Quantity Discount Contract for Supply Chain
Coordination with False Failure Returns
Ximin Huang∗∗∗
Sin-Man Choi†
Wai-Ki Ching
‡
31 March 2010
Abstract
Consumer return attracts more and more academic attention due to its rapidly expanding
size, while a large portion of it falls into the category of false failure return, which refers to
return without functional defect. In this paper, we exclusively consider profit results from
exerting costly effort to reduce false failure return in a reverse supply chain consisting of a
retailer and a supplier. The supply chain as a whole has strong incentive to reduce false failure
returns because it can avoid much re-processing cost associated. But since typically, retailers
enjoy a full credit provided by suppliers in case of returns, hence they may not have sufficient
incentive to exert enough effort for supply chain profit maximization. In some scenarios they
may even have motivation to actually encourage such returns. Given the similarity of this
problem with the one in forward supply chain that retailers may adopt a socially suboptimal
action when maximizing their own profit, we suggest using a coordination contract to resolve
such profit conflicts. The contract we propose is a quantity discount contract specifying a
payment to the retailer with an amount exponentially decreasing in the number of false
failure returns. We give explicit forms of such contracts given different assumptions about
distribution of the number of returns and we also prove that such contract is capable of
increasing both retailer’s and supplier’s profit simultaneously. Besides, when the contract
is used together with other forward supply chain coordination contracts in a closed-loop
chain, it is shown that it can act to deter retailer’s potential incentive to encourage false
failure returns. Moreover, some modifications of the contract may lead to easy allocation of
incremental profit within the supply chain.
Keywords: Consumer Returns; Closed-loop Supply Chains; Quantity Discount Contract; Supply Chain Coordination.
∗∗∗
Corresponding author. Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. E-mail: [email protected]
†
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of
Hong Kong, Pokfulam Road, Hong Kong. E-mail: [email protected]
‡
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of
Hong Kong, Pokfulam Road, Hong Kong. E-mail: [email protected]. Research supported in part by RGC
Grants, HKU CRCG Grants, Hung Hing Ying Physical Sciences Research Fund and HKU Strategic Research
Theme Fund on Computational Sciences.
1
1
Introduction
Many firms now offer liberal return policies allowing customers to return the products for any
reasons within some time (typically 90 days) after the purchase. The volume of such consumer
returns is rapidly increasing and already exceeded $100 billion per year in the U.S., see for
instance [6]. However, a large portion (95% in the electronic industry in U.S. by [5]) of them is a
result of reasons other than functional defects of the products. For example, it may be possible
that the customers regret over an impulse purchase, or they later find out that the product is
not suitable or too difficult to use. This kind of returns is referred to as “false failure returns”
in [1]. This is the main subject we are going to study in this paper.
Another closely related concept here is the closed-loop supply chain and reverse supply
chain. According to [2], the process of material flowing from supplier for manufacturing and
then to retailer for selling and finally purchased by customers is referred to as forward supply
chain. Then, in case of returns, the process of retailer accepting the returned products from
customers and transferring them back to supplier for possible re-manufacturing is referred to
as reverse supply chain. The forward and the reverse together form a closed-loop supply chain.
Usually, study on reverse supply chain is motivated by two reasons. The first one concerns about
sustainable development of the earth. Because re-collected products can be used for recycling,
hence reduce trash by converting them to treasure. Besides the ecological consideration, the
economical reason is that now material value involved in returns is too large to be ignored.
Studies (for example, [2]) show that proper design of reverse supply chain may even bring in
extra profit to firms in addition to lower the cost of it.
Although much existing literature focus on the planning or impact of re-manufacturing, this
paper studies the incentive conflicts in a reverse supply chain concerning exerting costly effort
to reduce false failure returns. Reducing such returns is usually highly beneficially to suppliers.
Suppliers who value much on their brand image typically provide a full credit to retailers and
take the returned products back. And by [3], during the take-back process, suppliers not only
bear goodwill cost but also have to pay for cost of possible test, refurbishment, re-manufacture,
recycle or loss in value of the products especially for innovative or fashionable ones, see for
instance, a discussion in [4]. The sum of these costs may be substantial but if the amount
of returns is reduced, such loss can be avoided. On the other hand, retailers do have to pay
something in case of a false failure return, for example, goodwill cost or fees of reprocessing and
transportation. But since they get full credit from suppliers, their incentives of reducing such
returns is usually lower. In some circumstances, they may even want to actively push the level
of such returns, for instance, if the product has poor demand.
In order to resolve this profit conflict, we propose a coordination contract. The design
of coordination contracts for a forward supply chain has been extensively studied and lots of
results are available. However, such findings may not be directly applied in the context of reverse
supply chain due to the differences in nature of the two, for example, in terms of timing or way
of transaction. However, certain results may offer high reference value. For instance, in [2], a
2
modification of the rebate contract which is a coordination contract for forward supply chain is
demonstrated to be able to resolve incentive conflicts in the context of a reverse supply chain.
