Transcript Quantity Discounts

Channel Coordination and
Quantity Discounts
Z. Kevin Weng
Management Science, Volume 41, Issue 9
(September, 1995), 1509-1522.
Prepared by: Çağrı LATİFOĞLU
Presentation Outline
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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:
Second Stream:
Operating costs are
functions of order quantities
and demand is treated as a
fixed constant.
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
Model
p: 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
 Supplier’s lot size
m=1,2,...
Q
mQ where
Model
 Holding cost of supplier
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
R=annual production capacity
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 that
satisfies 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 be
Gb(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