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Channel Coordination and
Quantity Discounts
Z. Kevin Weng
Presented byJing Zhou
Introduction
D(x)
Supplier Buyerxp
Q
c
mQ
0)(
dx
xdD
CS (Q)Operating Cost:
Cb (Q)
Channel
x, Q
Can be coordinated through the mechanisms of quantity discounts and franchise fees
The Role of Quantity Discounts in Channel Coordination
Economicliterature
Marketingliterature
Production managementliterature
Quantity Discounts
Price discrimination
Effect on the profit
Demand decreases in priceOperating cost is fixed
Effect on the operating costs
Demand is fixedOperating cost is a function of order quantities
Quantity discounts are effective and necessary mechanisms to achieve channel coordination
Assumptions
The buyer uses EOQ model as her inventory policies
The supplier offers the buyer or a group of homogeneous buyers an identical quantity discount policy
The supplier has complete knowledge of the buyer’s demands, holding costs and ordering costs
The demand decreases in selling price
The Model
D(x)
Supplier Buyerx
Channel
p
Q
c
mQ
xD(x)Revenue: pD(x)
0)(
dx
xdD
SSD(x)/Q+hSQ/2
Ordering & Holding Cost:
SbD(x)/Q+hbQ/2
Purchasing Cost:
cD(x) pD(x)
xD(x)
SJD(x)/Q+hJQ/2
cD(x)
hJ=hS+hb
SJ=SS+Sb
The Model (Con’t)
GS(p) = (p-c)D(x) - [SSD(x)/Q + hSQ/2]
Supplier’s profit:
Gb(x,Q) = (x-p)D(x) - [SbD(x)/Q + hbQ/2]
Buyer’s profit:
GJ(x,Q) = (x-c)D(x) - [SJD(x)/Q + hJQ/2]
Channel’s profit:
Scenario 1 (Decentralization)
Gb(x,Q) = (x-p)D(x) - [SbD(x)/Q + hbQ/2]
The buyer’s problem:
1. Given x, the buyer’s optimal order size is
the resulting ordering and holding cost is
2. With Qb(x), the buyer’s profit function is
3. For any p charged by the supplier, let denote the buyer’s
optimal selling price that maximizes her profit
the corresponding order quantity is
b
bb h
xDSxQ
)(2)(
)(2 xDhS bb
)(2)()()|( xDhSxDpxQxG bbbb
)( pxb
b
bbb h
pxDSpQ
))((2)(
Scenario 1 (Decentralization)
GS(p) = (p-c)D(x) - [SSD(x)/Q + hSQ/2]
The supplier’s problem:
1. With the buyer’s selling price , and the order quantity
, the supplier’s profit function is
Let denote the supplier’s unit selling price that maximizes
, let which is a lower bound on the
supplier’s profit
3. Accordingly, is the buyer’s minimum profit
and is the system’s profit without coordination
2
))(()())(()()(
pxDhS
h
h
S
SpxDcppG bbb
b
S
b
SbS
)( pxb
)( pQb
p
)( pGG SS)( pGS
))(( pxGG bbb
bS GG
Lemma 4.1
Buyer’s EOQ order quantity
b
bbb h
pxDSpQ
))((2)(
Supplier’s EOQ order quantity
S
bSS h
pxDSQ
))((2
Supplier’soperatingcost:
))((2)(2
1pxDhS
hS
hS
hS
hSbSS
bS
Sb
Sb
bS ))((2 pxDhS bSS
b
S
b
S
h
h
S
S when ""
The buyer’s EOQ order quantity also maximizesthe supplier’s profit only if b
S
b
S
h
h
S
S
Scenario 2 (Cooperation)
1. Given x, the joint operating cost is minimized by the
joint EOQ
order quantity
the resulting joint ordering and holding cost is
2. With , the joint profit function is
J
JJ h
xDSxQ
)(2)(
)(2 xDhS JJ
)(2)()())(|( xDhSxDcxxQxG JJJJ
GJ(x,Q) = GS(p) + Gb(x,Q)
= (x-c)D(x) - [SJD(x)/Q + hJQ/2]
Joint profit:
)(xQJ
Lemma 4.2
Joint EOQ order quantity
J
bJbJ h
pxDSpxQ
))((2))((
Buyer’s EOQ order quantity
b
bbb h
pxDSQ
))((2
)))((|)(( pxQpxG bJbJ bS GG
b
S
b
S
h
h
S
S when ""
With joint EOQ order quantity, the joint profit will be at least the system’s profit without joint coordination
Given ,
)( pxb
Profit:
Profit Impact of Joint Policy
1. The supplier can charge a p such that the resulting profit is higher
than his minimum profit, i.e.