This paper is devoted to designing a coordination mechanism for a reverse supply chain
in presence of false failure returns. Specifically, we consider a supply chain consisting of one
retailer and one supplier. Attention is exclusively focused on the profit and cost associated
with false failure returns and ignore the operations in forward supply chain in the first part of
discussion. We will then incorporate the two in a later section. Consider the following events
occurring in sequence: when a customer brings in a false failure return, typically a full money
refund is provided by the retailer who later gets a full credit of the wholesale price from the
supplier and returns the product. Meanwhile, we assume effort can be exerted to reduce the
volume of false failure returns. Given the above discussions, it is reasonable to add another
assumption that the supplier has already exerted all possible effort and all possible remaining
effort as well as the associated cost is to be taken by the retailers. We show that the profitmaximizing effort level chosen by the retailer is always less than the one chosen from a supply
chain’s point of view. This is sub-optimal because the supply chain profit is eventually to be
split between retailer and supplier. The origin of this problem is that retailer bears all the cost
yet only enjoys partial benefit of reducing amount of false failure returns. Hence here we aim
to design a contract specifying a way for the supplier to share the cost by making a payment to
the retailer, so as to induce the retailer to exert global optimal level of effort. In addition, we
prove under certain conditions, such contract is also capable of improving both parties’ profit
simultaneously. Specifically, the contract payment is exponentially decreasing in the returned
amount. So it is very similar to the quantity discount contract which is a coordination contract
in forward supply chain specifying a wholesale price decreasing in order quantity, so we call it
quantity discount contract throughout the paper. It is reasonable to have contract payment
decreasing on number of the false failure returns because it is hard to directly contract on effort
level since it is very difficult to verified. Yet on the other hand, a lower returns level is rather
credibly indicating that retailers are exerting effort, so they should be compensated more in
this case. In order to give the explicit formula for determining contract parameters, we consider
three cases of distribution of the number of false failure returns. In the first case, we consider the
number to be geometrically distributed. Since such return is a result of consumer’s individual
decision rather than because of the quality of the product itself, hence they are independent
among themselves. So the geometric distribution, which is memoryless, would be a reasonable
distribution for this random variable. In the second case, returned number is assumed to have a
Poisson distribution, which is more general. The results obtained here would be more accurate
if sales volume is large or the period under consideration is long. Lastly, we consider the case
of normal distribution, a continuous distribution and two numerical examples are given to show
the effectiveness of the contract.
The rest of this paper is organized as follows. In Section 2, we set up the model of the
problem, and optimal choices of effort level by supply chain and retailer are shown respectively.
The design of coordination contracts given different distributions of the number of false failure
3
returns are presented in Section 3. Section 4 is devoted to a discussion of the contract and a
possible modification of the quantity discount contract is presented. Concluding remarks are
then given in Section 5.
2
The Model
We study performance of a supply chain consisting of one retailer and one supplier in a
single-period setting using the model similar with that appeared in [1]. And we will see that the
basic results in this section is similar with the ones in [1]. For every unit of product, the supplier
bears the manufacturing cost c and then sells it to the retailer at wholesale price w. Finally
the product is sold in the market at retail price p. It is assumed that the consumers, after the
purchase, have the right to return the product to the retailer anytime within the period. In case
of a false failure return, consumer gets a full refund of p from the retailer, who later gets a full
credit of w from the supplier. Now we proceed to formulate the profit function of reducing one
unit of false failure return as follows: after returning the product, the consumer may choose to
re-purchase another one from the same supplier, with certain probability, or walk away. Hence
the expected future profit to the supplier after the false failure return can be written as δs (w−c).
Similarly, we write the expected future profit to the retailer as δr (p − w). Moreover, false failure
return comes at a cost, such as goodwill cost and the possible re-processing cost through the
reverse supply chain. We denote sum of all these cost to the supplier as s and the cost to the
retailer as r. Combining the cost saved and future profit, the profit of avoiding one false failure
return is S = δs (w − c) + s to the supplier and R = δr (p − w) + r to the retailer. On the other
hand, as in [1], number of false failure returns, which denoted as X(ρ), is modeled to be a nonnegative random variable depending on the effort level ρ. Denote its Cumulative Distribution
Function (CDF) as F (x|ρ) and Probability Mass Function (PMF) as f (x|ρ). It is assumed that
ρ ≥ 1 (at least a minimal of effort has to be exerted), and effort cost is
c(ρ) =
a 2
ρ
2
because quadratic function can approximate two features of effort cost: it is strictly increasing
and have decreasing marginal return. Assume that
E{X(ρ)} =
β
ρ
where β = E{X(ρ = 1)}
i.e. β is the number of false failure returns when minimal level of effort is in force. As in [1], we
confine our attention to impact of false failure returns on profit of the supply chain. The profit
of supply chain as a whole can be valued as
a
1
a
Π(ρ) = (S + R)(E{X(ρ)} − E{X(1)}) − ρ2 = (S + R)β(1 − ) − ρ2
2
ρ
2
4
(1)
Note that the first term is the product of expected profit of avoiding one false failure return and
expected number of reduced units with effort ρ, which is just the expected total profit. Then
cost is subtracted to make the net profit.
We want to find an effort level ρ to maximize the supply chain profit, to this end, we set
0=
1
∂Π(ρ)
= (S + R)β 2 − aρ
∂ρ
ρ
Solving it yields
ρC =
S+R
β
a
1/3
.
(2)
We check the second order condition as follows:
∂ 2 Π(ρ)
2
= −(S + R)β 3 − a < 0.
2
∂ρ
ρ
Hence Π(ρ) is actually concave in ρ and the ρC above is the effort level such that the supply
chain profit is maximized. ρC ≥ 1 is always assumed.
Note that since supply chain profit is the sum of profit earned by both retailer and supplier.
So with any transfer payment between the two, results in this section still hold.
2.1
Decentralized Case
Now we investigate the supply chain in absence of any contracts between the the retailer
and supplier concerning false failure returns. As in [1] and has been justified above, all effort
and associated cost of reducing returns is assumed to be taken by the retailer. Similar with
calculating supply chain’s profit, retailer’s profit is
1
1
ΠR (ρ) = Rβ(1 − ) − aρ2 .
