Download Wholesale Price Contract under the Newsvendor Problem

Wholesale Price Contract under the Newsvendor Problem
Li Jinghong, Luo Dingti
Management Science & Engineering Research Institute,
Hunan University of Technology, Zhuzhou, P.R.China, 412008
Abstract The application of contracts is one of the basic forms of supply chain coordination and profit
sharing. In a two-step supply chain with one supplier and one retailer, this paper studies the validity of
the wholesale price contract under newsvendor problem. Then based on the whole sale price contact, we
incorporate the compensation and punishment on the left-over inventory, and prove that the buy-back
contract and the wholesale price with default fine contract can make the supply chain achieve optimal
profit, and discuss the possible related problem.
Key words newsvendor problem, wholesale price contract, buy-back contract, default fine
1 Introduction
The application of supply chain contracts is an effective method to coordinate supply chains. Many
scholars over the world studied the supply chain contracts: Cachon [1] studied how the contracts, such as
the wholesale price contract, buy-back contract and so on, coordinate the supply chain under the
newsvendor problem, and studied the validity of different contracts. Lariviere & Porteus [2] study the
wholesale price contract under the newsvendor problem, and analyze the effect of different aberrance
coefficients on the efficiency of supply chain and the profit of the supplier. Lu Hongsheng and Wei
Zengxin[3] studied the flexible ordering strategy with default in a supply chain. Li Sufen and Tang Jiafu[4]
study the joint decisions of order quantity and pricing based on supply chain cooperation, and introduce
the second discount to share benefits more equitable with win-win fashion.
Based on Cachon [1], this paper further analyzes the problems in the wholesale price contract in
practice, and brings forward the solutions, then based on the wholesale price contract, studies the model
that can make supply chain achieve optimal profit, and analyses problems in practical implementation.
2 Basic model
There is a two-step supply chain with one supplier and one retailer. The retailer faces a newsvendor
problem: the retailer must choose an order quantity before the start of a single selling season that has
stochastic demand. Let D, ( D ≥ 0) be the demand during the selling season. Let F be the distribution
_
function of demand and f its density function: F is differentiable, strictly increasing. F (0) = 0 , F = 1 − F .
~
The cost of supplier is c s
~
，supply quantity is q ，the cost of retailer is c ，order quantity is
~
s
r
~
q r , the
~
retailer price is p , p > cs + c r . For each demand the retailer does not satisfy the retailer incurs a β
~
penalty for loss of goodwill. The retailer earns v (net of any salvage expenses) per unit unsold at the
~
~
end of season, v < c r .
The game of our analysis is as follows: the supplier offers the retailer a contract, the retailer accepts
or rejects the contract; assuming the retailer accepts the contract, the retailer submits an order quantity
q to the supplier; The supplier produces and delivers to the retailer before the selling season; season
demand occurs; and finally transfer payments are between the firms based upon the agreed contract. If
the retailer rejects the contract, the game ends.
Each firm is risk neutral, so each firm maximizes expected profit. There is full information, which
means that both firms have the same information at the start of the game. The supplier cannot force the
retailer to pay for units delivered in excess of the retailer’ s order quantity. But can the supplier deliver
less than the amount the retailer orders? There are many reasons, for example, unforeseen production
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difficulties or supply shortages for key components. The shortage may also be due to self’s interest. In
recognition of that motivation, the retailer could assume the supplier operates under voluntary
compliance, which means the supplier delivers the amount (not to exceed the retailer’s order) that
maximizes his profit given the terms of the contract. Alternatively, the retailer could believe the supplier
never chooses to deliver less than the retailer’s order because the consequences for doing so are
sufficiently great, e.g., court action. Call that regime forced compliance [1]. So does the retailer.
Let S (q) be expected sales, min(q, D)
S (q) = q(1 − F (q)) +
∫
q
yf ( y )dy = q −
∫
q
F ( y )dy
0
0
Let I (q) be the expected left over inventory, (q − D) +
I (q) =
∫
q
(q − y ) f ( y )dy = q − S (q)
0
Let L(q) be the lost of sales function
L( q ) =
∞
∫ ( y − q) f ( y)dy
， ( D − q)
+
= u − S (q)
q
（
~
）
Let T be the expected transfer payment from the retailer. see [1] The retailer’s profit function is
~
~
~
~
~
π r = pS (q) + v I (q) − β L(q) − T - c r q
~
~
~
~
~
~
= ( p − v − + β ) S (q) − T + v q − β µ - c r q
The supplier’s profit function is
~
~
~
π s = T − cs q
~
~
~
To simplify the model, because β µ is an external parameter, so let π r = π r + β µ
~ ~
~
~
~
~
， Let
~
p = p − v + β , T = T − v q , c r + c s = c r + c s − v . After simplification, profit functions are as follows
π r = pS (q) − T − c r q , π s = T − c s q
：
Supply chain integration
Supply chain profit can be optimal when supply chain members are integrated.
