Download Optimal Decisions in a Supply Chain with Different Suppliers’

Optimal Decisions in a Supply Chain with Different Suppliers’
Price-setting Power and Manufacturers’ Competition Structures1
WANG Chuanxu
School of Economy and Management, Shanghai Maritime University, Shanghai, 200135,P.R.China
Abstract In this paper, we focus our analysis on optimal decision models in an assembly supply chain
consisting of multiple suppliers and multiple manufacturers. We examine the optimal pricing models in
four different suppliers’ price-setting power structures under two types of manufacturers’ competition
structures (independent competition and Stacklberg competition) and investigate the impact of suppliers’
price-setting power structures on supply chain by performing comparison analysis. Our analysis shows
that the optimal retail price (the optimal production quantity) in the centralized supply chain is lower
(larger) than those in all three decentralized supply chains. In three decentralized supply chains, our
analysis shows that the production quantity, supply chain profit, manufacturers’ profit and suppliers’
profit (the supplier’s spare parts wholesale prices) when suppliers are integrated are larger (lower) than
those when suppliers are independent, the production quantity, supply chain profit, manufacturers’ profit
and suppliers’ profit (the supplier’s spare parts wholesale prices) when suppliers are independent are
larger (lower) than those when suppliers are unequal with a Stackelberg leader.
Key Words Price-setting Power, Independent Competition, Stackelberg Competition, Supply Chain
1 Introduction
In supply chain management, supply chain performance is increasingly investigated and compared
under centralized and decentralized decision making [1]. In a decentralized supply chain, there exists
competition among supply chain members, each member aims to achieve his own profit maximization
by optimizing his own decision. In this paper, we focus our analysis on optimal decisions in an assembly
supply chain consisting of multiple suppliers and multiple manufacturers. We examine the optimal
strategies in four different suppliers’ pricing-setting power structures with two types of manufacturers’
competition structures (independent and Stacklberg competition) and examine how these strategies
affect the wholesale price, order quantity, supply chain profit, and manufacturers’ profit as well as
suppliers’ profit.
Our work is related to the supply chain optimal pricing decision literature. Most of this work
considers pricing decision in a simple supply chain with one supplier(manufacturer) and one
retailer(distributor) [2][3]. In our paper, we look at the optimal pricing models in a supply chain with
multiple suppliers and multiple manufacturers.
Ha et al(2003)[4] consider the pricing strategies for suppliers in a supply chain consisting of two
suppliers and one customer. Bernstein and Federgruen(2003a, 2003b)[1][5], Padmanabhan and Png
(1997)[6] consider the pricing strategies for retailers in a supply chain consisting one manufacture and
competing retailers. The difference of our paper with previous research is that we consider the price
strategies for both suppliers and manufactures in a supply chain consisting of multiple suppliers and
multiple manufacturers. The other difference is that previous research treats only one type of
pricing-setting power (i.e. all participants set their individual retail price independently and
simultaneously), whereas our paper investigates four different pricing-setting power structures for
suppliers under two different manufacturer’s competing cases and analyzes the impact of different
scenarios on supply chain.
1
This research was supported by Shanghai Education Committee Foundation under Grant 05FZ11.
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2 Model and Notation
We consider a supply chain with n supplier selling spare parts to m manufacturers. We
assume the products manufactured have properties of oligopoly market structure. The notable
characteristic of oligopoly is that the number of manufacturers is small enough that actions taken by any
individual firm in the industry have a perceptible impact on the production quantity of other
manufacturers[7]. We consider four different suppliers’ price-setting strategies, i.e. suppliers and each
manufacturer are centralized, suppliers jointly decide on prices, suppliers decide on prices independently,
and suppliers decide on prices unequally with a stackelberg leader. Meanwhile, we study the oligopoly
models of manufacturer competition under both independence and Stackelberg conjectures.
Let p represent the unit selling price for manufacturers, Q represent market demand for all
manufacturers, qi represent market demand for manufacturer i ( i = 1,2,...m ). We assume that the
products offered by the manufacturers are substitutes. To simplify our analysis, we shall confine
ourselves to the case where the product demand function is of the linear form:
p = a − bQ with a > 0, b > 0
For manufacturer, define
c mi = The unit production cost for manufacturer i .
For supplier, define
c s j =The unit production cost for supplier j .
w j = The unit wholesale price for supplier j .