ρ
2
We can easily verify that
ρ=
Rβ
a
1/3
is the maximal solution by solving
Rβ
∂ΠR (ρ)
= 2 − aρ = 0
∂ρ
ρ
and check that ΠR (ρ) is concave in ρ because
∂ 2 ΠR (ρ)
2Rβ
= − 3 − a < 0.
2
∂ρ
ρ
5
(3)
But since the retailer has to exert at least minimal amount of effort, hence the effort level chosen
by the retailer to maximize his/her own profit would be
ρD = max{(
Rβ 1/3
) , 1}
a
It is obvious that ρC ≥ ρD , which means that if without any coordination, the retailer always
chooses an effort level that is lower than the supply chain optimum. This disagreement arises
because the retailer pays all the cost but only gets less expected profit than the supply chain
does. Some wealth is transferred to the supplier who is paying nothing. So it would be expected
that in order to induce the retailer to exert supply chain optimal effort, the supplier should
compensate the retailer for part of the cost.
And in the following discussion, the following condition
ρC =
S+R
β
a
1/3
>1
is always assumed. Because otherwise, ρC = ρD = 1, the supply chain is already in coordination.
Similar to the above, the supplier’s profit with effort level ρ would be
1
ΠS (ρ) = Sβ(1 − ).
ρ
3
(4)
The Quantity Discount Contract
In this section, we propose a quantity discount contract specifying a payment by supplier
to retailer to induce the later to exert supply chain optimal level of effort and hence to achieve
coordination. The payment amount is exponentially decreasing in the number of false failure
returns. Specifically, the value of the payment is
P (x) = T αx
(5)
where x is the number of false failure returns; T is a contract parameter with value equal to the
payment by supplier if there is no false failure return; α is another contract parameter satisfying
0 < α < 1. Hence
∞
X
(T αx )f (x)
E{P (x|ρ))} =
x=0
With such contract, retailer’s profit becomes
1
1
ΠR (ρ|(T, α))) = Rβ(1 − ) − aρ2 + E{P (x|ρ))}.
ρ
2
(6)
Similarly, supplier’s profit changes to
1
ΠS (ρ|(T, α))) = Sβ(1 − ) − E{P (x|ρ))}
ρ
6
(7)
We examine how this contract works under three scenarios, one with X(ρ) following a geometric
distribution, another with X(ρ) following a Poisson distribution and the third is for a normal
distribution case.
3.1
The Geometric Distribution Case
In this section, we discuss the case that the number of false failure returns follows a geometric
distribution. Specifically, assume that
f (x|ρ) = (
ρ
β x
)(
)
ρ+β ρ+β
with x = 0, 1, 2 . . . .
It can be easily checked that
E{X(ρ)} =
β
ρ
is satisfied here. The expected value of contract payment would be
E{P (x|ρ)} =
∞
X
x=0
ρ
β x
Tα (
)(
) =T
ρ+β ρ+β
x
ρ
ρ+k
(8)
with k = (1 − α)β. The following proposition shows the form of the coordination contract.
Proposition 1. Assuming the number of false failure returns is geometrically distributed, then
the quantity discount contract specifying a payment of T αx to the retailer with
T =
aS(k + ρC )2
ρC
k(S + R)
and
0<α<1
can coordinate the supply chain.
Proof. By (6) and (8), retailer’s profit now reduces to
aρ2
ρ
1
+T
ΠR (ρ|(T, α)) = Rβ(1 − ) −
ρ
2
ρ+k
We proceed by checking that ρC is indeed the maximal solution to this profit. First, it is easily
confirmed that first order condition is fulfilled with (2):
∂Π(ρ|(T,α)) ∂ρ
ρ=ρ
Moreover,
=
C
=
Rβ
k
( 2 − aρ + T
)
2
ρ
(k + ρ) ρ=ρC
k
aS (k + ρC )2
Rβ
ρC
−
aρ
+
= 0.
C
2
ρC
S+R
k
(k + ρC )2
∂ 2 Π(ρ|(T, α))
2Rβ
2k
=− 3 −a−T
< 0.
2
∂ρ
ρ
(k + ρ)3
Hence with T as specified above, ρC would be the unique maximal solution to (6). This means,
7
the contract can successfully induce the retailer to exert such level of effort that supply chain
profit is maximized.
Now we give a qualitative analysis of
T =
aS(k + ρC )2
ρC .
k(S + R)
Here we note that T increases in S and a yet decreases in R. The reason is that T measures the
payment size paid by supplier to induce retailer to exert more effort to reach supply chain optimal
level, hence if S is large, that means more benefit from the effort is transferred to supplier, so
supplier should have to compensate the retailer more. But retailers need less external incentive
since they benefit more from the reduction itself when R is large. However, more cost-sharing
is needed when a is larger because it implies higher effort cost. T is also positively related with
β because a larger β means that per unit effort exerted can bring down a larger number of false
failure returns hence it becomes more urgent and supplier is willing to pay more for it.
Proposition 2. With the quantity discount contract described above, the retailer’s profit is
always increased while supplier is better off if and only if
2
1
1
−
>
ρD
ρC
β
and
0<α≤1−
β( ρ1D
1
−
2 .
ρC )
Proof. First, note that since we have proved that the contract in Proposition 1 can induce the
retailer to choose ρC , hence now the payment from the contract, by (8) is
E{P (x|ρC )} =
=
=
=
T ρC
k+ρC
(k+ρC )2
ρC
aS
k+ρC S+R ρC
k
aS k+ρC 2
S+R k ρC
Sβ
Sβ
Sβ
aS 2
S+R ρC + k = ρC + k .
