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Channel Coordination and Quantity Discounts
Z. Kevin WengManagement Science, Volume 41, Issue 9
(September, 1995), 1509-1522.
Prepared by: Çağrı LATİFOĞLU
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Presentation Outline
Introduction Model Model Analyses Allocation of the Profits Quantity Discounts Conclusion
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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.
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Introduction
Joint decision policy is characterized by:
Unit selling prices
The order quantities (coordinated through the quantity discounts and franchies fees)
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Introduction
Annual Demand Rate Operating Costs(include purchase, ordering
and inventory holding costs)
are affected by:
Joint unit selling price Joint order quantity
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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.
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Introduction
This research is the generalized version of these two streams, considering channel coordination and operating cost minimization.
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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.
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Model
Annual demand rate is a decreasing function of buyer’s selling price
Operating costs of both parties depend on order quantities.
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Model
Buyers inventory policy is EOQ and quantity discount for buyers are same.
Demand increases with price reduction.
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Model
Quantity Discounts: to ensure the joint order quantity minimizes the operating costs.
Franchise fees: to enforce joint profit maximization
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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
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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,...
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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
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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
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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.
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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.
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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
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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
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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
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Scenario 1
p* maximizes Gs*(=Gs(p*))
xb(p*) maximizes Gb*(=Gb(xb(p*)))
Total profit maximum profitin case 1 = Gs*+ Gb*
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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.
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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
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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*
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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
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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
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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*)
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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)
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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*.
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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) .
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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
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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).
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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*)
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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
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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
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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
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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*)
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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.
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Q & A