0. Managing Economies of Scale in the Supply Chain: Cycle Inventory
Spring, 2014
Supply Chain Management: Strategy, Planning, and Operation
Chapter 10
Byung-Hyun Ha

1. Contents Introduction Economies of scale to exploit fixed costs
Economies of scale to exploit quantity discount
Short-term discounting: trade promotions
Managing multiechelon cycle inventory

2. Introduction Cycle inventory Notation
D: demand per unit time
Q: quantity in a lot or batch size (order quantity)
Cycle inventory management (basic)
Determining order quantity Q that minimizes total inventory cost with demand D given
inventory
level
time

3. Introduction Analysis of cycle
Average inventory level (cycle inventory) = Q/2
Average flow time = Q/2D
 Little’s law: (arrival rate) = (avg. number in system)/(avg. flow time)
Example
D = 2 units/day, Q = 8 units
Average inventory level
( )/4 = 4 = Q/2
Average flow time
( )/8 = 2 = Q/2D

4. Introduction Costs that are influenced by order quantity
C: (unit) material cost ($/unit)
Average price paid per unit purchased
 Quantity discount
H: holding cost ($/unit/year)
Cost of carrying one unit in inventory for a specific period of time
Cost of capital, obsolescence, handling, occupancy, etc.
H = hC
 Related to average flow time
S: ordering cost ($/order)
Cost incurred per order
Assuming fixed cost regardless of order quantity
Cost of buyer time, transportation, receiving, etc.
 10.2 Estimating cycle inventory-related costs in practice
SKIP!

5. Introduction ? ? Assumptions Cycle optimality regarding total cost
Constant (stable) demand, fixed lead time, infinite time horizon
Cycle optimality regarding total cost
Order arrival at zero inventory level is optimal.
Identical order quantities are optimal. ? ?

6. Introduction ? Determining optimal order quantity Q*
Economy of scale vs. diseconomy of scale, or
Tradeoff between total fixed cost and total variable cost Q1 D ? Q2 D

7. Economies of Scale to Exploit Fixed Costs
Lot sizing for a single product
Economic order quantity (EOQ)
Economic production quantity (EPQ)
Production lot sizing
Lot sizing for multiple products
Aggregating multiple products in a single order
Lot sizing with multiple products or customers

8. Economic Order Quantity (EOQ)
Assumption
Same price regardless of order quantity
Input
D: demand per unit time, C: unit material cost
S: ordering cost, H = hC: holding cost
Decision
Q: order quantity
D/Q: average number of orders per unit time
Q/D: order interval
Q/2: average inventory level
Total inventory cost per unit time (TC)
TO: total order cost
TH: total holding cost
TM: total material cost

9. Economic Order Quantity (EOQ)
Total cost by order quantity Q
Optimal order quantity Q* that minimizes total cost
Opt. order frequency
Avg. flow time TC Q Q*

10. Economic Order Quantity (EOQ)
Robustness around optimal order quantity (KEY POINT)
Using order quantity Q' = Q* instead of Q* 
0.5
0.6
0.7
0.8
0.9
1.0
1.2
1.4
1.6
1.8
2.0
1/2( + 1/)
1.250
1.133
1.064
1.025
1.006
1.000
1.017
1.057
1.113
1.178
TC' = 1.25TC*
TC* 
0.5 1 2

11. Economic Order Quantity (EOQ)
Robustness regarding input parameters
Mistake in indentifying ordering cost S' = S instead of real S
Misleading to 
0.5
0.6
0.7
0.8
0.9
1.0
1.2
1.4
1.6
1.8
2.0
1/2( + 1/)
1.061
1.033
1.016
1.006
1.001
1.000
1.004
1.014
1.028
1.043
TC' = 1.061TC*
TC*
 What does mean by robust? 
0.5 1 2

12. Economic Order Quantity (EOQ)
Sensitivity regarding demand (KEY POINT)
Demand change from D to D1 = kD
Opt. order frequency
Avg. flow time