The retailer will earn more requires the following holds
Rβ(1 −
1
1
1
1
) − aρC 2 + E{P (x|ρC )} ≥ Rβ(1 −
) − aρD 2 .
ρC
2
ρD
2
Rearrange terms to get an equivalent inequality:
Sβ Sβ
1
1
1
+ Rβ(
+
−
) − a(ρC 2 − ρD 2 ) ≥ 0.
ρC
k
ρD
ρC
2
There are two cases to be discussed.
1/3 } = ( Rβ )1/3 :
Case: 1 ρD = max{1, ( Rβ
a )
a
It suffices to prove
S
1
1
1
Sβ
≥ a(ρC 2 − ρD 2 ) −
− Rβ(
−
).
1−α
2
ρC
ρD
ρC
8
(9)
We proceed by proving the following:
S≥
Sβ
1
1
1
a(ρc 2 − ρD 2 ) −
− Rβ(
−
).
2
ρC
ρD
ρC
By the expressions of ρC and ρD , we have
a
R = ρ3D ,
β
S + R = ρ3C
a
β
and
a
S = (ρ3C − ρ3D ) ,
β
hence the above is equivalent to
(ρ3C − ρ3D )
1
a ρ3C − ρ3D
1
a
1
a
≥ a(ρc 2 − ρD 2 ) − β
− β ρD 3 (
−
).
β
2
β
ρC
β
ρD
ρC
Simplifying it yields:
2(ρ2C + ρD ρC + ρ2D )ρC ≥ −β(ρ2C + ρD ρC + 4ρ2D ).
The last inequality is obviously always holds. Moreover, since
combining with the above, we have shown that (9) holds.
1
1−α
> 1 because 0 < α < 1,
1/3 } = 1:
Case:2 ρD = max{1, ( Rβ
a )
1
1
E{P (x|ρC )} − a(ρ2C − 1) + Rβ(1 −
)
2
ρC
βS
1
1
1
1
(Sβ +
ρC − Sβ − Rβ + aρC + RβρC − Rβ)
=
ρC
k
2
2
2
1
a a
βS
1
ρC − Sβ − + + Rβ − Rβ)
(Sβ +
≥
ρC
k
2
2 2
1 Sβ
Sβ
=
+
ρC ) ≥ 0.
(
ρC 2
k
1/3 .
The second last inequality holds because ρC > 1 and a ≥ Rβ implied by 1 ≥ ( Rβ
a )
Hence retailer is always better off.
In order for supplier to earn more, it must be true that
Sβ(1 −
1
1
) − E{P (x|ρC )} ≥ Sβ(1 −
)
ρC
ρD
With Equation (8), it is equivalent to
Sβ(
Further simplification gives
1
1
1
1
−
) ≥ Sβ( +
).
ρD
ρC
k ρC
1
1
2
≤
−
.
β(1 − α)
ρD
ρC
9
(10)
To ensure the existence of the parameter α ∈ (0, 1), we need
2
1
1
−
> ,
ρD
ρC
β
while to ensure the inequality, we need
0<α≤1−
β( ρ1D
1
−
2 .
ρC )
To interpret the proposition, we note that E{P (x|ρC )}, which is the payment born by the
supplier, increases in α. If α is too large, the payment may outweigh the benefit of exerting
effort, so it may actually damage the supplier’s profit. Besides, it is very interesting to compare
the result here with the one in [1]. In that paper the condition for supplier to be better off is
1
2
−
>0
ρD
ρC
which is equivalent to S > 7R. But here we require
2
1
1
−
> ,
ρC
ρC
β
which implies S should be greater than an amount even larger than 7R. The reason is that,
although [1] is applying a target rebate contract under the assumption that X(ρ) is normally
distributed, which are all different from our settings, it has been shown that in coordination
E{P (x|ρC )} =
Sβ
ρC
in that paper while here we have
E{P (x|ρC )} =
Sβ Sβ
.
+
ρC
k
Hence in comparison, we are actually asking the supplier to pay more to the retailer. So
correspondingly, the profit of effort transferred to the supplier, which can be measured by S,
should be larger here to guarantee the supplier’s profit increment.
3.2
The Poisson Distribution Case
In this section, the number of false failure returns is assumed to follow a Poisson distribution
to model for a more general scenario. Specifically, we consider
f (x|ρ) =
(β/ρ)x −β/ρ
e
x!
10
with
x = 0, 1, 2 . . . .
It can be easily checked that
E{X(ρ)} =
β
ρ
holds as proposed. Expected value of contract payment is
E{P (x|ρ)} =
∞
X
(β/ρ)x
x=0
x!
β
e−β/ρ T αx = T e ρ
(α−1)
.
(11)
We now aim to find a coordination contract under this distribution.
Proposition 3. When the number of false failure returns has a Poisson distribution, the quantity
discount contract specifying a payment T αx by supplier can coordinate the supply chain with
T =
β
S
(1−α)
e ρC
1−α
and
α ∈ (max{1 −
3(S + R)
ρC , 0}, 1).
Sβ
Proof. By (6), now the retailer’s profit is
β
1
1
(α−1)
ΠR (ρ) = Rβ(1 − ) − aρ2 + T e ρ
ρ
2
β
When T =
(1−α)
S
ρC
,
1−α e
the first order condition is fulfilled at ρ = ρC :
∂ΠR (ρ|(T, α) ∂ρ
ρ=ρC
Moreover, with α ∈ (max{1 −
∂ 2 ΠR (ρ|(T, α)) ∂ρ2
ρ=ρC
=
=
β
β
βR
(α−1)
(− 2 )(α − 1))
( 2 − aρ + T e ρ
ρ
ρ
ρ=ρC
β
S
(1 − α) − (S + R) + R = 0.
