of the second selling period, then the expected profit at the beginning of the
second selling period is
EPT2C ( y, p2 ) p2 min{ D2 ( p2 , x2 ), x2 } (v h2 ) max{ 0, y D2 ( p2 , x2 )} g 2 min{ 0, y D2 ( p2 , x2 )}
For the simple, we replace the variable
above equation becomes
y
z 2 y y 2 ( p2 )
with a new variable
in this decision problem, the
EPT2C ( z2 , p2 ) p2 min{ D2 ( p2 , x2 ), x2 } (v h2 ) max{ 0, z2 } g 2 min{ 0, z2 }
Athwart to the beginning of the first selling period, the retailer makes an ordering quantity Q , to maximize
the expected profit of whole system. The expected profit at the beginning of the first selling period is:
EPT1C (Q, p1 ) p1 min{ D1 ( p1 , x1 ), x1} g1 min{ 0, Q D1 ( p1 , x1 )} h1 max{ 0, Q D1 ( p1 , x1 )}
EPT2C (max{ 0, Q D1 ( p1 , x1 )}) cQ
Also replace the variable Q with a new variable
z1 Q y1 ( p1 ) in this decision problem, the above
equation becomes:
EPT1C ( z1 , p1 ) p1 min{ D1 ( p1 , x1 ), x1} g1 min{ 0, z1} h1 max{ 0, z1 )} c( z1 y1 ( p1 )) EPT2C (max{ 0, z1})
We can get the expected profit at the beginning of the first selling period:
EPT1C ( z1 , p1 , p2 ) ( p1 c) [ y1 ( p1 ) 1 ] p2 [ y2 ( p2 ) 2 ] (h1 c) I1 ( z1 ) ( p1 g1 c) L1 ( z1 )
(h2 v) I 3 [ z1 y2 ( p2 )] ( p 2 g 2 ) L3 [ z1 y 2 ( p 2 )]
Where
(1)
I i ( zi ) i ( zi xi ) f i ( xi )dxi is for the expected residual quantity at the end of i -th selling period,
z
i 1,2 ; Li ( z i ) zi ( xi z i ) f i ( xi )dxi is the expected shortage quantity at the end of
z1 y2 ( p2 ) x1
i 1,2 ; I 3 ( z1 y 2 ( p 2 ))
z1
[ z1 y 2 ( p2 ) x1 x2 ] f 2 ( x 2 ) f 1 ( x1 )dx2 dx1
i -th
selling period,
is the expected residual
quantity at the end of selling cycle when the residual quantity at the end of selling period is postponed to the
second selling period; L3 ( z1
y 2 ( p 2 )) 1 z1 y2 ( p2 ) x1 [ x1 x 2 y 2 ( p 2 ) z1 ] f 2 ( x 2 ) f 1 ( x1 )dx2 dx1
z
is the
expected shortage quantity at the end of selling cycle when the residual quantity at the end of selling period is
postponed to the second selling period.
We can get the relationship among
probability theories:
I 1 ( z1 ) , L1 ( z1 ) , I 3 ( z1 y2 ( p2 )) and L3 ( z1 y 2 ( p2 )) based on
L1 ( z1 ) 1 z1 I 1 ( z1 ) ; L3 ( z1 y 2 ( p 2 )) y 2 ( p 2 ) 2 I 1 ( z1 ) I 3 ( z1 y 2 ( p 2 ))
C
Furthermore, we can get the expected profit function EPT1 ( z 1 , p1 , p 2 ) through rational transform:
4
Supply chain coordination with CCPP for a short-life-cycle product
EPT1C ( z1 , p1 , p2 ) ( p1 c) y1 ( p1 ) g 2 [ y2 ( p2 ) 2 ] g1 1 1 I1 ( z1 ) 2 I 3 ( z1 y2 ( p2 )) 3 z1
Where 1
(2)
p1 g1 h1 p2 g 2 , 2 p2 g 2 h2 v , 3 p1 g1 c
EPT1C ( z1 , p1 , p2 )
Then, the optimal problem for the whole system becomes Max
( z1 , p1 , p2 )
(3)
C
Namely, the key problem is how to find the optimal selling price in the first selling period p1 , the optimal
selling price in the second selling period p 2 , and the optimal ordering quantity Q z1 y1 ( p1 ) , which will
maximize the expected profit of whole system.
C
C
C
C
C
Lemma: Under the stochastic additive demand, the expected profit function EPT1 ( z1 , p1 , p2 ) is
concave on the decision variable z 1 , p1 and p 2 respectively.
Proof: we can find this lemma by suing for the derivative of the expected profit function
EPT1C ( z1 , p1 , p 2 ) on decision variables respectively, and prove all of them less than 0.