We assume that one product needs N j spare parts provided by supplier j . Let
n
N
cs = ∑ N j c s j , w = ∑ N j w j .
j =1
j =1
We assume that p > c mi > w, w j > c s j (i = 1,2,...m, j = 1,2,...n) because the profits of
manufacturers and suppliers are non-negative.
3 Optimal Decisions under Manufacturers Independent Competition
In this section, we consider four different suppliers’ pricing strategies, i.e. suppliers and each
manufacturer make decisions together, suppliers jointly decide on prices first and then manufacturers
decide their production quantities, suppliers decides on prices independently first and then manufacturer
decide their production quantities, and suppliers decides on prices unequally with a Stackelberg leader
and then manufacturer decide their production quantities. Meanwhile, we study the oligopoly model of
manufacturer competition under Bertrand conjectures.
3.1 Suppliers and each manufacturer are centralized
In this section, we analyze the case in which suppliers and manufacturer i decide on retail price
jointly, assuming that a central decision maker decides on decision variables so as to achieve supply
chain wide profit maximization. Because the supplier’s wholesale price is irrelevant to the formulation
of the profit function. Hence, each supply chain’s only decision is the total production quantity and
individual production quantity. In this case, each supply chain profit function is given by
(1)
Max Π SCii = ( p − c m − c s )qi
i
where the subscript SCi denotes the supply chain consisting of all suppliers and manufacturer i .
It can be shown that the above profit function is concave. We can use the first order condition and
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get the unique solution to the above profit function. The optimal total production quantity Q *B1 , and total
1
can be obtained, where the superscript B1 stands for the case 1 under
supply chain profits Π *B
SC
independent competition.
3.2 Suppliers jointly decide on prices
In this model, we present the following sequence of moves. In the first stage, the suppliers jointly
set the wholesale price, w . In the second stage, given the suppliers’ wholesale price, the manufacturers
decide their individual production quantity simultaneously and independently. In this case, the model
can be given by
n
MaxΠ S = ∑ ( w j − cs j ) N jQ
,
j =1
(2)
s.t. MaxΠ M i = ( p − cmi − w)qi = ( a − bQ − cmi − w) qi
where the subscript S denotes suppliers, subscript M i denotes manufacturer i .
It can be easily shown that the above profit functions are concave. By solving the first order condition,
the optimal wholesale price for unit product w
*B 2
, the optimal total production quantity Q *B1 , the
2
and total supply chain profit
optimal supplier profit Π *SB 2 , the optimal manufacturer profit Π *B
M
2
can be obtained, where the superscript B2 stands for Case 2 under independent competition, the
Π *B
SC
subscript SC stands for the total supply chain.
3.3 Suppliers decide their individual prices independently
In this section, we present the following sequence of moves. In the first stage, all suppliers set their
individual wholesale price independently and simultaneously. In the second stage, given the suppliers’
wholesale prices, each manufacturer simultaneously and independently decides individual production
quantity. In this case, the model can be given by
Max Π S j = ( w j − cs j ) N jQ
(3)
s.t. MaxΠ M i = ( p − cmi − w) qi = ( a − bQ − cmi − w) qi
where the subscript S j denotes supplier j .
Similarly, It can be shown that above profit functions are concave. In this case, the optimal unit spare
*B 3
part j ’s wholesale price w j
production quantity Q
*B 3
, the optimal wholesale price for unit product w
*B 3
, the optimal supplier profit Π S
*B 3
M and
*B 3
, the optimal total
, the optimal manufacturer profit
*B 3
SC can
Π
total supply chain profit Π
be obtained, where the superscript B3 stands for Case 3
under independent competition.