ρ2C 1 − α
3(S+R)
Sβ ρC , 0}, 1),
we have
β
2β
2β
β2
(α−1)
− a + (α − 1)( 3 + (α − 1) 4 )T e ρC
3
ρC
ρC
ρC
a
β
S
= −
(3R + S + (1 − α)(2 + (α − 1) ))
))
S+R
ρC 1 − α
a
Sβ
= −
(3(S + R) + (α − 1)
) < 0.
S+R
ρC
= −
So it is proved that ρC is the unique solution to maximize the retailer’s profit when the contract
is in force, hence supply chain coordination is achieved.
Comparing the form of the quantity discount contract ensuring supply chain coordination in
the last section with the one here, it can be easily seen that with geometrically distributed X(ρ),
α can be any number in (0, 1), while in the case of X(ρ) being a Poisson distributed random
11
variable α cannot be falling into too low a range under some conditions. Possible explanation
for it may be that with the same mean ρβC , geometrically distributed X(ρ) tends to cluster at a
low range while Poisson distributed one is more dense around the mean, so given the payment
amount is of the form T αx , α cannot be too small in the later case because the retailer gets
small payments with higher probability than in the first case. Take a close look into
T =
β
S
(1−α)
e ρC
1−α
we conclude that value of T increases in a and β while decreases in R. The analysis and the
rationale is similar with the one in the last section.
Proposition 4. Under the assumption that X(ρ) has a Poisson distribution, if the contract
specified in Proposition 3 is adopted, the retailer will be earning more if
max{0, 1 −
2ρC (ρ2C + ρC ρD + ρ2D )
}<α<1
β(ρ2C + ρC ρD − 2ρ2D )
max{0, 1 −
2ρC
}<α<1
β
when
when
ρD = (
ρD = 1
while supplier will be better off if
0<α≤1−
β( ρ1D
1
−
1 .
ρC )
Proof. By Equation (11) and Proposition3,
E{P (x|ρC )} = (
β
β
S
S
(1−α)
(α−1)
)(e ρC
)=
e ρC
.
1−α
1−α
First we focus on the retailer’s side.
1/3 } = ( Rβ )1/3 :
Case: 1 ρD = max{1, ( Rβ
a )
a
By Equation (9), the retailer is better off if and only if
1
1
1
S
a(ρ2C − ρ2D ) − Rβ(
.
−
)≤
2
ρD
ρC
1−α
Substituting S = (ρ3C − ρ3D ) βa and R = ρ3D βa in and reduce it to
1−α≤
2(ρ3C − ρ3D ) βa
D
a(ρ2C − ρ2d ) − 2ρ3D βa ρρcC−ρ
ρD
Further simplification yields
1−
2ρC (ρ2C + ρC ρD + ρ2D )
≤ α.
β(ρ2C + ρc ρD − 2ρ2D )
12
.
Rβ 1/3
)
a
1/3 } = 1:
Case: 2 ρD = max{1, ( Rβ
a )
≥
≥
S
1−α −
S
1−α −
2
S
2 ( 1−α
1
1
2
2 a(ρC − 1) + Rβ(1 − ρC )
1 1 S+R
a
ρC ( 2 a a β − 2 ρC )
− ρβC ) ≥ 0
when α > 1 − 2ρβC , by making use of the facts that ρC > 1 and Rβ ≤ a implied by ρD = 1.
Then let’s look at the supplier’s side. By Equation (10), the supplier is better off if
S
1
1
≤ Sβ(
−
).
1−α
ρD
ρC
After some simplification we get
α≤1−
1
β( ρ1D −
1 .
ρC )
Proposition 5. If the contract in Proposition 3 were to both coordinate the supply chain and
guarantee Pareto improving for supplier and retailer,
max{0, 1−
2ρC (ρ2C + ρC ρD + ρ2D )
3(S + R)
1
, 1−
ρC } < α ≤ 1−
Sβ
β(ρ2C + ρc ρD − 2ρ2D )
β( ρ1D −
max{0, 1 −
3(S + R)
1
2ρC
,1 −
ρC } < α ≤ 1 −
1
β
Sβ
β( ρD −
when
1
ρC )
1
ρC )
when
ρD = (
Rβ 1/3
) .
a
ρD = 1.
And to guarantee the existence of such α, it is required that
β(
1
1
−
)>1
ρD
ρC
holds in any case. Moreover, if ρD = 1, then it is also required that
ρC ≥ 3/2
and
(4S + 3R)3
β
>
α
27(S + R)4
must hold.
Proof. First, for Pareto improving to exists for both supplier and retailer, we should have
1−
2ρC (ρ2C + ρC ρD + ρ2D )
1
≤α≤
1
2
2
β(ρC + ρc ρD − 2ρD )
β( ρD −
1−
2ρC
1
≤α≤
β
β( ρ1D −
13
1
ρC )
1
ρC )
when ρD = (
when ρD = 1
Rβ 1/3
) .
a
Besides, 0 < α < 1 must be satisfied by the model assumption and
1−
3(S + R)
ρC < α
Sβ
is required by Proposition 3. We write
D=
2ρC (ρ2C + ρC ρD + ρ2D )
β(ρ2C + ρc ρD − 2ρ2D )
and
U=
1
−
β( ρ1D
and
D′ =
2ρC
β
1 .
ρC )
1/3 , the α satisfying 1 − D ≤ α ≤ 1 − U always exists since U ≤ D,
1. (a) If ρD = ( Rβ
a )
because
ρ2C (2ρC + ρD )
≥0
D−U =
β(ρc − ρD )(ρC + 2ρD )
since ρC ≥ ρD .
