Transcript Channel Coordination and Quantity Discounts

Channel Coordination and
Quantity Discounts
Z. Kevin Weng
Presented by
Jing Zhou
Introduction
Channel
Supplier
Buyer
c
p
x
mQ
Q
D(x)
Operating Cost:
Cb (Q)
CS (Q)
dD ( x )
0
dx
x, Q
Can be coordinated through the mechanisms
of quantity discounts and franchise fees
The Role of Quantity Discounts
in Channel Coordination
Quantity Discounts
Economic
literature
Marketing
literature
Price
discrimination
Effect on the
profit
Demand decreases in price
Operating cost is fixed
Production management
literature
Effect on the
operating costs
Demand is fixed
Operating 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
Channel
Supplier
Buyer
c
p
x
mQ
Q
D(x)
Revenue:
Purchasing Cost:
Ordering &
Holding Cost:
pD(x)
cD(x)
SSD(x)/Q+hSQ/2
dD ( x )
0
dx
xD(x)
xD(x)
pD(x)
cD(x)
SbD(x)/Q+hbQ/2
SJD(x)/Q+hJQ/2
SJ=SS+Sb
hJ=hS+hb
The Model (Con’t)
Supplier’s profit:
GS(p) = (p-c)D(x) - [SSD(x)/Q + hSQ/2]
Buyer’s profit:
Gb(x,Q) = (x-p)D(x) - [SbD(x)/Q + hbQ/2]
Channel’s profit:
GJ(x,Q) = (x-c)D(x) - [SJD(x)/Q + hJQ/2]
Scenario 1 (Decentralization)
The buyer’s problem: Gb(x,Q) = (x-p)D(x) - [SbD(x)/Q + hbQ/2]
1.
Given x, the buyer’s optimal order size is
Qb ( x) 
2 S b D( x)
hb
the resulting ordering and holding cost is 2Sb hb D( x)
2.
With Qb(x), the buyer’s profit function is
Gb ( x | Qb )  ( x  p) D( x)  2Sb hb D( x)
3.
For any p charged by the supplier, let xb ( p) denote the buyer’s
optimal selling price that maximizes her profit
the corresponding order quantity is Qb ( p)  2Sb D( xb ( p))
hb
Scenario 1 (Decentralization)
The supplier’s problem:
1.
GS(p) = (p-c)D(x) - [SSD(x)/Q + hSQ/2]
With the buyer’s selling price xb ( p) , and the order quantity
Qb ( p) , the supplier’s profit function is
GS ( p)  ( p  c) D( xb ( p))  (

S S hS Sb hb D( xb ( p))
 )
Sb hb
2

Let p denote the supplier’s unit selling price that maximizes
GS ( p) , let GS  GS ( p  ) which is a lower bound on the
supplier’s profit
3.
Accordingly, Gb  Gb ( xb ( p  )) is the buyer’s minimum profit
and GS  Gb is the system’s profit without coordination
Lemma 4.1
Buyer’s EOQ order quantity Supplier’s EOQ order quantity
Qb ( p) 
2Sb D( xb ( p))
hb
Supplier’s 1 S S hb
Sh
(
 b S ) 2S S hS D( xb ( p))
operating 2 Sb hS
S S hb
cost:
QS 

2S S D( xb ( p))
hS
2S S hS D( xb ( p))
" " when
S S hS

Sb hb
The buyer’s EOQ order quantity also maximizes
the supplier’s profit only if S S  hS
Sb
hb
Scenario 2 (Cooperation)
Joint profit:
GJ(x,Q) = GS(p) + Gb(x,Q)
= (x-c)D(x) - [SJD(x)/Q + hJQ/2]
1.
Given x, the joint operating cost is minimized by the joint EOQ
order quantity QJ ( x) 
2 S J D( x )
hJ
the resulting joint ordering and holding cost is 2S J hJ D( x)
2.
With QJ (x) , the joint profit function is
GJ ( x | QJ ( x))  ( x  c) D( x)  2S J hJ D( x)
Lemma 4.2

Given xb ( p ) ,
Joint EOQ order quantity
Buyer’s EOQ order quantity
2S J D( xb ( p  ))
QJ ( xb ( p )) 
hJ
2Sb D( xb ( p  ))
Qb 
hb

Profit:
GJ ( xb ( p  ) | QJ ( xb ( p  )))

" " when
GS  Gb
S S hS

Sb hb
With joint EOQ order quantity, the joint profit
will be at least the system’s profit without joint
coordination
Profit Impact of Joint Policy
Given a joint policy ( x, QJ ( x)) , if
The supplier can charge a p such that the resulting profit is
1.
higher than his minimum profit, i.e. GS ( p | QJ ( x))  GS
pmin ( x)
and
2.
This p leads the buyer’s profit is higher than her minimum

profit, i.e. Gb ( x, QJ ( x))  Gb
pmax ( x)
Then both the supplier and the buyer would
accept the joint policy
Profit Impact of Joint Policy (Con’t)
With pmin ( x) and pmax ( x) , we have
GJ ( x | QJ ( x))  (GS  Gb )  g ( x) D( x) The increased profit
as a result of joint
coordination
The joint profit increases if the joint unit selling price x
where g ( x)  pmax ( x)  pmin ( x)
1.
satisfies x  xb ( p  )
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.
x  xb ( p  ) also leads to an increase in the demand rate

from D ( xb ( p )) to D(x)
Dividing the Profits

Suppose x* maximizes the increased total profit g(x)D(x) and


both parties agree to employ the optimal joint policy ( x , QJ ( x )

If the buyer’s unit purchase price
p J  pmin ( x  )  (1   ) pmax ( x  )
then the buyer’s profit increases by g ( x ) D( x )
and the supplier’s profit increases by (1   ) g ( x ) D( x )
Implementation of the Optimal
Joint Policy

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 QJ ( x )  2S J D( x ) / hJ

A control mechanism that make both parties choose the
decision policies that maximize their individual profits as
well as the joint profit simultaneously
a) a quantity discount policy with an average unit purchase
price pJ will induce the buyer to order QJ ( x )
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 QJ ( x  ) and the average
unit purchase price PJ , the buyer’s profit function is
QJ ( x )
D( x)
Gb ( x)  ( x  p J ) D( x)  ( Sb
 hb
), let x ( p)  max Gb ( x)

x
QJ ( x )
2
1.
2.
x ( p)  x
Identical when x = x*
There exists a unit purchase price (1   ) pJ , 0    1 , such that


the buyer’s optimal unit selling price x ((1   ) p J )  x
3.
If the buyer make a fixed payment  p J D ( x  ) to the supplier,
then the buyer’s profit function is
QJ ( x )
D( x)

Gb ( x)  ( x  p J ) D( x)  ( Sb

h
)


p
[
D
(
x
)  D( x)]
b
J
QJ ( x )
2
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 twovariable maximization problem
The author used some results without
necessary proofs. These results may
depend on the demand distribution.
Thank you!