Channel Coordination and Quantity Discounts

Z. Kevin WengManagement Science, Volume 41, Issue 9

(September, 1995), 1509-1522.

Prepared by: Çağrı LATİFOĞLU

Presentation Outline

Introduction Model Model Analyses Allocation of the Profits Quantity Discounts Conclusion

Introduction

This paper represents a model analyzing the impact of joint decision policies on a channel coordination in a system consisting of a supplier and group of homogenous buyers.

Introduction

Joint decision policy is characterized by:

Unit selling prices

The order quantities (coordinated through the quantity discounts and franchies fees)

Introduction

Annual Demand Rate Operating Costs(include purchase, ordering

and inventory holding costs)

are affected by:

Joint unit selling price Joint order quantity

Introduction

Past studies on this problem is branched into two streams:

First Stream:

Operating costs are functions of order quantities and demand is treated as a

fixed constant.

Second Stream:

Demand is a decreasing function of buyer’s selling prices and operating costs are assumed to be fixed.

Introduction

This research is the generalized version of these two streams, considering channel coordination and operating cost minimization.

Model

There is one supplier and one buyer (or a group of homogenous buyers who are all treated same)

It is difficult to extend the model for heterogenous customers since it is difficult to find the avarage inventory in this case.

Model

Annual demand rate is a decreasing function of buyer’s selling price

Operating costs of both parties depend on order quantities.

Model

Buyers inventory policy is EOQ and quantity discount for buyers are same.

Demand increases with price reduction.

Model

Quantity Discounts: to ensure the joint order quantity minimizes the operating costs.

Franchise fees: to enforce joint profit maximization

Modelp: buyer’s unit purchase price-charged by supplier

x: buyer’s unit selling price-charged by buyer

hb: buyer’s yearly unit inventory holding cost

hs’: supplier’s yearly unit inventory holding cost

Sb: buyer’s fixed ordering cost per order

Sp: supplier’s fixed order processing cost

Ss’: supplier’s setup cost for each machine

Model

Supplier procures the material by either manufacturing or purchasing where cost of procurement c < p.

Buyer’s lot size Q

Supplier’s lot size mQ where m=1,2,...

Model

Holding cost of supplier

R=annual production capacity

Proc. by purc. :hsQ/2 where hs=Mhs

’

M=m-1

Proc. by mfg. :hsQ/2 where hs=Mhs

’

M=m-1-(m-2)*D(x)/R

Model

Supplier‘s order processing and setup ordering cost SsD(x)/Q where Ss=Sp+Ss

’/m

Supplier’s yearly profit:Gs(p)=(p-c)D(x)- SsD(x)/Q- hsQ/2

revenue # of setups inv.holding

Model

Buyer’s yearly profit:Gb(x,Q)=(x-p)D(x)- SbD(x)/Q- hbQ/2

As we also see in the profits supplier can only control p, while buyer controls Q and x.

Model Analyses

In the scenario 1, supplier & buyer will try to maximize their profits by optimizing the decision varibles that are under their control.

In the scenario 2, objective is to maximize the joint profit of both supplier & buyer s.t. both of their profits are greater than the first case.

Scenario 1

For supplier’s unit selling price p, xb(p) denotes the buyer’s optimal selling price.

Buyer’s optimal order size is (EOQ): Qb(p)=(2SbD(xb(p))/hb)½

where holding & ordering cost is (2SbhbD(xb(p)))½

Scenario 1

Gb(xb) is the corresponding buyer‘s profit:Gb(xb|Qb)= (x-p)D(x) - (2SbhbD(x))½

The corresponding supplier’s profit: Gs(p) = (p-c)D(xb(p))–(Ss/Sb+ hs/hb) * (SbhbD(xb(p))/2)½

Scenario 1

Lemma 1:With buyer’s EOQ order quantity, Qb(p), supplier’s yearly profit is never higher than the maximum that can be achieved by supplier’s EOQ order quantity.

(Sshb/Sbhs+ Sbhs/Sshb) >= 2

Buyer’s EOQ will also maximize this profit if Ss/Sb= hs/hb

Scenario 1

p* maximizes Gs*(=Gs(p*))

xb(p*) maximizes Gb*(=Gb(xb(p*)))

Total profit maximum profitin case 1 = Gs*+ Gb*

Scenario 2

In this case, the joint policies which enables both supplier & buyer to achieve higher profits, are analyzed, given that they are willing to cooperate.