The Lemma implies that the decision vector
( z1 , p1, p2 )
to maximize the expected profit of the whole
C
system is exclusive. Though we cannot prove that the expected profit function EPT1 ( z 1 , p1 , p 2 ) is concave
on the decision vectors ( z1 , p1, p2 ) , we can get the optimal variables ( z1 , p1 , p2 ) with the iterative search
C
arithmetic according to the concavity of EPT1 ( z 1 , p1 , p 2 ) on
z 1 , p1
and
C
iterative search arithmetic is designed to find the optimal decision vectors ( z1
expected profit of the whole system
p2 , respectively. Furthermore, the
, p1C , p2C ) , which will maximize the
EPT1C ( z1 , p1, p2 ) .
Theorem 1: Under the stochastic additive demand, the optimal prices to maximize the expected profit of the
p1 p1C _ 0
whole system are satisfied:
p2 p
C _0
2
L1 ( z1 )
2 1
{
L3 [ z 1 y2 ( p 2 )]
2 2
(4)
( p2 g 2 h2 v) F3 [ z 1 y2 ( p 2 )}
Where 1 F1 ( z 1 ) 2 F3 [ z 1 y 2 ( p 2 )] 3
(5)
p1C _ 0
1 c 1 1
is the optimal selling price in the first period without random factor, and
21
p2C _ 0
2 g 2 2 2
is the optimal selling price in the second selling period without random factor.
2
Proof: from the Lemma, we can know the expected profit function
EPT1C ( z 1 , p1 , p 2 )
is concave
on z1 , p1 and p2 respectively. So, based on the first order condition of optimal problem, we can get this
theorem.
5
Supply chain coordination with CCPP for a short-life-cycle product
4. The jointed decision on ordering and pricing under decentralized retailer with contact
4.1 Describing the combined contract with pending prices
Analyzing the setting and marketing mode, several classic contracts, such as return policy, quantity flexible,
et al., can not be suitable for the objective in this paper. In such setting, with drastic competitive market and
improved high-requirement people, new and substitutable product will impact the original product in its selling
price and its customer demand, and then the original selling price can not uphold its present value. At this time, it
is best for the retailer to markdown his product. So, there are two types of risks confronted by the retailer, and
these risks will affect the retailer’s ordering decision badly though the retailer can make the selling price based on
the dynamic market.
Based on the assumption of rational retailer, and following the thoughts of combined contact in literature
(LIU, 2006), a new type of contact is designed as follows: Firstly, the selling prices are determined by forecasting
the market, and the retailer makes his ordering based on the wholesale price w offered by supplier and the
market forecasting made by retailer. In the first period in the selling cycle, the retailer sells the product at a higher
price p1 to the end customer, and the supplier contracts the markdown money agreement with the retailer, which
means that the supplier gives the retailer rebate
m for unsold units at the end of the first period; Then, all unsold
units are dropped in the selling price. In the second period, the retailer sells the product at a lower price p 2 , and
the supplier contracts the return policy with the retailer, which means that the supplier agrees that the retailer
returns his unsold units at the end of the second period with the price b . We call this contact formed as
( w, m, b, p1 , p 2 ) is the combined contact with pending prices (CCPP).
By the above contact, alike the consequence of the expected profit of centralized system, we can get the
expected profit under decentralized system with contact:
EPR1DC ( z1 , m, b, p1 , p2 ) ( p1 w) y1 ( p1 ) g1 1 g 2 [ y2 ( p2 ) 2 ]
6 z1 4 I1 ( z1 ) 5 I 3 ( z1 y2 ( p2 ))
where
(6)
4 p1 g1 h1 p2 g 2 m , 5 p2 g 2 h2 b , 6 p1 g1 w
It should be emphasized that the retailer will set the selling prices when ordering, and then the pricing of
retailer will affect the customer demand. So, the return price
b and compensation price m are depended on the
pricing decision of retailer.
4.2 The necessary condition to coordinate the channel with CCPP
Because the expect profit of decentralized retailer with contract
EPR1DC () is similar to the expected
EPR1C () , it is easy to know that EPR1DC () is concave on the decision variable
z 1 . At the same time, because the wholesale price w , the compensation price m , and the return price b all are
profit of centralized system
6
Supply chain coordination with CCPP for a short-life-cycle product
depended on the selling prices p1 and p 2 , and this dependent relationship is pending, it cannot make sure the
concavity of the expect profit function
EPR1DC () on the selling prices p1 and p 2 . So, we consider the
derivative of EPR1 () on z 1 , and can get the necessary condition to coordinate the channel system by
comparing it with that one of the expect profit of centralized system.
