Download Research on Stackelberg Game between the Supplier and the Retailer

Research on Stackelberg Game between the Supplier and the Retailer
under VMI
Cai Jianhu, Zhou Gengui
College of Business and Administration, Zhejiang University of Technology, P.R.China, 310032
Abstract
Now many small and mid-sized businesses are selling perishable products, and it is
important to decide inventory quantity before the selling season. The two-echelon supply chain in our
paper comprises one supplier and one retailer, and the products in our paper are perishable and sold over
a single season. The paper discusses the vendor managed inventory (VMI) tactics. Under VMI, the
supplier is in charge of the inventory and he should decide optimal inventory quantity before the selling
season. A demand function is introduced, and wholesale price is assumed to be fixed. The retailer acts as
a Stackelberg leader by setting retail price, and the supplier then decides his optimal inventory quantity.
The equilibrium solutions of the Stackelberg game are gained.
Key words Supply chain, Stackelberg game, VMI
1 Introduction
Now many small and mid-sized businesses are selling perishable products, and the products’
selling seaon is always very short. Thus the businesses must decide inventory quantity before the selling
season under uncertainty. The classical newsvendor model perfectly reflects the above decision
environment and has offered some basic research conclusions. In classical model, there is only one
decision maker, and there is only one decision variable, i.e. wholesale price. Thus, it is comparatively
simply and easy to find the optimal inventory quantity. And newsvendor model has been discussed by
many researchers [1-5]
In reality, many enterprises are the members of a supply chain, and they must consider the other
members’ responses. The two-echelon supply chain in our paper comprises one supplier and one retailer,
and the supply chain’s competition structure must be analyzed. Traditionally, the retailer orders products
from the supplier before the selling season, and the retailer should bear all inventory risk. And the main
decision variable are wholesale price and inventory quantity. Lariviere and Porteus[6] constructs a
Stackelberg game in which the supplier acts as the leader by setting wholesale price, and the retailer
should choose his optimal order quantity. The equilibrium solutions are gained, and they find the
supplier always can gain most profit of the whole supply chain. And the retailer can gain more profit by
giving more demand information to the supplier. Many researches always discuss the cases in which the
retailer bears all inventory risk. Now VMI is playing a more and more important role in supply chain
management. According to some investigations, VMI tactics are more important than the JIT (Just in
Time). Some retailers, such as Wal-Mart, Kmart, Dillard, JCPenney, succeed in applying VMI. Disney
and Towill[7] find the bullwhip effect can be effectively reduced by applying VMI. It is worthwhile for
us to study the decision models under VMI, and the supply chain’s competition structure should be
especially discussed.
In this paper, we discuss the inventory decisions models when the demand is relevant to the retail
price. The supply chain operates under VMI, and the supplier takes charge of the inventory. The
Stackelberg game is introduced, and the competition processes are analyzed. We find the equlibrium
solutions are unique.
2 The Model
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Consider a supply chain with one supplier and one retailer, and the products are perishable and sold
over a single selling season. The supply chain operates under VMI, and the supplier controls inventory
and bears all inventory risk. Let w denote wholesale price per unit and v denote salvage value of every
unsold unit and c denote marginal production cost for the supplier. Let d denote stochastic demand,
which is relevant with the retail price p . The demand function can be expressed as:
d = X − bp
，b > 0
(1)
where X is the market potential for the products during the selling period. And we assume X is a
normal distribution mean µ , variance σ , namely, X ~ N ( µ , σ ) . Because market potential is not
negative, we always assume that µ is sufficiently large and σ is sufficiently small. Thus we
2
2
have d ~ N ( µ − bp, σ ) . Let f ( x ) and F ( x) denote the density and distribution functions
2
of d respectively. We further assume wholesale price w is exogenously fixed. Thus the decision
variables are retail price p and inventory quantity q . The supplier incurs a shortage cost when the
products are out of stock, and this shortage cost is s per unit. Assume v < c < w < p .
+
Define t = max(0, t ) . The game process is following:
the retailer first decides unit retail price as the Stackelberg leader;
①
②the supplier then chooses inventory quantity.