Scenario 2

Joint profit function:Gj(x,Q) = Gs(p) + Gb(x,q)

Qj(x) = (2SjD(x)/hj)½ where

Sj=Ss+Sb and hj=hs+hb

Scenario 2

Joint profit function:Gj(x|Qj(x)) =(x-c)D(x) - (2SjD(x)hj)½

For buyer’s unit selling price xb(p*) and Qj(xb(p*)) = (2SjD(xb(p*))/hj)½

Lemma 2:Gj(xb(p*)|Qj(xb(p*))) >= Gs*+ Gb*

Scenario 2

For a given policy (x, Qj(x))

Gs(p|Qj(x))= (p-c)D(x)-SsD(x)/Qj(x)- hsQj(x)/2

Let pmin(x) is the smallest price thatsatisfies Gs(p|Qj(x))>= Gs*

pmin(x) = c +{Gs*/D(x) + (Ss/Sj+ hs/hj) * (Sjhj/2D(x))½

Scenario 2

In that case buyer’s profit will beGb(x, Qj(x))= (x-p)D(x)-SbD(x)/Qj(x)-

hbQj(x)/2

Let pmax(x) is the largest buyers purchasing

price that satisfies Gb(x, Qj(x)) >= Gb*

pmax(x) = x -{Gb*/D(x) + (Sb/Sj+ hb/hj) * (Sjhj/2D(x))½

Scenario 2

Gj(x|Qj(x)) - (Gs*+ Gb*) =

D(x)*[pmax(x) - pmin(x)]

Increased Unit Profit

Yearly increase in Profit

For achiving

this buyer

should select

x rather than

xb(p*) where

x<= xb(p*)

Allocation of the Profits

For the joint optimal policy (x*, Qj(x*))

If the d percentage of the increased profit goes to buyer, (1-d) percentage will go to supplier and so the price that will be charged by the supplier will be:

pj=d pmin(x)+(1-d) pmax(x)

Allocation of the Profits

To make buyer choose the joint optimum order quantity(rather than the amount that maximizes its profit alone) quantity discounts are offered.

For making him choose the joint optimum unit selling price, franchise fees are used.

Once a year buyer pays the supplier ß pj D(x*) and in return supplier charges (1-ß) pj avarage unit selling price. In this case the buyer’s optimal selling price x*((1-ß) pj) is equal to optimal joint selling price x*.

Quantity Discounts

All unit: If buyer orders an amount Qx (>Qi) , the discount is applied to whole order(Qx).

Incremental: If buyer orders an mount Qx (>Qi) , the discount is applied to additional units (Qx-Qi) .

Quantity Discounts – All Unit

Qai is a price breakpoint where the

corresponding all-unit discount price is rai p*

If Ss/Sb= hs/hb then Qb(rai p*) = Qai

Else Qb(rai p*) ≠ Qai

Quantity Discounts – All Unit

It is also proposed that there should be only one price breakpoint and it should be at joint optimal order quantity(since it is unique).

Quantity Discounts – All Unit

Buyer’s yearly profit increase λ % (>=0) (which satisfies Gb(x*(rap*))>= Gb*)

Supplier’s yearly profit increase ß % (>=0)

In that case;

rap* =pj = pmax(x*) - λGb*/ D(x*)

Qa = Qj(x*) = [2SjD(x*)/hj]½

λ Gb* + ß Gs* =[pmax(x*) - pmin(x*)]D(x*)

Quantity Discounts – All Unit

From the formulations we can see that all unit discount percentage and buyer’s profit increase percentage have a linear relationship due to the fact that pj linearly affects purchase cost but it has no impact on the other costs.

Another observation is the negative linear relation between supplier percentage profit increase and all-unit quantity discount

Incremental Quantity Discount

In this policy, the discount is applied to the units that are over the price breakpoint Q.

r1’=r1(1-Q/Q1) + Q/Q1

Gb(xb(r1’p*)|Q)=(xb(r1

’p*)- r1’p*) D(xb(r1

’p*)) - Sb D(xb(r1

’p*))/Q1- hbQ1/2

Incremental Quantity Discount

Q1= [2(Sb+p*(1-r1’Q) D(xb(r1

’p*))/hb]

Gs1( r1’p*|Q)= (r1

’p*-c) D(xb(r1’p*)) - Ss

D(xb(r1’p*))/Q1- hsQ1/2

Equivalence of AQD and IQD

Given that both AQD and IQD increase buyer’s profit by an equal amount (since they have the same unit selling price, x*) the increase in supplier’s profits should be same. (details are in the paper)

It is found that ra= r1’p*=pj and Qa= Q1=

Qj(x*)

Conclusion

Quantity discounts alone are not sufficient to guarantee joint profit maximization, franchise fees should be implemented as a control mechanism

Whether the demand is constant or not, AQD and IQD perform identically,

Dependency of demand on unit selling price and operating cost dependency on order quantities is more critical.

Q & A