(b) If ρD = 1, with ρC ≥ 3/2, the α satisfying 1 − D ′ ≤ α ≤ 1 − U always exists since
U ≤ D ′ , because
D′ − U =
1
2ρC
−
β
β(1 −
1
ρC )
=
1 2ρC − 3
ρC
≥ 0.
β
ρC − 1
2. To make sure the α from (1 − D, 1 − U ) and (1 − D ′ , 1 − U ) both have intersections with
(0, 1), we need 1 − D < 1, 1 − D ′ < 1 and 1 − U > 0, which is requiring D > 0 D ′ > 0 and
U < 1. The first two conditions are obviously satisfied. To have
U=
β( ρ1D
we need
1
−
1
ρC )
< 1,
1
1
−
) > 1.
ρD
ρC
β(
3. To also incorporate the requirement that
1−
3(S + R)
ρC < α
Sβ
we need
U<
3(S + R)ρC
.
Sβ
1/3 , we have
(a) If ρD = ( Rβ
a )
R = ρ3D
a
β
a
and S + R = ρ3C .
β
14
Hence the inequality reduces to proving
ρ3C − ρ3D < 3ρ4C (
1
1
−
)
ρD
ρC
which is equivalent to
ρD ρ2C + ρC ρ2D + ρ3D < 3ρ3C .
And the last inequality always holds since ρC < ρD .
(b) If ρD = 1, the inequality reduces to (4S + R) < 3(S + R)ρC , which is just equivalent
to requiring
β
(4S + 3R)3
>
.
a
27(S + R)4
3.3
The Normal Distribution Case
In this section, the number of false failure returns is assumed to be a normal distributed
random variable. And throughout this section, φ(.) and Φ(.) are used to denote the PDF and
the CDF of a standard normal distribution respectively. To ensure E{(X(ρ)} = βρ as has been
specified above, we consider f (x|ρ) to be
(x − µ)2
1
} with µ = βρ .
f (x|ρ) = √ exp{
−2σ 2
σ 2π
Hence the expected value of total payment is
∞
∞
1
(x − µ)2
T exp{x ln α} √ exp{
}dx
−2σ 2
σ 2π
0
0Z
∞
(x − µ − σ 2 ln α))2
1
1
√ exp{−
} exp{ (σ 2 ln2 α + 2µ ln α)}dx
= T
2
2σ
2
σ 2π
0
1 2 2
µ + σ 2 ln α
= T exp{ (σ ln α + 2µ ln α)}Φ(
).
2
σ
E{P (x|ρ)} =
Z
T αx f (x|ρ)dx =
Z
By (6), retailer’s profit is now given by
1
1
1
µ + σ 2 ln α
ΠR (ρ) = Rβ(1 − ) − aρ2 + T exp{ (σ 2 ln2 α + 2µ ln α)}Φ(
).
ρ
2
2
σ
Proposition 6. When the number of false failure returns X(ρ) is a normal distributed random
variable, the quantity discount contract specifying a payment T αx by the supplier to the retailer
can coordinate the supply chain with
T =
S exp{− 12 (σ 2 ln2 α + 2 ρβC ln α)}
ln α1 Φ( β/ρC +σ
σ
2
ln α
) − σ1 φ( (β/ρC +σ
σ
15
2
ln α)
)
.
Moreover, the other contract parameter α should be chosen such that
∂ 2 ΠR (ρ) ∂ρ2 < 0.
ρ=ρC
Proof. For maximization, retailer’s choice of ρ is the solution to the following equation:
0=
1
2
∂ΠR (ρ)
Rβ
β
µ + σ 2 ln α
1
µ + σ 2 ln α 1
2
= 2 − aρ − 2 T e 2 (σ ln α+2µ ln α) (φ(
) − Φ(
) ln )
∂ρ
ρ
ρ
σ
σ
σ
α
For the T given as above, we have
∂ΠR (ρ) 1
= 2 (Rβ − (S + R)β + Sβ) = 0
∂ρ
ρC
ρ=ρC
Moreover, to make sure ρC is indeed a maximizer of retailer’s profit, the following is required so
as to satisfy the second order condition:
∂ 2 E{(P (x|ρ)} ∂ρ2
1
1
θ
1
β
β
ln α)(Φ(θ) ln − φ(θ) ) + φ(θ)( 2 − ln α ))
(−(2 +
3
ρ
α
σ
σ
σ
ρC
C
S
1
1
θ
1
β
β
ln α)(Φ(θ) ln − φ(θ) ) + φ(θ)( 2 − ln α ))
(−(2 +
ρC
α
σ
σ
σ
(Φ(θ) ln α1 − φ(θ) σ1 ) ρ3C
1
= T e2
ρ=ρC
=
(σ2 ln2 α+2 ln α ρβ )
C
where
θ=
β
ρC
− σ 2 ln α1
σ
.
Hence, we need
Rβ
∂ 2 E{(P (x|ρ)} ∂ 2 ΠR (ρ) = 2 − aρC +
< 0.
∂ρ2 ρ=ρC
∂ρ2
ρC
ρ=ρC
This is equivalent to say that
3(R + S) −
Sβ
1
Sφ(θ)
β 2
ln +
) > 0.
(
ρC
α σ ln α1 Φ(θ) − φ(θ) σρC
We remark that for this contract T > 0 always holds. Because for
g(x) = kΦ(x − k) − φ(x − k)
we have
g′ (x) = kφ(x − k) + (x − k)φ(x − k) = xφ(x − k) > 0
always holds for positive x, which means g(x) increases in x. Moreover, g(0) = 0. Hence it can
16
be concluded that g(x) > 0 holds for ∀x > 0 and ∀k. Let
σ ln
1
=k
α
and
β
= x > 0.
