第五單元：Inventory Management: Cycle Inventory-II
Inventory Management:
Cycle Inventory-II
郭瑞祥教授
【本著作除另有註明外，採取創用CC「姓名標示
－非商業性－相同方式分享」台灣3.0版授權釋出】
1
Lessons From Aggregation



2
Aggregation allows firm to lower lot size without increasing cost
Complete aggregation is effective if product specific fixed cost is a
small fraction of joint fixed cost
Tailored aggregation is effective if product specific fixed cost is a
large fraction of joint fixed cost
Holding Cycle Inventory for
Economies of Scale
► Fixed costs associated with lots
► Quantity discounts
► Trade Promotions
3
Quantity Discounts
► Lot size based
》Based on the quantity ordered in a single lot
> All units
► Volume based
> Marginal unit
》Based on total quantity purchased over a given period
Total Material Cost
Average Cost per
Unitshould buyer react? How does this decision affect the supply chain
 How
in terms of lot sizes, cycle inventory, and flow time?
C0
 What
C1are appropriate discounting schemes that suppliers should offer?
C2
Quantity Purchased
4
q1
q2
q3
Order Quantity
q1
q2
q3
Evaluate EOQ for All Unit Quantity Discounts

Evaluate EOQ for price in range qi to qi+1 , Qi = 2DS
hCi
》 Case 1:If qi  Qi < qi+1 , evaluate cost of ordering Qi
Q
TCi = D S + i hCi + DCi
2
Qi
》 Case 2:If Qi < qi, evaluate cost of ordering qi
q
TCi = D S + i hCi + DCi
2
qi
》 Case 3:If Qi  qi+1 , evaluate cost of ordering qi+1
qi+1
D
S+
TCi =
hCi + DCi+1
qi+1
2

5
Choose the lot size that minimizes the total cost over all price ranges.
Marginal Unit Quantity Discounts
Marginal Cost per Unit
Total Material Cost
C0
C1
C2
Quantity
Purchased
q1
q2
q3
Order Quantity
q1
q2
q3
If an order of size q is placed, the first q1-q0 units are priced at C0, the
next q2-q1 are priced at C1, and so on.
6
Evaluate EOQ for Marginal Unit Discounts
► Evaluate EOQ for each marginal price Ci (or lot size between qi and qi+1)
》 Let Vi be the cost of order qi units. Define V0 = 0 and
Vi=C0(q1-q0)+C1(q2-q1)+‧‧‧+Ci-1(qi-qi-1)
》 Consider an order size Q in the range qi to qi+1
Total annual cost = ( D/Q )S
+[Vi+(Q-qi)Ci] h/2
+ ( D/Q ) [Vi+(Q-qi)Ci]
(Annual order cost)
(Annual holding cost)
(Annual material cost)
Optimal lot size
Qi =
7
2D(S+Vi-qiCi)
hCi
Evaluate EOQ for Marginal Unit Discounts
► Evaluate EOQ for each marginal price Ci, Qi=
2D(S+Vi-qiCi)
hCi
》 Evaluate EOQ for each marginal price Ci
– Case 1 :If qi  Qi < qi+1 , calculate cost of ordering Qi
h D
TCi = D S +[ Vi+(Qi-qi)Ci] 2 + Q [ Vi+(Qi-qi)Ci]
i
Qi
– Case 2 and 3 : If Qi < qi or Qi > qi+1 , the lot size in this range
is either qi or qi+1 depending on which has the
lower total cost
 D

h D
D
h D
TCi = Min
S + Vi + Vi ,
S + Vi +1 +
Vi +1 
2 qi
qi +1
2 q i +1
 qi

》 Choose the lot size that minimizes the total cost over all price ranges.
8
The Comparison between All Unit and
Marginal Unit Quantity Discounts


