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!