Download Modeling for Rebate and Penalty Contract with Retailer’s Combined Decision Bias

Modeling for Rebate and Penalty Contract with Retailer’s Combined
Decision Bias
ZHU Jinlou1, SONG Fugen2
Glorious Sun School of Business and Management, Donghua University, Shanghai, P.R.China, 200051
1. [email protected], 2. [email protected]
Abstract: This paper considers a supply chain model in which a single supplier sells a single product to a
single retailer who faces the newsvendor problem. We establish the supply chain with retailer’s combined
decision bias, and analyze how the retailer’s bias affects his order quantity and the contract parameter. The
results show that the supply chain can achieve channel coordination with rebate and penalty contract, the
rebate and penalty quotiety increases as the retailer’s waste-averse preferences increases and decreases as
stockout-averse preferences increases. With the numerical examples, we demonstrate that the retailer’s
expected profit and utility of rebate and penalty are more than that of the decentralized decision making
system.
Keywords: Supply Chain Management; Rebate and Penalty Contract; Coordination; Combined Decision
Bias.
1 Introduction
Optimal Supply chain performance requires the execution of a precise set of action. Unfortunately,
those actions are not always in the best interest of the members in the supply chain, i.e., the supply chain
members are primarily concerned with optimizing there own objectives, and that self serving focus often
results in poor performance. However, optimal performance can be achieved if the firms coordinate by
contracting on a set of transfer payments such that each firm’s objective becomes aligned with the supply
chain’s objective[1].
Cachon [1] reviewed and summarized the supply chain contract in detail. The existing supply chain
contracts are focused on risk-neutral, and with the maximization of expected profit as single objective. There
are a lot of reverse examples in realistic decision-making to show why a decision maker may order an
inventory quantity that differs from the expected profit-maximizing quantity. Fisher and Raman[2] provide
some evidence to indicate that managers’ decisions do not correspond to the expected profit-maximizing
order quantity. Tsay and Nahmias[3] called for analyses of supply chain contracts where various players are
allowed to have objective functions other than profit maximization. Khouja[4] took the newsvendor’s
extension of different objective or utility function as the principal question of future research for supply
chain management. Wu, etc[5] regarded the researching for science of behavior in supply chain management
as a very important and challenge task. Schweitzer and Cachon[6] studied the newsvendor model with
decision bias.
In this paper, we define the newsvendor problem, present the expected profit-maximizing solution, and
describe utility functions that could influence the inventory decision process. And employ the rebate and
penalty contract to coordinate the supply chain.
2 Question, assumptions and notation
2.1 Question
There is one supplier and one retailer that face a newsvendor problem: the retailer must choose an order
quantity before the start of a single selling season that has stochastic demand, and no further replenishments
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are possible. If the order quantity is greater than realized demand, the retailer must dispose of the remaining
stock at a loss. If the order quantity is lower than realized demand, the retailer forgoes some profit.
Therefore, in choosing an order quantity the retailer must balance the costs of ordering too little against the
costs of ordering too much.
The supplier is assigned to make the contract offer, rather than the retailer. If the retailer rejects the
contract, the game ends and each firm earns a default pay. We consider the case that the retailer accepts the
contract. And the following sequence of events occurs in this game:
a) the supplier offers the retailer a contract;
b) the retailer accepts the contract;
c) the retailer accepts the contract and submits an order quantity to the supplier;
d) the supplier produces and delivers to the retailer before the selling season;
e) season demand occurs;
f) Transfer payments are made between the firms based upon the agreed contract.
2.2 Assumptions
The assumptions used in the models are as follows:
a) All the firms possess the same information when making their decisions (Asymmetric information);
b) Demand is stochastic, with the determinate wholesale price and selling price;
c) Both supplier and retailer are risk- neutral.
d) Retailer has both waste-averse preferences and stockout-averse preferences;
e) Retailer uses utility- maximizing as well as profit-maximizing to make order decision.