9
The order quantity of all unit quantity discounts is less than the order
quantity of marginal unit quantity discounts.
The marginal unit quantity discounts will further enlarge the cycle
inventory and average flow time.
Why Quantity Discounts?
► Quantity discounts are valuable only if they result in:
► Coordination:
max total profits
of supply
suppliers
and retailers
》 Improved coordination
in the
chain
》 Extractioninofthe
surplus
through
► Coordination
supply
chain price discrimination
► Use price discrimination to max supplier’s profits
》Quantity discounts for commodity products (in
the perfect competition market, price is fixed)
》Quantity discounts for products for which the
firm has market power (in the oligopoly market,
the determined price can influence demand)
>Two-part tariffs
> Volume discounts
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Coordination for Commodity Products
► Assume the following data.
– Retailer: D =120,000/year , SR=$100 , hR=0.2 , CR=$3
– Suplier: SS =$250 , hS =0.2 , CS =$2
► Retailer cost
Q* =
2 x 120,000 x 100 =
6,324
0.2 x 3
TC = 100 x 120,000
6,324
+ 6,324
2
x 0.2 x 3 = $3,795
► Supplier’s cost is based on retailer’s optimal order size.
TC = 250 x 120,000 + 6,324 x 0.2 x 2 = $6,009
6,324
2
》 Supply chain total cost = 3,795+6,009=$9,804
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Coordination for Commodity Products
► Consider a coordinated order size=9,165.
$3,795
Suppler's TC=100x120,000 + 9,165 x0.2x3 =$4,059 (Increased by $264)
9,165
2
$6,009
Suppler's TC=250x120,000 + 9,165x0.2x2 =$5,106 (decreased by $903)
9,165
2
》Supply chain total cost=4,059+5,106 =$9,165(decreased by $639)
► Coordination through all unit quantity discounts.
– $3 for lots below 9,165
$2.9978 for lots of 9,165 or higher
– Increase in retailer’s holding cost and order cost can be compensated
by the reduction in material cost. 120,000(3-2.9978)=$264
– Decrease in supplier’s cost = supply chain savings = 903–264=$639
(can be further shared between two parties)
12
Coordination for Commodity Products




