Inventory Management: Cycle Inventory-II

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Inventory Management Cycle Inventory-II
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Lessons From Aggregation

• 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

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Holding Cycle Inventory for Economies of Scale

• Fixed costs associated with lots
• Quantity discounts
• Trade Promotions

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Quantity Discounts

• Lot size based
• Volume based

• Based on the quantity ordered in a single lot

• All units
• Marginal unit

• Based on total quantity purchased over a given
  period

• How should buyer react? How does this decision
  affect the supply chain
• in terms of lot sizes, cycle inventory,
  and flow time?

? What are appropriate discounting schemes that
suppliers should offer?
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Evaluate EOQ for All Unit Quantity Discounts

• Evaluate EOQ for price in range qi to qi1 ,
• Case 1If qi ? Qi lt qi1 , evaluate cost of
  ordering Qi
• Case 2If Qi lt qi, evaluate cost of ordering qi
• Case 3If Qi ? qi1 , evaluate cost of ordering
  qi1
• Choose the lot size that minimizes the total cost
  over all price ranges.

Qi
D



DCi
hCi
S
TCi
2
Qi
qi
D



DCi
hCi
S
TCi
2
qi
qi1
D



DCi1
hCi
S
TCi
qi1
2
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Marginal Unit Quantity Discounts
Marginal Cost per Unit
Total Material Cost
C0
C1
C2
Quantity Purchased
Order Quantity
q1
q2
q3
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.
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Evaluate EOQ for Marginal Unit Discounts

• Evaluate EOQ for each marginal price Ci (or lot
  size between qi and qi1)

• Let Vi be the cost of order qi units. Define V0
  0 and
• ViC0(q1-q0)C1(q2-q1)???Ci-1(qi-
  qi-1)

• Consider an order size Q in the range qi to qi1
• Total annual cost ( D/Q )S
  (Annual order cost)
• Vi(Q-qi)Ci h/2
  (Annual holding cost)
• ( D/Q ) Vi(Q-qi)Ci
  (Annual material cost)

2D(SVi-qiCi)
Qi
hCi
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Evaluate EOQ for Marginal Unit Discounts

• Evaluate EOQ for each marginal price Ci,

• Evaluate EOQ for each marginal price Ci
• Case 1 If qi ? Qi lt qi1 , calculate cost of
  ordering Qi
• Case 2 and 3 If Qi lt qi or Qi gt qi1 , the lot
  size in this range
• is either qi or qi1
  depending on which has the
• lower total cost
• Choose the lot size that minimizes the total cost
  over all price ranges.

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The Comparison between All Unit and Marginal Unit
Quantity Discounts

• 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.

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Why Quantity Discounts?

• Quantity discounts are valuable only if they
  result in

• Improved coordination in the supply chain
• Extraction of surplus through price discrimination

• Coordination max total profits of suppliers and
  retailers

• Coordination in the supply chain

• Use price discrimination to max suppliers 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)

gtTwo-part tariffs gt Volume discounts
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Coordination for Commodity Products

• Assume the following data.
• Retailer D 120,000/year , SR100 , hR0.2 ,
  CR3
• Suplier SS 250 , hS 0.2 , CS 2
• Retailer cost
• Suppliers cost is based on retailers optimal
  order size.
• Supply chain total cost 3,7956,0099,804

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Coordination for Commodity Products

• Consider a coordinated order size9,165.

(Increased by 264)
(decreased by 903)

• Supply chain total cost4,0595,106
  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 retailers holding cost and order
  cost can be compensated
• by the reduction in material cost.?
  120,000(3-2.9978)264
• Decrease in suppliers cost supply chain
  savings 903264639 (can be further shared
  between two parties)

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Coordination for Commodity Products

• Since the price is determined by the market,
  supplier can use lot-
• size 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.

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Quantity Discounts When Firm has Market Power

• No inventory related costs.
• Assume the following data
• Demand curve 360,000-60,000p (p is
  retailers sale price)
• CS 2 (cost of supplier).
• Need to determine CR (Supplers charge on
  retailer) and p.

p
CR
CS 2
Retailer
Supplier
Demand 360,000-60,000p
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Scenario 1 No Coordination

• Maximize individual profits and make pricing
  decision independently
• Demand 360,000-60,000(5)60,000
• Profit for retailer (5-4)(60,000)60,000
• Profit for supplier (4-2)(60,000)120,000
• Profit for supply chain 60,000120,000180
  ,000

(p-CR)
ProfitR
(360,000-60,000p)
ProfitS
(CR-2)
(360,000-60,000p)
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Quantity Discounts When Firm has Market Power

• No inventory related costs.
• Assume the following data
• Demand curve 360,000-60,000p (p is
  retailers sale price)
• CS 2 (cost of supplier).
• Need to determine CR (Supplers charge on
  retailer) and p.

p
CR
CS 2
Retailer
Supplier
Demand 360,000-60,000p
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Maximize Supply Chain Profits

• Profit for supply chain
• Demand 360,000-60,000(4)120,000
• Profit for supple chain (4-2)(120,000)240,000
  gt 180,000

(p-Cs)
(360,000-60,000p)
(p-2) (360,000-60,000p)
Þ p4
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How to increase the total profit through
coordination ?
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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).