2.3 Notation
The notation used in the models is as follows:
p : selling price per item,
c : production cost per item,
q : order quantity of retailer,
w : unit wholesale price of supplier,
v : unit salvage value which is unsold at the end of season,
x : random variable for single-period demand,
f ( x) : probability density function (pdf) of single-period demand,
F ( x) : cumulative distribution function (cdf) of single-period demand, F is differentiable, strictly
increasing and F (0) = 0 . Let F ( x) = 1 − F ( x) ,
u = E ( x) = ∫
+∞
0
xf ( x)dx : The expected demand,
q
S (q) = E min(q, x) = q − ∫ F ( x)dx : expected sales,
0
I (q ) = E (q − x) + = q − S (q) : the lost sales function,
L(q) = E ( x − q)+ = u − S (q) : expected left over inventory,
T : expected transfer payment from the retailer to the supplier,
ΠT : supply chain’s expected profit function,
Π S : the retailer’s expected profit function,
Π R : the supplier’s expected profit function,
EU R : the retailer’s expected utility function,
Supposed: p > w > c > v ;
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3 Model and analysis
When the order quantity of retailer is more than the demand, there will be remaining stock after selling
season, and the salvage value is very low. We assume that a waste-averse decision maker particularly
dislikes salvaging excess inventory, so incurs an additional penalty α (α > 0) for each unit of inventory that
must be salvaged at the end of the season.
When the order quantity of retailer is less than the demand, there will be losing potential sales. We refer
to these preferences as stockout-aversion. For example, stocking out could lead to irate customers or a loss
in market share[10]. We assume that a stockout-averse decision maker particularly dislikes losing sales or
market share, so incurs an additional penalty β ( β > 0) for each demand that the retailer does not satisfy.
We consider the model in which the retailer has both waste-averse and stockout-aversion preferences in
this paper. According to Schweitzer M E, Cachon G.(2000), we can get the retailer’s expected utility as
follow:
EU R = Π R − α I (q ) − β L(q ) or
q
+∞
0
q
，
EU R = Π R − α ∫ (q − x) f ( x)dx − β ∫ ( x − q) f ( x)dx
(1)
3.1 Centralized decision making
The supply chain coordination of contract targets centralized decision making, and try to realize
profit-maximizing of the whole supply chain. So, we consider the centralized problem where the supplier
and retailer are owned by one firm. The firm’s objective is to maximize its own expected profits for the
known information and it does not consider allocating the profits of the supply chain system. We assume the
demand and sale price are exogenous, so the problem, now, is that the firm’s owner chooses the production
quantity which follows the optimization problem to meet his objective.
Supply chain’s profit function:
(2)
Π T = ( p − v ) S ( q ) + (v − c ) q
：
According to the Leibniz rule, We can get the optimal order quantity
p−c
q0 = F −1 (
)
p−v
(3)
q0 Is the optimal order quantity to meet the centralized supply’s profit-maximizing object. And it is
the aim of contract coordination for supply chain.
3.2 Decentralized decision making
Decentralized system means that the supplier and retailer are belong to different firms. The supplier
offers the good at a per unit wholesale price w , the retailer submits its orders in response to the supplier's
price and the demand. And the retailer retains possession of any excess stock. We also regard it as wholesale
contract. Although it can not coordinate the supply chain, it is worth studying because it is commonly
observed in practice. 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 profit increase[1]. Lariviere and Porteus[12]
gave a more complete analysis of this contract in the context of the newsvendor problem.
So according to the decentralized decision making or wholesale price contract, we can come to the
expressions as follows
(4)
a) supply chain’s expected profit Π S ( w) = ( w − c) q
：
b) the transfer payment
：
： T (q, w) = wq
(5)
w
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c)
d)
：Π
the retailer’s expected utility：
the retailer’s expected profit
R ( w)
(q, w) = ( p − v) S (q) − ( w − v) q
q
+∞
0
q
(6)
EU R ( w) = Π R ( w) − α ∫ (q − x) f ( x)dx − β ∫ ( x − q ) f ( x) dx
Proposition 1
(7)
There exists a unique optimal order quantity qwπ for the retailer to maximize his
profit with the function Π R ( w) . And qwπ meets:
qwπ = F −1 (
p−w
)
p−v
(8)
Proposition 2 There exists a unique optimal order quantity qw for the retailer to maximize his
utility with the function EU R ( w) . And qw meets:
qw = F −1 (
p − w+ β
)
p − v +α + β
(9)
Since F −1 (⋅) is strictly increasing, we can deduce conclusion 1 from proposition 1 and proposition 2.
Conclusion 1
when
p−w+β
p−w
>
p − v +α + β
p−v
，the retailer’s optimal order quantity with his combined
decision bias is more than without decision bias, otherwise the situation is the opposite.
From Equation (9), we could come to the conclusion 2
Conclusion 2 Given the retailer’s waste-averse preferences quotiety, the retailer’s optimal order
quantity increases as the stockout-averse preferences increases and decreases as waste -averse preferences
increases.