13
Since the price is determined by the market, supplier can use lotsize based quantity discounts to achieve coordination in supply
chain and decrease supply chain cost.
Lot size-based quantity discounts will increase cycle inventory.
In theory, if supplier reduces its setup or order cost, the discount it
offers will change and the cycle inventory is expected to decrease.
In practice, the cycle inventory does not decrease in the supply
chain because in most firms, marketing and sales department
design quantity discounts independent of operations department
who works on reducing the order cost.
Quantity Discounts When Firm has Market Power
► No inventory related costs.
► Assume the following data
Demand curve = 360,000-60,000p (p is retailer’s sale price)
CS = $2 (cost of supplier).
► Need to determine CR (Suppler’s charge on retailer) and p.
CS =$2
p
CR
Supplier
Retailer
Demand
=360,000-60,000p
14
Scenario 1: No Coordination
► Maximize individual profits and make pricing decision independently
ProfitR =(p-CR)(360,000-60,000p)
C
 (ProfitR)
= 0 p=3+ R
2
p
CR
ProfitS= (CR-2)(360,000-60,000p)=(CR-2)[360,000-60,000( 3+
)]
2
 (ProfitS)
CR
CS =$2
= 0  CR=4; p=5
CR
p
Supplier
Retailer
► Demand = 360,000-60,000(5)=60,000
Demand
Profit for retailer = (5-4)(60,000)=$60,000
=360,000-60,000p
Profit for supplier = (4-2)(60,000)=$120,000
Profit for supply chain = 60,000+120,000=$180,000
15
Quantity Discounts When Firm has Market Power
► No inventory related costs.
► Assume the following data
Demand curve = 360,000-60,000p (p is retailer’s sale price)
CS = $2 (cost of supplier).
► Need to determine CR (Suppler’s charge on retailer) and p.
Variation
CS =$2
Fix
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p
CR
Supplier
Retailer
Demand
=360,000-60,000p
Maximize Supply Chain Profits
► Profit for supply chain
=(p-Cs) (360,000-60,000p)
=(p-2) (360,000-60,000p)
 (Profit)
p
= 0  p=4
► Demand = 360,000-60,000(4)=120,000
Microsoft。
► Profit for supple chain = (4-2)(120,000)=$240,000 > $180,000
Microsoft。
Microsoft。
How to increase the total profit through coordination ?
17
Scenario 2: Coordination through Two-Part Tariff -I
► Supplier charges his entire profit as an up-front franchise fee.
Supplier sells to the retailer at production cost (CS).
► Proof:Assume demand function = a-bp
(a, b are constants)
Then retailer’s profit = (p-cR)(a-bp)-F (F: franchise fee)
The supply chain’s profit = (p-cS)(a-bp)
Maximize both profits will obtain
18
Scenario 2: Coordination through Two-Part Tariff-II
► Supplier charges his entire profit as an up-front franchise fee.
Supplier sells to the retailer at production cost (CS).
► Proof:Assume demand function = a-bp
(a, b are constants)
Then retailer’s profit = (p-cR)(a-bp)-F (F: franchise fee)
The supply chain’s profit = (p-cS)(a-bp)
Maximize both profits will obtain
19
Scenario 2: Coordination through Two-Part Tariff-III
► Supplier charges his entire profit as an up-front franchise fee.
Supplier sells to the retailer at production cost (CS).
► Proof:Assume demand function = a-bp
(a, b are constants)
Then retailer’s profit = (p-cR)(a-bp)-F (F: franchise fee)
The supply chain’s profit = (p-cS)(a-bp)
Maximize both profits will obtain
a CR a C S
P = 2b +
= +
2 2b 2
\ CR= CS
In our example, CR =CS =2 , p =4, demand=120,000
Assume a franchise fee of 180,000
Retailer’s profit =(4-2)(120,000)-180,000=$60,000 (same as before)
Supplier’s profit = F = $180,000
Supply chain’s profit = 60,000+180,000=$240,000
20
Scenario 3: Coordination through Volume- based
Quantity Discounts
► The two-part tariff is really a volume-based quantity discounts.
► Supplier offers the volume discounts at the break point of optimal
demand.
► Supplier offers the discount price so that the retailer will have a profit 
the profit of no coordination and no discount.
In our example, design the volume discounts
CR =$4 (for volume < 120,000)
CR =$3.5 (for volume  120,000)
To sell 120,000, the retailer sets price at p = 4.(from the demand function)
Retailer’s profit =(4-3.5)(120,000)=$60,000 (same as before)
Supplier’s profit = (3.5-2)(120,000) = $180,000
Supply chain’s profit = 60,000+180,000=$240,000
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Lessons From Discounting Schemes