DC
EPR1DC ( z1 , w, m, b, p1 , p2 )
4 F1 ( z1 ) 6 5 F3 ( z1 y2 ( p2 ))
z1
EPR1DC ( z1 , w, m, b, p1 , p2 )
0,
From the first-order necessary condition of optimal problem, we know
z1
and get the following result by comparing it to the optimal decision condition of centralized system in Theorem 1.
p1 g1 h1 p2 g 2 m p2 g 2 h2 b p1 g1 w
p1 g1 h1 p2 g 2
p2 g 2 h2 v p1 g1 c
and
(7)
p 2DC p 2C
Furthermore, from Q y1 ( p1 ) z1 , the other necessary condition is listed as follows:
(8)
p1DC p 1C
(9)
So, summing up above analysis can get Theorem 2.
Theorem 2: When the retailer and the supplier covenant the combined contract with pending prices
( w, m, b, p1 , p 2 ) under decentralized system, the necessary condition of channel coordination with the
optimal jointed decision on ordering and pricing of the retailer confronted by the stochastic additive demand is:
p1 g1 h1 p2 g 2 m p2 g 2 h2 b p1 g1 w
p1 g1 h1 p2 g 2
p2 g 2 h2 v p1 g1 c
p1 p1C
(10)
p2 p2C
It is easy to find that Equation (10) is really the transmutation of the necessary condition to coordinate the
channel system. When the selling prices p1 and p 2 is determined by market, namely, the market is perfect
competitive, the Theorem 2 is same to the result in literature (LIU, 2006). And the same time, when the two
selling prices are decision variables, a ratio gotten from Equation (10) can coordinate the channel system after the
C
C
optimal selling prices p1 , p 2 being found.
5. Examples and analysis
Assume that a company operates a short-life-cycle product and it contracts the two-period dynamic combined
contract with pending contact with its supplier. We assume that the relative cost parameters are:
7
Supply chain coordination with CCPP for a short-life-cycle product
c 6 , g 1 1 , g 2 0.5 , h1 0.05 , h2 0.02 , v 1
and
y1 ( p1 ) 1000 30 p1 , y 2 ( p2 ) 200 10 p2 , x1 ~ N (0,30) , x2 ~ N (0,15)
Table 1
Cases
System performance and optimal contract parameters under different system status
Q
p1C
p C2
EPR
( w, m, b)
EPT
Ⅰ
-
421.57
21.16
13.51
4560.72
5825.43
(9, 0, 0)
Ⅱ
-
491.69
19.67
12.50
-
5929.12
-
0
491.69
19.67
12.50
4560.72
5929.12
(9.15, 1.66, 3.58)
0.5
491.69
19.67
12.50
4612.57
5929.12
(9.03, 1.59, 3.48)
1
491.69
19.67
12.50
4764.42
5929.12
(8.91, 1.53, 3.38)
Ⅲ
Notes: Ⅰ. Independent without coordination; Ⅱ. Centralized system; Ⅲ. Independent with coordination.
Based on the above dynamic model and the CCPP, we can obtain the optimal contract parameters of different
allocation profit scheme under different conditions and their corresponding expected profits of retailer and
supplier (Table 1) are listed as follows.
From Table 1, we can obtain that the optimal jointed decision under decentralized system without contact is
less efficient than that of centralized system and the optimal prices under decentralized system is higher than that
under centralized system. When the combined contact with pending prices is covenanted, the expected profit of
the retailer will decrease when the supplier gives him more markdown money and higher return price. The reason
for this counter-intuition conclusion is that the wholesale price will increase with the increasing of markdown
money and the return price. Actually, this conclusion depicts well the risk market profit allocation principle, “the
more the expected profit is, the greater the risk is”. This conclusion is allied to the conclusion in literature.
6. Conclusion
This paper investigates a channel problem of supply chain under price-dependent demand. Based on the
practices of a short-life-cycle product, the combined contract with pending prices is designed, and it can unite the
decision under decentralized system to the framework of centralized system. Summarily, it is shown in this paper
that CCPP can depict the practices under different market policy in different selling period well, at the same time,
the combined contract with pending contract designed by us, can ensure the system coordination and allocate the
coordination profit arbitrarily.
In future research, we will consider the expected profits of the retailer and the supplier under different market
conditions, and address into the sensitive analysis of optimal contract parameters on the cost structure;
Furthermore, we will develop the dynamic model under the relative demand, the two-purchasing opportunities
dynamics model with updating information to depict the more realistic practices in IT industry, and mine out
many significant management views to improve the competition ability and the performance of every company on
the supply chain.
(to be continued on Page 19)
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