σρC
We can easily see that T > 0.
Since the this quantity discount contract is capable of coordinating the supply chain, the
contract payment thus reduces to
E{P (x|ρC )} =
=
S exp{− 12 (σ 2 ln α + σ 2 ln2 α)}
2
ln α1 Φ( µ+σσ ln α ) − σ1 φ( (µ+σσ ln α) )
SΦ(θ)
.
1
ln α Φ(θ) − σ1 φ(θ)
2
!
1
µ + σ 2 ln α
exp{ (σ 2 ln2 α + 2µ ln α)}Φ(
)
2
σ
The condition for both retailer and supplier to have profit growth, by (6) and (7), is
1
1
1
1
1
a(ρ2C − ρ2D ) − Rβ(
−
) ≤ E{P (x|ρC )} ≤ Sβ(
−
).
2
ρD
ρC
ρD
ρC
3.4
Numerical Examples
Here two numerical examples are given to show how much incremental profit can be obtained in presence of the contract compared with a decentralized case when X(ρ) is normally
distributed.
Parameters set for Figure 1 and Figure 2 are similar except that Figure 2 has a much larger
β, which indicates the size of the sales, and correspondingly, value of σ is larger. However, they
provide consistent qualitative insights. Increase in S leads to increase in all profit while a larger
R only results in a higher retailer profit but others’ profit is lower. Since α is contract parameter,
it does not effect the supply chain profit. However, it is shown that its value is actually effecting
the allocation of incremental profit within the chain.
4
Discussion
In this section, we discuss the quantity discount contract designed above from the aspect of
implementation as well as its impact on incentive when combined with a forward supply chain
contract. We then discuss some modifications of the contract.
First, we believe that the quantity discount contract here is attractive to retailers because by
the design of it, there will always be some payment to the retailer. And this payment is larger
in size if the number of false failure returns is lower, hence it should be able to provide direct
incentive for inducing effort.
17
90
60
Retailer
Supplier
Supply Chain
80
70
Retailer
Supplier
Supply Chain
50
60
40
50
30
40
30
20
20
10
10
0
20
25
30
35
40
45
50
55
0
10
60
(a) Dependence of Incremental Profit On S
14
16
18
20
22
24
26
28
(b) Dependence of Incremental Profit On R
35
30
12
28
Retailer
Supplier
Supply Chain
26
Retailer
Supplier
Supply Chain
24
22
25
20
20
18
16
15
14
12
10
10
5
50
55
60
65
70
8
75
2
3
4
5
6
7
8
9
10
(c) Dependence of Incremental Profit On β (d) Dependence of Incremental Profit On ln
1
α
Figure 1: Dependence of Incremental Profit of Supply Chain, Retailer And Supplier On Different
Parameters. Details as follows: (a) a = 1, β = 60, R = 20, ln α1 = 6, σ = 2; (b) a = 1, β = 60,
S = 30, ln α1 = 6, σ = 2; (c) a = 1, S = 30, R = 20, ln α1 = 6, σ = 2; (d) a = 1, β = 60, S = 30,
R = 20, σ = 2.
4.1
Incentives to Encourage False Failure Returns
Now we proceed to see how the contract works in cooperation with forward supply chain contract in a closed-loop supply chain context. As as been discussed in [1], retailers sometimes have
incentive to actually encourage sales with awareness of the high probability that the products
may end up being a false failure return if the supplier is providing a full credit, it actually works
as converting their inventory back to the suppliers. For example, retailers may want to do so if
they are selling something with poor demand. Here we show how the quantity discount contract
helps to deter retailers from such behaviors. Note that when the contract is absent, profit of a
failure failure return includes the full credit of wholesale price w, and cost of it would be the
sum of goodwill cost, re-processing cost of the returns, the salvage value of unsold inventory
and possible refund from supplier for leftover inventory from forward supply chain contract (for
example, buy-back contract or quantity flexibility contract), we sum them up and denote it as
b. Retailer encourages returns whenever it is beneficial to do so, i.e., when w > b. However,
since quantity discount contract gives the retailer a compensation d, which is higher for lower
18
200
80
Retailer
Supplier
Supply Chain
180
160
Retailer
Supplier
Supply Chain
70
60
140
50
120
100
40
80
30
60
20
40
10
20
0
20
25
30
35
40
45
50
55
0
20
60
(a) Dependence of Incremental Profit On S
24
26
28
30
32
34
36
38
(b) Dependence of Incremental Profit On R
70
60
22
65
Retailer
Supplier
Supply Chain
60
Retailer
Supplier
Supply Chain
55
50
50
45
40
40
30
35
30
20
25
10
240
250
260
270
280
290
300
310
320
20
330
2
3
4
5
6
7
8
(c) Dependence of Incremental Profit On β (d) Dependence of Incremental Profit On ln
1
α
Figure 2: Dependence of Incremental Profit of Supply Chain, Retailer And Supplier On Different
Parameters. Details as follows: (a) a = 1, β = 300, R = 25, ln α1 = 6, σ = 5; (b) a = 1, β = 300,
S = 30, ln α1 = 6, σ = 5; (c) a = 1, S = 30, R = 25, ln α1 = 6, σ = 5; (d) a = 1, β = 300, S = 30,
R = 25, σ = 5.
level of returns, hence returns is costing the retailer more, especially when return level is high.
Hence the condition of beneficially pushing false failure returns rises to w > b + d. Hence the
contract reduces the incentive of encouraging false failure returns all the time.