SJS GxQpG ))(|(
Given a joint policy , if ))(,( xQx J
Then both the supplier and the buyer would accept the joint policy
bJb GxQxG ))(,(
and
2. This p leads the buyer’s profit is higher than her
minimum profit, i.e.
)(min xp
)(max xp
Profit Impact of Joint Policy (Con’t)
)()()())(|( xDxgGGxQxG bSJJ
With and , we have)(min xp
)()()( minmax xpxpxg
)(max xp
where
1. The joint profit increases if the joint unit selling price x satisfies
2. If x is chosen such that g(x) > 0, then g(x) represents the
increased unit profit due to the joint EOQ order quantity
3. also leads to an increase in the demand rate
from to
)( pxx b
)( pxx b
))(( pxD b )(xD
The increased profit as a result of joint coordination
Dividing the Profits
If the buyer’s unit purchase price
then the buyer’s profit increases by
and the supplier’s profit increases by
Suppose x* maximizes the increased total profit g(x)D(x) and
both parties agree to employ the optimal joint policy)(,( xQx J
)()1()( maxmin xpxppJ
)()( xDxg
)()()1( xDxg
Implementation of the Optimal Joint Policy
A control mechanism that make both parties choose the
decision policies that maximize their individual profits as
well as the joint profit simultaneously
To maximize the joint profit, both conditions should be met:
a) the buyer chooses the selling price as x*
b) the buyer chooses order quantity as JJJ hxDSxQ /)(2)(
)( xQJ
a) a quantity discount policy with an average unit
purchase price pJ will induce the buyer to order
b) but a quantity discount policy is not sufficient to
induce the buyer to choose the optimal unit selling price x*
Implementation of the Optimal Joint Policy (Con’t)
Given a QD policy with order quantity and the average
unit purchase price PJ , the buyer’s profit function is
)( xQJ
)2
)(
)(
)(()()()(
xQ
hxQ
xDSxDpxxG J
bJ
bJb , let )(max)( xGpx bx
10 ,)1( Jp2. There exists a unit purchase price , such that
the buyer’s optimal unit selling price
xpx J ))1((
3. If the buyer make a fixed payment to the supplier,
then the buyer’s profit function is )]()([ )2
)(
)(
)(()()()( xDxDp
xQh
xQ
xDSxDpxxG J
Jb
JbJb
)( xDpJ
Identical when x = x*
1. xpx )(
Quantity discounts and franchise fees
Quantity discounts and franchise fees can coordinate the channel
The role of quantity discounts is to ensure that the joint order quantity selected by both parties minimizes the joint operating costs
The role of franchise fees is to enforce the joint profit maximization for both parties
Equivalence of AQD and IQD
As long as the average unit discount rate and the order size are the same for either types of quantity discount schemes, the increased benefits due to quantity discounts are identical
The selection of the type of quantity discount has no effect on achieving channel coordination
Discussion
Contribution Generalize the two streams of research on
the roles of quantity discounts in channel coordination
Investigate the role and limitation of quantity discounts in channel coordination• Quantity discounts alone are not sufficient to
guarantee joint profit maximization
• AQD policy and IQD policy perform identically in benefiting both the supplier and the buyer
Discussion (Con’t)
Limitation
Should discuss the partial concavity property when sequentially solving a two-variable maximization problem
The author used some results without necessary proofs. These results may depend on the demand distribution.
Thank you!