• ProofAssume demand function a-bp (a, b
  are constants)

Then retailers profit (p-cR)(a-bp)-F (F
franchise fee)
The supply chains profit (p-cS)(a-bp)
Maximize both profits will obtain
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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).

• ProofAssume demand function a-bp (a, b
  are constants)

Then retailers profit (p-cR)(a-bp)-F (F
franchise fee)
The supply chains profit (p-cS)(a-bp)
Maximize both profits will obtain
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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).

• ProofAssume demand function a-bp (a, b
  are constants)

Then retailers profit (p-cR)(a-bp)-F (F
franchise fee)
The supply chains profit (p-cS)(a-bp)
Maximize both profits will obtain
\ CR CS
In our example, CR CS 2 , p 4,
demand120,000 Assume a franchise fee of
180,000 Retailers profit (4-2)(120,000)-180,000
60,000 (same as before) Suppliers profit F
180,000 Supply chains profit
60,000180,000240,000

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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 lt 120,000)
CR 3.5 (for volume ? 120,000) To sell
120,000, the retailer sets price at p 4.(from
the demand function) Retailers profit
(4-3.5)(120,000)60,000 (same as before)
Suppliers profit (3.5-2)(120,000) 180,000
Supply chains profit 60,000180,000240,000
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Lessons From Discounting Schemes

• 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.

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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 ?

Þ CR 3
Demand200,000-50,000(3)50,000
Profit (3-2)(50,000)50,000
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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
p4
p3
Marginal cost 2
p2
200,000
Demand
50,000
100,000
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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
• Profit of supplier F 100,000

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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.

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Holding Cycle Inventory for Economies of Scale

• Fixed costs associated with lots
• Quantity discounts
• Trade Promotions

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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)

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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
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Forward Buying Decisions

• Goal
• Assumptions

• Identify Qd that maximizes the reduction in total
  cost (material cost order cost holding cost)

• Discount will only be offered once.
• Order quantity Qd is a multiple of Q.
• The retailer takes no action to influence the
  demand.

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Decision on Qd

• Assume the following data
• Normal order quantity EOQ
• 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

(C-d)Qd
S
Qd/2 (C-d)h Qd/D
(C-d)Qd S (Qd/D)2 (C-d)h / 2D
Note Discount period Qd/D
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Decision on Qd

• Now estimate the total cost of ordering Q in the
  discount period

Annual TC(Q) material cost order cost
inventory cost
CD
Discount period TC (Q) Qd/D Annual TC(Q)

• Define the cost reduction in the discount period
• F(Qd) TC(Qd) Discount period TC(Q)

• Forward buy Qd Q

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Decision on Qd

• Now estimate the total cost of ordering Q in the
  discount period

Annual TC(Q) material cost order cost
inventory cost
CD
Discount period TC (Q) Qd/D Annual TC(Q)

• Define the cost reduction in the discount period
• F(Qd) TC(Qd) Discount period TC(Q)

• Forward buy Qd Q

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Decision on Qd

• Now estimate the total cost of ordering Q in the
  discount period

Annual TC(Q) material cost order cost
inventory cost
CD
Discount period TC (Q) Qd/D Annual TC(Q)

• Define the cost reduction in the discount period
• F(Qd) TC(Qd) Discount period TC(Q)

• Forward buy Qd Q

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Example

• Assume the following data without promotion.
• D 120,000/year , C 3 , h 0.2 , S 100
• then ?Q 6,324
• Cycle inventory Q/2 3,162
• Average flow time Q/2D
• 0.02635(year) 0.3162 (month).

• Forward buy 38,236 6,324 31,912

• Trade promotions lead to a significant increase
  in lot size
• and cycle inventory because of forward
  buying by the
• retailer.

• Trade promotions generally result in reduced
  supply chain
• profits unless the trade promotions reduce
  demand
• fluctuations.

• Assume a promotion is offered (d 0.15)

38,236
Cycle inventory Qd/2 19,118 Average flow time
Qd/2D 0.1593(year) 1.9118 (month).
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Promotion Pass through to Customers

• Assume demand function a-bp (a, b are
  constants)
• Then retailers profit p-CRa-bp
• Maximizing retailers profits will obtain
• If a discount d is offered, the new C RCR-d
• Then the new
• Retailers optimal response to a discount is to
  pass only 50 of the discount to the customers.

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Example-I

• Demand curve at retailer 300,000 60,000p
• Normal supplier price, CR 3.00
• Promotion discount 0.15
• Retailer only passes through half the promotion
  discount

• Optimal retail price 4.00
• Customer demand 60,000

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Example-II

• Demand curve at retailer 300,000 60,000p
• Normal supplier price, CR 3.00
• Promotion discount 0.15
• Retailer only passes through half the promotion
  discount

• Optimal retail price 4.00
• Customer demand 60,000

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Example-III

• Demand curve at retailer 300,000 60,000p
• Normal supplier price, CR 3.00
• Promotion discount 0.15
• Retailer only passes through half the promotion
  discount

• Optimal retail price 4.00
• Customer demand 60,000

• Optimal retail price 3.925
• Customer demand 64,500

• Demand increases by only 7.5
• Cycle inventory increases significantly

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Trade Promotions
Goal is to discourage forward buying in the
supply chain

• Counter measures

• EDLP
• Scan based promotions
• Customer coupons

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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

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