4.2
Modification of the Contract
It has been demonstrated that the quantity discount contract discussed above can coordinate
a supply chain with false failure returns and also ensures Pareto improving in the sense that both
retailer and supplier are better off with the contract. However, how the increased supply chain
profit is eventually split between the two parties is not immediately clear. Here we introduce one
possible way of modifying the original contract that results in arbitrary division of supply chain
incremental profit between retailer and supplier by adding a fixed payment into the contract
term.
To this end, define the following:
19
 ′
1
1

 ∆ ΠS = Sβ( ρD − ρC )
∆′ ΠR = Rβ( ρ1D − ρ1C ) − 12 a(ρ2c − ρ2D )


∆Π
= (R + S)β( ρ1D − ρ1C ) − 21 a(ρ2c − ρ2D )
Proposition 7. A contract specifies a transfer payment with an amount of T αx − Q by supplier
to retailer with (T, α) as in Proposition1 when X(ρ) is geometrically distributed or with (T, α)
as in Proposition 3 when X(ρ) has a Poisson distribution; or with (T, α) as in Proposition 6
when X(ρ) has a normal distribution; and
Q = θ∆′ ΠR − (1 − θ)∆′ ΠS + E{P (x|ρC )}
is a coordination contract for the supply chain with 0 ≤ θ ≤ 1. Moreover, this contract helps
to allocate θ of increased supply chain profit to supplier and (1 − θ)∆Π to retailer and hence
guarantees the existence of Pareto improving for retailer and supplier.
Proof. Denote increase of profit for retailer and supplier under this contract is ∆ΠR and ∆ΠS
respectively.
∆ΠR = ∆′ ΠR + (E{P (x|ρC )} − Q)
= ∆′ ΠR + E{P (x|ρC )} − θ∆′ ΠR + (1 − θ)∆′ ΠS − E{P (x|ρC )}
= (1 − θ)∆Π
Hence, ∆ΠS = ∆Π − ∆ΠR = θ∆Π.
5
Concluding Remarks
In this paper, we design a quantity discount contract to resolve profit conflicts arise in a
reverse supply chain. Here we focus exclusively on cost and revenue result from exerting effort
to reduce false failure returns, which refers to customer returns with no functional defects of the
products. Since from a supply chain’s point of view, reducing such returns is highly beneficial
because substantial goodwill cost and re-processing cost can be saved. But to the retailer,
incentive of doing so may be distorted by having to pay all the cost of effort while enjoying a full
credit provided by the supplier in case of return. As a consequence, retailer always chooses an
effort level that is lower than the supply chain’s optimum. To tackle this problem, we propose
to apply a quantity discount contract so as to induce retailer to exert more effort. Specifically,
the payment specified by the contract is exponentially decreasing in the number of false failure
returns. We give explicit formulations of the contracts under different assumptions about the
distribution of the number of false failure returns. The distribution is assumed to be geometric
in the first case, Poisson in the second case and normal in the last case. It is proved that
the contract is able to successfully coordinate the supply chain, inducing retailer to choose
desired effort level. We also prove that with the contract parameters properly set, retailer’s and
20
supplier’s profits are both increased at the same time. It is also demonstrated that the contract
also has the function to deter retailer from pushing false failure returns in the forward supply
chain. In addition, we show how the contract may be modified so that it leads to easy allocation
of incremental profit within the supply chain.
Study of reverse supply chain is attracting more and more attention due to both ecological
and economical reasons. And besides concerning the operational aspects, for example, to investigate the planning and impacts of re-manufacturing process, studying reverse supply chain from
a strategic aspects, for example, to resolve incentive conflicts in decentralized channel should
also be beneficial. Since coordination contract in forward supply chain has been studied in depth
and there are a lot of available results, while on the other hand, reverse supply chain is relatively unexplored. So, in spite of the fact that most results from forward supply chain cannot
be directly applied in the new context, they would server as a valuable reference. The design
of quantity discount contract here is an example. We suggest this may be a further research
direction. Some other research directions may also include to generalize our result in a more
general setting, for example, to see how quantity discount contract may work with different
return policies offered by retailer to customers or to consider how to modify the contract to be
applied in multiple periods setting etc.
References
[1] Ferguson, M., Guide,V.D.R,Jr.,Souza,G., Supply Chain Coordination for False Failure Returns, Manufacturing and Service Operations Management. 4, pp376-393, 2006.
[2] Guide, V. D. R., Jr., L. N. Van Wassenhove, eds. Business Aspects of Closed-Loop Supply
Chains. Carnegie Mellon University Press, Pittsburgh, PA. 2003.
[3] Guide, V. D. R., Jr., T. Harrison, and L. N. Van Wassenhove. The challenge of closed-loop
supply chains. Interfaces 33(6) pp3C6. 2003.
[4] Guide, V. D. R., Jr., G. Souza, L. Van Wassenhove, J. Blackburn. Time value of commercial
product returns. Management Science 52(8). 2006.
[5] Lawton, C., The War on Returns, Wall Strict Journal. Page D1, May 8, 2008.
[6] Stock, J.,Speh T., Shear H., Many Happy(product) Returns, Harvard Bus. Rev. 80(7), pp1617, 2002.
[7] Cachon, G.P., “Supply chain coordination with contracts”, In: Graves, S., T. de Kok, eds.
Handbooks in Operations Research and Management Science: Supply Chain Management.
North-Holland, Amsterdam, The Netherlands, 2003.
[8] Su, X., “Consumer returns policies and supply chain performance”, Manufacturing and
Service Operations Management, Vol 11, Issue 4, pp 595-612. 2009.
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