：
3.3 The rebate and penalty contract
With a rebate and penalty contract, the supplier charges w per unit, but then gives the retailer a τ
rebate pur unit above a threshold T . At the same time, the supplier punishes the retailer also a τ pur unit
under the threshold T . Taylor and Krishan[7] study this contract in the case that the retailer’s effort is chosen
simultaneously with the order quantity. Kapuscinski and Butz[8] studies this contract in the case that the
retailer chooses an order quantity, a signal of demand is observed and then effort is exerted.
According to the contract, we can come to the expressions as follows
(12)
a) the supplier’s expected profit Π S (τ ) = ( w − c) q − τ ( S ( q) − T )
b)
c)
d)
：
：
the transfer payment： T (q, w,τ ) = wq − τ ( S (q) − T )
the retailer’s expected profit： Π (q, w,τ ) = ( p − v + τ ) S (q ) + (v − w)q − τ T
the retailer’s expected utility： EU = Π − α ∫ (q − x) f ( x )dx − β ∫ ( x − q) f ( x)dx
τ
R (τ )
R (τ )
R (τ )
q
+∞
0
q
(13)
(14)
(15)
Ref the related literature, we know that the supply chain’s optimal rebate and penalty quotiety is
( w − c)( p − v)
, when it is under newsvendor problem and using this contract.
τ0 =
c−v
Proposition 3
(16)
EU R (τ ) is concave for q , There exists a unique optimal order quantity qτ for the
retailer to maximize his utility with the function EU R (τ ) . And qτ meets:
qτ = F −1 (
p−w+β
)
p − r +α + β
Proof Considering first derivative and second derivative of EU R (τ ) on the order quantity, we get
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(17)
dEU R (τ )
= ( p + τ − w + β ) − ( p − v + τ + α + β ) F (q )
dq
d 2 EU R (τ )
= −( p − v + τ + α + β )( f ( q))
dq 2
From
Equation (19) we could prove :
(18)
d 2 EU R (τ )
dq 2
<0
，so
(19)
EU R (τ ) is strictly concave in q, we assume
that qτ is the optimal order quantity. So it meets:
dEU R (τ )
dq
= ( p + τ − w + β ) − ( p − v + τ + α + β ) F ( q) = 0
q = qτ
F (qτ ) =
p +τ − w + β
p − v +τ + α + β
：
And we get the retailer’s optimal order quantity with rebate and penalty contract
p +τ − w + β
qτ = F (
)
p +τ − v +α + β
−1
【
】
Proof complete
We could prove the Proposition 4 in the same way:
Proposition 4 Π R (τ ) is concave for q , There exists a unique optimal order quantity qπ for the
τ
retailer to maximize his utility with the function Π R (τ ) . And qπ meets:
τ
p − w +τ
)
q =F (
p − v +τ
π
τ
−1
(20)
Proposition 5 Given the supplier’s wholesale price w , there exist an appropriate rebate and penalty
quotiety τ which makes the revenue sharing contract coordinate the supply chain with the retailer’s
combined decision bias.
Proof From Equation (15), we could come to the retailer’s expected utility expression
(21)
EU R( r ) = ( p − v + τ + α ) S (q) + (v − w − α + β )q − τ T − β u
：
 p − v + τ + α = λ ( p − v)
v − w − α + β = λ (v − c)
Let 
：
，and λ > 0 ,
So, we compare to Equation (1) and come to
EU R( r ) = λΠT − τ T − β u
It follows immediately that qτ = q0 is the optimal for the retailer. When T <
coordinates the supply chain.
λΠT − β u
, EU R ( r ) > 0 it
τ
【Proof complete】
With qτ = q0 , come to:
p +τ − w + β
p−c
=
p +τ − v +α + β p − v
τ=
Since
∂r
p −c
∂r
=
> 0,
= −1
∂α c − v
∂β
p−c
( p − v)( w − c)
α −β +
c −v
c −v
，come to conclusion 3：
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(22)
Conclusion 3 Given the retailer’s waste-averse preferences quotiety α , the supplier’s rebate and
penalty quotiety τ decreases as the retailer’s stockout-averse preferences increases ; Given the retailer’s
stockout-averse preferences quotiety β , τ increases as waste -averse preferences α increases.