Let π (q) be the supply chain profit
π (q) = π r + π s = pS (q) − (c r + c s )q
Let q 0 be the optimal order quantity of the supply chain, q 0 = arg max π (q ) . F is strictly
increasing, so π (q) is strictly concave. The optimal order quantity is unique, and q0 satisfies
_
S ' ( q 0 ) = F ( q 0 ) = (c r + c s ) / p
(1)
Let q *r be the retailer’s optimal order quantity, q r* = arg max π r (q) . Further, retailer’s order
quantity depends on the transfer payment that is decided in the contract.
3 The wholesale price contract
In this contract, the supplier decides the transfer payment T (q, w) = wq . π r (q, w) is concave in
q , the retailer’s unique optimal order quantity satisfies pS ' (q) − w − c r = 0 .
Since S ' (q) is decreasing. q r* = q 0 , only when w = c s , that is to say, wholesale price can
coordinate the supply chain only if the supplier uses the marginal cost pricing, but the supplier earns
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zero profit. So supplier clearly prefers to choose a higher wholesale price. Hence, the wholesale price
contract is generally considered a non-coordinating contract. But it is worth studying because it is
commonly observed in practice, that fact alone suggests the contract has redeeming qualities. For
instance, the wholesale price contract is simple to administer. As a result, a supplier may prefer the
wholesale price contract over a coordinating contract if the additional administrative burden associated
with the coordinating contract exceeds the supplier’s potential increase. [1]
The retailer’s profit function is
π r (q, w) = pS (q) − wq − c r q
Let the first order of above formula equals to zero, then retailer’s optimal order quantity satisfies
F (q *r ) = 1 − ( w + c r ) / p
F is strictly increasing and continuous , there is a one-for-one mapping between w and q *r .
−
w( q ) p F ( q ) − c r
The supplier’s profit function is
π s (q.w(q)) = ( w(q) − c s )q
(2)
Apparently, given w(q) > c s , it is voluntary compliance for the supplier to produce and deliver q
units. Supplier’s marginal profit is
∂π s (q, w(q)) / ∂q = w(q) + w ' (q) − c s
_
= w(q)(1 − qf (q) /( F (q)− c r / p)) − c s
The first part is the marginal revenue, the second part is the marginal cost c s which is constant, if
the marginal revenue is decreasing, the supplier’s profit function is concave that is to say, w(q ) and
，
−
_
1 − qf (q ) /( F (q )− c r / p ) is decreasing. Apparently, w(q ) satisfies, 1 − qf (q ) / F (q ) has already be
proved to be true
[ 2]
_
, c r / p is constant, so 1 − qf (q ) /( F (q )− c r / p ) is decreasing, too. Hence, there is
*
a unique quantity q s for supplier to maximize profit. The game is: the supplier chooses optimal
wholesale price w(q *s ) firstly, then the retailer chooses order quantity q *s .
The supplier and the retailer’s profits are as follows
π s (q) = w(q *s )q *s − c s q s*
π r (q) =
pS (q −) w(q *s )q *s
− cr q
(3)
(4)
q *s
is optimal for the supplier. But is it voluntary compliance for the retailer? The retailer’s
marginal profit function can be written as
∂π r (q, w(q )) / ∂q = − w ' (q )q = pf (q )q > 0
It is clearly that retailer’s profit function is increasing in q . So retailer attends to deviation q *s ,
e.g. When the retailer’s opportunity cost is higher than π s (q s* , w(q s* )) , it may buy other products which
is latency threaten to the supplier. To solve this problem, the supplier can increase q through
decreasing the wholesale price. The rationality lies in that the supplier increases q , q ∈ (q *s , q 0 ) , which
not only increases the retailer’s profit, but also increase the supply chain profit. However, that
improvement comes about at the supplier’s expense. [1]
A solution to solve the deviation is: the wholesale price contract can permit a price discount [4].
When order quantity exceeds a certain quantity, the supplier can choose a wholesale price which is
lower than before, that is to say, the supplier chooses a wholesale price w(q *s ) which maximizes its
。
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own profit for the first q *s products, when the order quantity q exceeds q *s , the whole sale price for
the additional ones is wl ( wl > c s ). Assume that the retailer’s order quantity is q , q ∈ (q *s , q 0 ) .