3.4 Suppliers decide on price unequally with a Stackelberg leader
In this model, we consider the suppliers’ price-setting structure in a supply chain in which one
supplier has more wholesale price setting power than any other supplier. Without loss of generality, we
assume supplier 1 wields more wholesale price setting power than any other supplier. We present the
following sequence of moves. In the first stage, the supplier1decides its wholesale price. In the second
stage, given the supplier 1’s wholesale price, the other suppliers decide their wholesale price
independently and simultaneously. In the third stage, given the suppliers’ wholesale prices, each
manufacturer simultaneously and independently decides individual order quantity. In this case, the
model can be expressed as
MaxΠ S1 = ( w1 − c1 ) N1Q
s.t. MaxΠ S j = ( w j − cs j ) N jQ ,( j ≠ 1)
s.t. MaxΠ M i = ( p − cmi − w)qi = ( a − bQ − cmi − w) qi
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(4)
Similarly, It can be shown that the above profit functions are concave. In this case, the optimal unit
*B 4
spare part 1’s wholesale price w1
*B 4
, the optimal unit spare part j ’s ( j ≠ 1) wholesale price w j
,
the optimal unit product wholesale price w *B 4 , the optimal total production quantity Q *B 4 , the
*B 4
*B 4
optimal supplier 1’s profit Π S1 , the optimal supplier j ’s ( j ≠ 1) profit Π S j , and the optimal
*B 4
supply chain profit Π SC
can be obtained by solving the first order condition, where the superscript
B 4 denotes Case 4 under independent competition.
4 Optimal Decisions under Manufacturers’ Stackelberg Competition
In this section, we consider the same suppliers’ pricing strategies as section 3. However, we study
the oligopoly model of manufacturer competition under Stackelberg conjectures here. Without loss of
generality, we assume manufacturer 1 wields more decision power than any other manufacturer.
4.1 Suppliers and each manufacturer are centralized
In this case, the model can be expressed as
Max
Π SCi = ( p − cm − cs ) q1
(5)
1
s.t. MaxΠ SCi = ( p − cm − cs )qi (i ≠ 1)
i
Similarly, it can be shown the above profit functions are concave. In this case, the optimal total
*S 1
*S 1
production quantity Q
and the optimal supply chain profit Π SC can be obtained, where the
superscript S1 denotes Case 1 under Stackelberg competition.
4.2 Suppliers jointly decide on prices
As in Section 3.2, the model can be given by
n
MaxΠ S = ∑ ( w j − cs j ) N jQ
j =1
(6)
s.t. MaxΠ M1 = ( p − cm1 − w)q1 = ( a − bQ − cm1 − w)q1
s.t MaxΠ M i = ( p − cmi − w)qi = ( a − bQ − cmi − w)qi , (i ≠ 1)
It can be shown that the above profit functions are concave. In this case, the optimal unit product
wholesale price w
Π
*S 2
M1
*S 2
, the optimal total production quantity Q
, the optimal manufacturer i ’s (i ≠ 1) profit Π
*S 2
SC
*S 2
Mi
and the optimal supply chain profit Π
can be obtained,
under Stackelberg competition.
4.3 Suppliers decide on prices independently
As in Section 3.3, the model can be expressed as
*S 2
, the optimal manufacturer 1’s profit
*S 2
, the optimal all suppliers’ profit Π S
,
where the superscript S2 denotes Case 2
MaxΠ S j = ( w j − cs j ) N jQ
(7)
s.t. MaxΠ M1 = ( p − cm1 − w)q1 = (a − bQ − cm1 − w) q1
s.t. MaxΠ Mi = ( p − cmi − w)qi = ( a − bQ − cmi − w)qi ,(i ≠ 1)
Similarly, the profit function can be proved to be concave. In this case, the optimal unit
*S 3
*S 3
product wholesale price w
, the optimal total production quantity Q
, the optimal
*S 3
*S 3
manufacturer 1’s profit Π M 1 , the optimal manufacturer i ’s (i ≠ 1) profit Π M i ,
suppliers’ profit Π
*S 3
S ,
and the optimal supply chain profit Π
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*S 3
SC
the optimal all
can be obtained,
where the
superscript S3 denotes Case 3 under Stackelberg competition.
4.4 Suppliers decide on price unequally with a Stackelberg leader
As in Section 3.4, the model can be expressed as
MaxΠ S1 = ( w1 − c1 ) N1Q
MaxΠ S j = ( w j − cs j ) N jQ , ( j ≠ 1)
(8)
s.t. MaxΠ M1 = ( p − cm1 − w)q1 = ( a − bQ − cm1 − w) q1
s.t. MaxΠ M i = ( p − cmi − w)qi = ( a − bQ − cmi − w) qi ,(i ≠ 1)
The above profit function can be shown to be concave. In this case, the optimal unit product
*S 4
*S 4
wholesale price w , the optimal total production quantity Q , the optimal manufacturer 1’s
*S 4
*S 4
profit Π M 1 , the optimal manufacturer i ’s (i ≠ 1) profit Π M i ,
the optimal all suppliers’ profit
4
Π *S
S , and the optimal
*S 4
supply chain profit Π SC
competition.
can be obtained, where the superscript S4 denotes Case 4 under Stackelberg
5 Comparison Analysis
Based on the results of Section 3 and Section 4, we have the following propositions.