4 Numerical Analysis and Discussions
Demand density function
：
 1 4000 , 0 ≤ x ≤ 4000
f ( x) = 
0,else
1, x > 4000

F ( x) =  x 4000 , 0 ≤ x ≤ 4000

0, x < 0
w=3
v = 1 ; and 0 ≤ α ≤ 1,0 ≤ β ≤ 1
：
More assume： p = 5 ， c = 2 ，
Demand distribution function
，
，
From Equation (22), we could come to:
。
rebate and penalty quotiety
(23)
τ (α , β ) = 3α − β + 4
The relation of supplier’s rebate and penalty quotiety τ to the retailer’s waste-averse preferences
quotiety α and retailer’s stockout-averse preferences quotiety β is as following figure:
7
6
5
4
3
1
1
0.5
stock-out averse
0.5
0
0
waste-averse
Fig 1 The relation between supplier’s revenue sharing quotiety τ and retailer’s decision bias
From the figure, we know that the minimal rebate and penalty quotiety is τ min = r (0,1) = 3 , and the
maximal revenue sharing quotiety is τ max = r (1,0) = 7 .
，
In the standard “Newsvendor Problem” model with rebate and penalty contract the supply chain’s
contract parameter is τ 0 = 4 , so τ min < τ 0 < τ max .
We assume that the retailer’s decision bias quotiety is determinate, i.e. α = 0.4, β = 0.6 . So, the
retailer’s expected profit and expected utility with wholesale price contract:
2
Π Opt
(24)
R ( w ) ( w) = 500(5 − w)
：
( w) = −
EU ROpt
(w)
(4480 − 800w)2 28
+(
− w) * (4480 − 800w) − 1200
5
2000
The retailer’s expected profit and expected utility with revenue sharing contract:
80
(25)
2000 w
3
Opt
EU R ( r ) (w) = 7650 − 1500 w
Π Opt
R (φ ) ( w) =
(26)
(27)
Taking w as independent variable, we draw the figures for the function of the four above figures in
different group.
6000
5000
the retailer's expected profit
the retailer's expected utility
retailer's expected profit
retailer's expected utility
4500
4000
4000
expected profit\utiliy
expected profit/utility
5000
3000
2000
1000
0
3500
3000
2500
2000
1500
1000
-1000
-2000
500
2
2.5
3
3.5
wholesale price
4
4.5
0
5
Fig 2 Expected profit/ utility with wholesale
2
2.5
3
3.5
wholesale price
4
4.5
5
Fig 3 Expected profit/utility with rebate and penalty
Fig 2 shows the difference between the retailer’s expected profit and utility with wholesale contract.
The wholesale price elasticity of expected profit is less than that of expected utility. And we could know that
the relation of optimal order quantity with different decision-making rule: when 0 ≤ w ≤ 2.6 , the optimal
order quantity with profit-maximization rule is equal to or more than that with utility-maximization; when
2.6 ≤ w ≤ 5 , the optimal order quantity with profit-maximization rule is less than that of with
utility-maximization.
Fig 3 shows the difference between the retailer’s expected profit and utility with rebate and penalty
contract. The wholesale price elasticity of expected profit is equal to that of expected utility, but the expected
utility always higher than expected profit. And they decrease as the wholesale price increases. The optimal
order quantity for profit and utility are the same, i.e. 2000.
6000
4500
expected profit of wholesale contract
expected profit of rebate and penalty
4000
3500
4000
3000
3000
2500
utility
expected profit
Utility of wholeslae contract
Utility of rebate and penalty contract
5000
2000
2000
1000
1500
0
1000
-1000
500
0
-2000
2
2.5
3
3.5
wholesale price
4
4.5
5
Fig 4 Expected profit of wholesale /rebate and penalty
2
2.5
3
3.5
4
wholesale price
4.5
5
Fig 5 Expected utility of wholesale/ rebate and penalty
Fig 4 shows the different expected profit of retailer between wholesale contract and rebate and penalty
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contract. From the figure, we know that the expected profit of rebate and penalty contract is always higher
than that of wholesale contract.
Fig 5 shows the different expected utility of retailer between wholesale contract and rebate and penalty
contract. From the figure, we know that the expected utility of wholesale contract is higher than that of
rebate and penalty contract at first, but it changes to the contrary as the wholesale price increasing farther.
In a word the expected profit and utility of rebate and penalty contract is higher than that of wholesale
contract basically.
，
5 Conclusions
In this paper, we study the supply chain rebate and penalty contract with the retailer’s combined
decision bias. And the result shows that there are difference in rebate and penalty contract between with
retailer’s combined decision bias and without retailer’s combined decision bias: (1) The supplier’s optimal
rebate and penalty quotiety which can achieve channel coordination has close relation with the retailer’s
decision bias. (2) The rebate and penalty quotiety increases as the retailer’s waste-averse preferences
increases and decreases as stockout-averse preferences increases.
Further research: The supply chain contract coordination model with the supplier’s decision bias, or
with price dependent demand at the same time.
References
，
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