The supplier’s profit function is
π s (q) = w(q *s )qs* + wl (q − q s* ) − c s q
(5)
The retailer’s profit function is
π r (q) = pS (q) − w(q s* )q s* − wl (q − q s* ) − c r q
(6)
Compared with the contract before and after, the improvement of the supplier’s profit is (5)-(3)
π s (q) − π s (q *s ) = (wl − c s )(q − q *s )
Since wl > c s , q > q *s , the above formula ≥ 0 , it is feasible for the supplier to retain the retailer
and increase profit as well.
The improvement of the retailer’s profit is (6)-(4)
π r (q) − π r (q s* ) = p( S (q) − S (q *s )) − (w − c r )(q − q *s )
Since, S (q) = q −
∫
q
yf ( y )dy :
0
π r ( q ) − π r (q *s ) = ( p − wl − c r )(q − q *s ) + p
∫
q
q *s
F ( y )dy
Since p > w + c s + c r > wl + c s + c r , q > q *s , F ( y ) is strictly increasing, the above formula ≥ 0 , it
is feasible for the retailer to satisfy its own needs and increase profit as well.
This contract can make the supplier, the retailer and the supply chain profits better than before, it is
feasible to implement. However, this cannot make the supply chain optimal, why? The discussion is
described as follows.
4 The optimal supply chain contract
4.1 Left-over inventory Compensation
Wholesale price can not coordinates the supply chain, but plus left over inventory compensation,
just let parameter b be the buy-back price per unit, can coordinate the supply chain. This contract is
called buy-back contract [5]. Assume the supplier’s net salvage value is greater than the retailer’s net
salvage value, the supplier can obtain the number of the left over inventory without incurring a cost.
In the buy-back contract, the supplier decides the wholesale price wb , and buy back the left over
inventory at the end of the season, assume the retailer cannot profit from left over inventory, b < wb
the transfer payment is T (q, w, b) = wb q − bI (q ) = bS (q) + ( wb − b)q
Given parameter ( wb , b) , the retailer’s profit function is
π r (q, wb , b) = ( p − b) S (q) − (wb − b + c r )q
(7)
The retailer’s optimal order quantity is
S ' (q r* ) = ( wb − b + c r ) /( p − b)
(8)
Compared (1) with (8), the retailer and the supply chain profits achieve optimal simultaneously
when (1)=(8), that is to say, S ' (q *r ) = S ' (q 0 )
( wb − b + c r ) /( p − b) = (c r + c s ) / p
wb = b − c r + (c r + c s )(1 − b / p)
(9)
Replace it in (7)
π r (q, wb , b) = ( p − b)S (q) − (c s + c r )(1 − b / p)q
，
π r (q, wb , b) = (1 − b / p)π (q)
Hence, through the buy-back contract, the optimal supply chain profit is shared between the
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supplier and the retailer, the retailer shares 1 − b / p .
Does the supplier implement the contract with voluntary compliance? The supplier’s profit depends
on the actual sale quantity, so it may not deliver q 0 to the retailer. The supplier’s profit function is:
π s (q, wb , b) = bS (q) + (wb − b)q − c s q
(10)
Replace it in (10)
π s (q, wb , b) = bS (q) + (c r + c s )qp / b
Compared with π (q)
π s (q, wb , b) = (b / p)π (q )
The supplier shares b / p of the optimal supply chain profit, so q 0 is also the optimal quantity
for the supplier. Hence, the supplier is voluntary compliance to implement the contract.
Hence, buy-back contract can coordinate supply chain. From (9), parameter b has no connection
with market demand, so the decision of b need not to consider it. Interestingly, voluntary compliance
actually increases the robustness of the supply chain. Suppose the retailer is not rational and orders
q > q 0 . Since the supplier is allowed to deliver less than the retailer’s quantity the supplier corrects the
retailers mistake by delivering only q 0 units. However, because the retailer can refuse to accept more
than his order, the supplier cannot correct the retailer’s mistake if he deviate q 0 and orders q < q 0 .
4.2 wholesale price contract with default fine
In order to solve the above problem, the supplier stores the left-over inventory itself, and signs a
contract with the retailer, which contains a wholesale price and order quantity, then, the retailer decides
the order quantity according to its actual demand, if the actual order quantity is less than the quantity in
the contract, the retailer has to pay default fine.
The trade course is as follows: the supplier and the retailer sign a contract with wholesale
price w and the order quantity q . The supplier produces q unit products, the retailer orders products
according to the actual demand, and the left-over inventory is stored by the supplier (marginal inventory
cost is c ), the retailer pays default fine according to the supplier ( B is the default fine of each unit)[3].