22
Lot size-based discounts increase lot size and cycle inventory in
the supply chain.
Lot size-based discounts are justified to achieve coordination for
commodity products.
Volume-based discounts with some fixed cost passed on to retailer
are more effective in general
Volume-based discounts are better using rolling horizon to avoid the
“hockey stick phenomenon”.
Price Discrimination to Max Supplier Profits
► Price discrimination is the practice which a firm charges differential
prices to maximize profits.
► Price discrimination is also a volume-based discount scheme.
► Consider an example
Demand curve (supplier sells to retailer)=200,000-50,000CR
CS =2
Profit of supplier = (CR-2)(200,000-50,000CR)
► What is the optimal “ fixed ” price CR to maximize profit ?
 (Profit)
CR
= 0 CR =$3
Demand=200,000-50,000(3)=50,000
Profit =(3-2)(50,000)=$50,000
23
Demand Curve and Demand at Price of $3
► The fixed price of $3 does not maximize profits for the supplier.
The profit is only the shaded area in the following figure.
► The supplier could obtain the entire area under the demand curve
above his marginal cost of $2 (the triangle within the solid lines) by
pricing each unit differently.
Price
p=4
p=3
Marginal cost = $2
p=2
50,000 100,000
24
200,000 Demand
An Equivalent Two-Part Tariff to
Price Discrimination
► The entire triangle under the demand curve (above the marginal
cost of $2) = franchise fee = 1/2(4-2)(100,000)=$100,000
► The selling price to retailer: CR =CS =2.
► Demand = 200,000-50,000(2)=100,000
Price
50,000
Profit of
supplier
=
F
=
$100,000
p=4
p=3
Marginal cost = $2
p=2
50,000
50,000 100,000
25
200,000 Demand
Demand Curve and Demand at Price of $3
► The fixed price of $3 does not maximize profits for the supplier.
The profit is only the shaded area in the following figure.
► The supplier could obtain the entire area under the demand curve
above his marginal cost of $2 (the triangle within the solid lines) by
pricing each unit differently.
Price
p=4
50,000
p=3
Marginal cost = $2
p=2
50,000
50,000 100,000
26
200,000 Demand
Holding Cycle Inventory for
Economies of Scale
► Fixed costs associated with lots
► Quantity discounts
► Trade Promotions
27
Trade Promotion
► Goals:
– Induce retailers to spur sales
– Shift inventory from manufacture to retailer and the customer
– Defend a brand against competition
► Retailer options:
– Pass through some or all of the promotion to customers to
spur sales
– Pass through very little of the promotion to customers but
purchase in greater quantity to exploit temporary reduction
in price (forward buying)
28
Inventory Profile for Forward Buying
I(t)
Qd: lot size ordered at the discount price
Q* : EOQ at normal price
Qd
Q*
Q*
Q*
Q*
Q*
t
29
Forward Buying Decisions
► Goal:
– Identify Qd that maximizes the reduction in total cost
(material cost + order cost + holding cost)
I(t)
Qd: lot size ordered at the discount price
Q* : EOQ at normal price
► Assumptions:
–
–
–
Discount
will only be offered once.
Qd
Order quantity Qd is a multiple of Q*.
The retailer takes no action to influence the demand.
Q*
30
Q*
Q*
Q*
Q*
t
Decision on Q*d
► Assume the following data
Normal order quantity = EOQ = Q* = 2DS
hC
The discount = $d.
The discounted material cost = $(C-d )
► Now estimate the total cost of ordering Qd in the discount period
TC(Qd) = material cost + order cost + inventory cost
I(t)
d: lot sized ordered at the discount price
=(C-d)Qd + S + Qd/2 Q
(C-d)h
[ Q /D ]
* : EOQ at normal price
Q
Qd d + S + (Qd/D)2 (C-d)h / 2D
=(C-d)Q
Note: Discount period =Qd/D
Q*
Q*
Q*
Q*
Q*
t
31
Decision on Q*d
► Now estimate the total cost of ordering Q* in the discount period
Annual TC(Q*) = material cost + order cost + inventory cost
=CD +(D/ 2DS )S + 2DS hC/2 =CD+ 2hCDS
hC
hC
Discount period TC (Q*) = Qd/D [Annual TC(Q*) ]=Qd/D [CD+ 2hCDS ]
► Define the cost reduction in the discount period
F(Qd) = TC(Qd) – Discount period TC(Q*)
F(Qd)
Qd
=0  Qd=
► Forward buy = Qd – Q*
32
CQ*
dD
+
[C-d]h C-d
Decision on Q*d
► Now estimate the total cost of ordering Q* in the discount period
Annual TC(Q*) = material cost + order cost + inventory cost
=CD +(D/ 2DS )S + 2DS hC/2 =CD+ 2hCDS
hC
hC
Discount period TC (Q*) = Qd/D [Annual TC(Q*) ]=Qd/D [CD+ 2hCDS ]
► Define the cost reduction in the discount period
F(Qd) = TC(Qd) – Discount period TC(Q*)
F(Qd)
Qd
=0  Qd=
► Forward buy = Qd – Q*
33
CQ*
dD
+
[C-d]h C-d
Decision on Q*d
► Now estimate the total cost of ordering Q* in the discount period
Annual TC(Q*) = material cost + order cost + inventory cost
=CD +(D/ 2DS )S + 2DS hC/2 =CD+ 2hCDS
hC
hC
Discount period TC (Q*) = Qd/D [Annual TC(Q*) ]=Qd/D [CD+ 2hCDS ]
► Define the cost reduction in the discount period
F(Qd) = TC(Qd) – Discount period TC(Q*)
F(Qd)
Qd
=0  Qd=
► Forward buy = Qd – Q*
34
CQ*
dD
+
[C-d]h C-d
Example
Assume the following data without promotion.
 D =120,000/year , C =$3 , h =0.2 , S =$100
► Forward buy = 38,236 – 6,324 =31,912
 then →Q* = 6,324
► Trade promotions
lead to a significant increase in lot size
*
 Cycle inventory = Q /2 = 3,162
and cycle inventory
because of forward buying by the
*
 Average flow time = Q /2D =
retailer.= 0.3162 (month).
0.02635(year)
► Trade promotions generally result in reduced supply chain
► Assume a promotion
is offered
(d =$0.15)
profits unless
the trade
promotions reduce demand
fluctuations.
dD
CD*
0.15X120,000 3(6,324)
=38,236
Qd =
+
=
+
[C-d]h C-d
[3-0.15][0.2]
3-0.15
Cycle inventory = Qd/2 = 19,118
Average flow time = Qd/2D = 0.1593(year) = 1.9118 (month).
35
Promotion Pass through to Customers
► Assume demand function =
a-bp (a, b are constants)
Then retailer’s profit = [p-CR][a-bp]
a + CR
Maximizing retailer’s profits will obtain P =
2b 2
p
If a discount d is offered, the new C R=CR-d
Then the new p = a + CR - d = p - d
2 2
2b
2
► Retailer’s optimal response to a discount is to pass only 50% of the
discount to the customers.
36
Example-I
Demand curve at retailer: 300,000 – 60,000p
► Normal supplier price, CR = $3.00