We will analyze the above game by backward induction. Given retail price, the supplier should decide
his optimal inventory quantity. And the supplier’s expected profit function is:
π s = E[(w − c ) min( q, d ) + (v − c )(q − d )+ − s(d − q )+ ]
q
= ( w + s − c )q − ( w + s − v ) ∫ F ( x)dx − s µ ,
(2)
0
We have
∂π s
= ( w + s − c) − ( w + s − v ) F ( q )
∂q
∂ 2π s
= −( w + s − v ) f ( q ) < 0
∂q 2
Thus
π s is concave in q , let
Φ(
∂π s
w+ s −c
= 0 , and we have F (q ) =
, which can be expressed as:
∂q
w+ s −v
q − ( µ − bp)
σ
)=
w+s−c
w+s−v
So that the optimal inventory quantity that the supplier will choose given retail price is:
q1 = µ − bp + L1σ
Where
Φ ( L1 ) =
(3)
w+ s−c
w+ s−v
(4)
the retailer determines his optimal retail price anticipating the response of the supplier, and the objective
profit function of the retailer is:
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π r = ( p − w) E min(q1 , d ) − sE(d − q1 )+
(5)
q1
= ( p + s − w)(q1 − ∫ F ( x)dx) − sµ
0
q1
∂π r
= q1 − ∫ F ( x)dx + ( p + s − w)(−b + bF (q1 ))
0
∂p
We have
∂ 2π r
= −2b(1 − F (q1 )) − b2 ( p + s − w) f (q1 ) ≤ 0
∂p2
Thus π r is concave in p , let
∂π r
= 0 , we have:
∂p
q1
µ − bp + L1σ − ∫ F ( x)dx + ( p + s − w)(−b + b
0
w+s−c
)=0
w+s−v
(6)
where
∫
q1
0
q1
F ( x)dx = ( xF ( x ))0q1 − ∫ x f ( x)dx = L1σ
0
w+s−c
+ σφ ( L1 )
w+s−v
(7)
substitute (7) to (6) , and we can gain the retailer’s optimal retail price as following:
p* =
1
[( w + s − v)( µ − σφ ( L1 )) + (c − v )( L1σ + bw − bs)]
b(w + s + c − 2v)
(8)
*
We further substitute p to (3), and we can gain the supplier’s optimal inventory quantity as following:
q1* = µ − bp* + L1σ
(9)
*
*
Thus the equilibrium solution of this Stackelberg game is ( p , q1 ) , and we also can gain the supplier’s
equilibrium expected profit π s and the retailer’s equilibrium expected profit π r .
*
*
3 Conclusion
In this model, the retailer transfers inventory risk to the supplier. And the retailer also gains the
Stackelberg leader status by first setting retail price and he always chooses a high retail price to gain
most profit. Thus the supplier is under the high pressure. In reality, the supplier will try his best to
improve his competition ability for more profit. For example, the supplier may set a profit level, under
which the supplier will not take part in the trade with the retailer. Thus the retailer must consider the
supplier’s competition ability while he chooses his optimal retail price. In this model, we only consider
the situation in which there is only one retailer. And we should consider inventory transshipment among
the retailers if there are multiple retailers, and the retailers also should compete with each other for more
profit. And wholesale price is fixed in our paper,, the supplier has no chance to set wholesale price in
this paper. In fact, the supplier always has the ability to set wholesale price, under which there is a
three-stage game between the supplier and the retailer. In first stage, the supplier decides wholesale
price. In second stage, the retailer chooses retail price. And in third stage, the supplier will choose his
optimal inventory quantity. This game will be difficult to analyze by using backward induction. We
should explore above new problems in the future.
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References
[1] Gallego G, Moon I. The Distribution Free Newsboy Problem: Review and Extensions. Journal of
the Operational Research Society, 1993, 44: 825~834
[2] Moutaz Khouja. The Single-Period (News-Vendor) Problem: Literature Review and Suggestions for
Future Research. Omega, 1999, 27: 537~553
[3] Kabak I, Schiff A. Inventory Models and Management Objectives. Sloan Management Review,
1978, 19(2): 53~59
[4] Lau H. The Newsboy Problem under Alternative Optimization Objectives. Journal of Operational
Research Society, 1980, 31: 525~535
[5] Li J, Lau H, Lau AH. A Two-Product Newsboy Problem with Satisfying Objective and Independent
Exponential Demands. IIE Transactions, 1991, 23: 29~39
[6] Martin A. Larivier, Evan L. Porteus. Selling to the Newsvendor: An Analysis of Price-Only
Contracts. Manufacturing & Service Operations Management, 2001, 3(4): 293~305
[7] S. M. Disney, D. R. Towill. Vendor-Managed Inventory and Bullwhip Reduction in a Two-Level
Supply Chain. International Journal of Operations & Production Management, 2003, 23(6):
625~651
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