So the basic model does not include the net salvage I (q) of the retailer’s left over inventory, that is
~
v I (q) , the inventory cost of the supplier is cI (q) , simplifying the profits function, we have
π r = pS (q) − T − c r q − cI (q) ,
π s = T − cs q
The transfer payment T = BI (q ) , I (q) = q − S (q) , the supply chain’s profit function is:
π (q, w, B) = pS (q) − c s q − c r q − cI (q) = ( p + c) S (q ) − (c + c s + c r )q
The first order is ∂π (q, w, B ) / ∂q = ( p + c) S ' (q) − (c + c s + c r )
The second order is ∂ 2 π (q, w, B ) / ∂q 2 = −( p + c) f (q) < 0 , we find π (q, w, B ) is concave in q ,
Let the first order is zero, S ' (q) = (c + c s + c r ) /( p + c ) , i.e. the unique optimal quantity is
q0 = F −1 ( p − c r − c s ) /( p + c)
The retailer’s profit function is
π r (q, w, B ) = pS (q) − wS (q) − BI (q) − c r q = ( p − w + B )S (q) − wS (q) − ( B + c r )q
(11)
The first order is ∂π r (q, w, B ) / ∂q = ( p − w + B )S ' (q ) − ( B + c r ) Let the first order is zero,
S r ' (q) = ( B + c r ) /( p − w + B) , i.e. the unique optimal order quantity is
q *r = F −1 ( p − c r − c s ) /( p + c )
The supplier’s profit function is
(12)
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π s (q, w, B ) = wS (q) + BI (q) − cI (q) − c r q = ( w − B + c )S (q) + ( B − c − c r )q
The first order is ∂π s (q, w, B ) / ∂q = ( p − w + B) S s ' (q) − ( B + c r ) Let the first order be zero,
S s ' (q) = ( B + c r ) /( p − w + B ) , the unique optimal quantity is
q *s = F −1 (w − c s ) /( w + c − B ) .
(13)
S r' (q)
'
In order to make supply chain achieve optimal profit, let (11)= (12), i.e.
= S (q) , we obtain
B0 = [( p − w)(c + c s ) − c r ( w − c )] /( p − c s − c r ) .
(14)
The retailer and the whole supply chain achieve optimal profits simultaneously, we validate the
enforceability of the supplier as follows.
Let q *s = q r* , we have Bsr = [( p − w)(c + c s ) − c r ( w − c )] /( p − c s − c r ) = B0 , and the supplier, the
retailer and the whole supply chain achieve optimal profit simultaneously, i.e. under this contract, the
supplier implements it with voluntary compliance.
From above, we find that default fine B = [( p − w)(c + c s ) − c r (w − c )] /( p − c s − c r ) has no
relationship with market demand, hence the decision of B need not to consider it. This contract can
solve the problem that the retailer deviates the optimal order quantity under the buyback contract,
making supply chain achieve optimal profit.
5 Conclusion
This paper studies the coordination of supply chain with a supplier and a retailer under newsvendor
problem, firstly, it analyses the wholesale price contract, which cannot optimize the supply chain, but
has quality in practice. When combined with price discount, it can increase both parties’ and supply
chain’s profit. Secondly, this paper analyzes the optimal profit from the aspects of left-over inventory
compensation and punishment, studies two contracts: the buy-back contract and the wholesale price
contract with default fine. Although the former can coordinate the supply chain, the retailer’s actual
order quantity is less than the optimal order quantity, the supplier bears a lose, the supply chain is
sub-optimal as well; the latter deal with the problem, where the supplier stores the left-over inventory,
and gives the retailer punishment—default fine. It solves the problem in the buyback contract, and can
achieve supply chain optimal profit.
References
[1] Cachon, G. P. Supply chain coordination with contracts. Working paper, The Wharton school,
university of Pennsylvania. 2001
[2] Lariviere, M. A., porteus. E. L. Selling to the newsvendor: an analysis of price-only contracts.
Manufacturing & service operations management, 2001.3(4): 293-305
[3] Lu Hongsheng, Wei Zengxin. The flexible ordering strategy with default in the supply chain.
Chinese journal of management science, 2002, 10:111-113(in Chinese)
[4] Li Sufen, Tang Jiafu. The joint decisions of order quantity and pricing based on supply chain
cooperation, Annual academic conference of Chinese papers of controlling and decision-making,
2004(in Chinese)
[5] Cachon, G. P., Lariviere, M. A. Supply chain coordination with revenue-sharing contracts:
strengths and limitations, Management science, 2005. 51(1): 30-44
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