》Optimal retail price = $4.00
》Customer demand = 60,000
► Promotion discount = $0.15
► Retailer only passes through half the promotion discount
37
Example-II
Demand curve at retailer: 300,000 – 60,000p
► Normal supplier price, CR = $3.00

》Optimal retail price = $4.00
》Customer demand = 60,000
► Promotion discount = $0.15
► Retailer only passes through half the promotion discount
38
Example-III
Demand curve at retailer: 300,000 – 60,000p
► Normal supplier price, CR = $3.00

》Optimal retail price = $4.00
》Customer demand = 60,000
► Promotion discount = $0.15
》Optimal retail price = $3.925
》Customer demand = 64,500
► Retailer only passes through half the promotion discount
》Demand increases by only 7.5%
》Cycle inventory increases significantly
39
Trade Promotions
Goal is to discourage forward buying
in the supply chain
Counter measures
– EDLP
– Scan based promotions
– Customer coupons
Wikipedia
Microsoft。
Microsoft。
40
Levers to Reduce Lot Sizes Without Hurting Costs
Cycle Inventory Reduction
► Reduce transfer and production lot sizes
》Aggregate fixed cost across multiple products, supply points,
or delivery points.
► Are quantity discounts consistent with manufacturing and
logistics operations?
》Volume discounts on rolling horizon
》Two-part tariff
► Are trade promotions essential?
》EDLP
》Base on sell-thru rather than sell-in
41
版權聲明
頁碼
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作品
授權條件
作者/來源
17
本作品轉載自Microsoft Office 2007多媒體藝廊，依據Microsoft 服
務合約及著作權法第46、52、65條合理使用。
40
本作品轉載自WIKIPEDIA (http://en.wikipedia.org/wiki/File:WalMart_logo.svg)，瀏覽日期2012/2/20。
40
本作品轉載自Microsoft Office 2007多媒體藝廊，依據Microsoft 服
務合約及著作權法第46、52、65條合理使用。
40
本作品轉載自Microsoft Office 2007多媒體藝廊，依據Microsoft 服
務合約及著作權法第46、52、65條合理使用。