Proposition 1 Under Manufacturers’ independent competition,
(a) Q
* B1
(b) w
(c) p
> Q *B 2 > Q *B 3 > Q *B 4 ;
*B 2
* B1
* B1
< w* B 3 < w * B 4
< p *B 2 < p *B 3 < p *B 4
*B 2
*B 3
*S 4
(d) Π SC > Π SC > Π SC > Π SC
*B 2
(e) Π S
*B 2
> Π *SB 3 > Π *SB 4
*B 3
*B 4
(f) Π M > Π M > Π M
The proof of Proposition 1 is straightforward from algebra, therefore omitted.
Proposition 2 Under manufacturers’ Stackelberg competition,
(a) Q
(b) w
(c) p
*S1
*S 2
*S1
*S1
> Q *S 2 > Q * S 3 > Q * S 4
< w *S 3 < w *S 4
< p *S 2 < p *S 3 < p *S 4
*S 2
*S 3
*S 4
(d) Π SC > Π SC > Π SC > Π SC
*S 2
(e) Π S
*S 2
> Π *SS 3 > Π *SS 4
*S 3
*S 4
(f) Π M > Π M > Π M
The proof of Proposition 2 is straightforward from algebra, therefore omitted.
Proposition 1 and Proposition 2 indicate that suppliers’ pricing strategies will affect the
performance of a supply chain with oligopoly market. Under manufacturer’s independent and
Stackelberg competition scenarios, The production quantity and supply chain profit(the product retail
price) in the centralized supply chain is larger(lower) than those in all three decentralized supply chains.
In the decentralized supply chains, the supplier’s spare parts wholesale prices when suppliers are
integrated are lower than those when suppliers are independent , the supplier’s spare parts wholesale
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prices when suppliers are independent are lower than those when suppliers are unequally with a
Stackelberg leader. However, the production quantity, supply chain profit, manufacturers’ profit and
suppliers’ profit when suppliers are integrated are larger than those when suppliers are independent. This
implies that the benefits in efficiency from the supplier integration effort can be effectively shared with
the downstream stages in the supply chain to increase demand [8]. The production quantity, supply chain
profit, manufacturers’ profit and suppliers’ profit when suppliers are independent are larger than those
when suppliers are unequally with a Stackelberg leader. This implies that the lower supplier’s spare parts
wholesale prices lead to the more production quantity, and accordingly result in the increase of supply
chain profit as well as manufacturers’ profit and suppliers’ profit.
6 Conclusions and Future Research
In this paper, we have investigated how the different suppliers’ price-setting power structures affect
a supply chain with oligopoly Market. We have compared four different suppliers’ pricing-setting power
structures under two types of manufacturers’ competition structures (independent and Stacklberg
competition). Our comparison analysis shows that the optimal product retail price(the optimal
production quantity) in the centralized supply chain is lower (larger) than those in all three decentralized
supply chains.
In three decentralized supply chains, our analysis shows that production quantity, supply chain
profit, manufacturers’ profit and suppliers’ profit (the supplier’s spare parts wholesale prices) when
suppliers are integrated are larger (lower) than those when suppliers are independent. The production
quantity, supply chain profit, manufacturers’ profit and suppliers’ profit (the supplier’s spare parts
wholesale prices) when suppliers are independent are larger(lower) than those when suppliers are
unequally with a Stackelberg leader. This means that the lower suppliers’ wholesale prices will lead to
the higher production quantity, and higher supply chain profit as well as suppliers’ profit and
manufacturers’ profit.
Future research will relax some assumptions made in this paper to develop more comprehensive
models. First, other demand functions (non-linear demand functions) can be considered in the profit
functions. Second, other competition such as service competition can be incorporated in the suppliers’
profit functions to determine the optimal spare parts wholesale prices and their production quantity.
Finally, our work considers only one type of supply chain power structure, i.e. the suppliers wield more
power than the manufacturers. Other different power structures will be included in our future research.
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The author can be contacted from e-mail : [email protected]
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