13. Economic Order Quantity (EOQ)
Reducing flow time by reducing ordering cost (KEY POINT)
Efforts on reducing S to S1 = S
Hoping Q1* = kQ*
How much should S be reduced? (What is ?)
  = k2 (ordering cost must be reduced by a factor of k2)

14. Economic Production Quantity (EPQ)
Production of lot instead of ordering
P: production per unit time
Total cost by production lot size Q
Optimal production quantity Q*
 When P goes to infinite, Q* goes to EOQ. Q x D
(P – D)
Q/P
Q/D – Q/P = Q(1/D – 1/P)
1/(D/Q) = Q/D

15. Aggregating Products in a Single Order
Multiple products
m products
D: demand of each product
S: ordering cost regardless of aggregation level
All the other parameters across products are the same.
All-separate ordering
All-aggregate ordering
 Impractical supposition for analysis purpose

16. Lot Sizing with Multiple Products
Multiple products with different parameters
m products
Di, Ci, hi: demand, price, holding cost fraction of product i
S: ordering cost each time an order is placed
Independent of the variety of products
si: additional ordering cost incurred if product i is included in order
Ordering each products independently?
Ordering all products jointly
Decision
n: number of orders placed per unit time
Qi = Di /n: order quantity of item i
Total cost and optimal number of orders

17. Lot Sizing with Multiple Products
Example 10-3 and 10-4
Input
Common transportation cost, S = $4,000
Holding cost fraction, h = 0.2
Ordering each products independently
ITC* = $155,140
Ordering jointly
n* = 9.75
JTC* = $155,140 i
LE22B
LE19B
LE19A Di Ci si
12,000
$500
$1,000
1,200
120 i
LE22B
LE19B
LE19A
Qi*
ni* TCi*
1,095
11.0
$109,544
346
3.5
$34,642
110
1.1
$10,954 i
LE22B
LE19B
LE19A
Qi*
1,230
123
12.3

18. Lot Sizing with Multiple Products
How does joint ordering work?
Reducing fixed cost by enjoying robustness around optimal order quantity
Is joint ordering is always good?
No!
Then, possible other approaches?
Partially joint
NP-hard problem (i.e., difficult)
A heuristic algorithm
Subsection: “Lots are ordered and delivered jointly for a selected subset of the products”
SKIP!

19. Exploiting Quantity Discount
Total cost with quantity discount
Types of quantity discount
Lot size-based
All unit quantity discount
Marginal unit quantity discount
Volume-based
Decision making we consider
Optimal response of a retailer
Coordination of supply chain
TO: total ordering cost
TH: total holding cost
TM: total material cost

20. All Unit Quantity Discount
Pricing schedule
Quantity break points: q0, q1, ..., qr , qr+1
where q0 = 0 and qr+1 = 
Unit cost Ci when qi  Q  qi+1, for i=0,...,r
where C0  C1    Cr
 It is possible that qiCi  (qi + 1)Ci
Solution procedure
1. Evaluate the optimal lot size for each Ci.
2. Determine lot size that minimizes the overall cost by the total cost of the following cases for each i.
Case 1: qi  Qi*  qi+1 , Case 2: Qi*  qi , Case 3: qi+1  Qi*
average cost per unit C0 C1 C2
... Cr
... q0 q1 q2 q3
... qr

21. All Unit Quantity Discount
Example 10-7
r = 2, D = 120,000/year
S = $100/lot, h = 0.2
Q* = 10,000 i 1 2 qi Ci
$3.00
5,000
$2.96
10,000
$2.92

22. All Unit Quantity Discount
Example 10-7 (cont’d)
Sensitivity analysis
Optimal order quantity Q* with regard to ordering cost
(no discount)
C = $3
(discount)
(original) S = $100/lot
6,324
10,000
(reduced) S' = $4/lot
1,256

23. Marginal Unit Quantity Discount
Pricing schedule
Quantity break points: q0, q1, ..., qr , qr+1
where q0 = 0 and qr+1 = 
Marginal unit cost Ci when qi  Q  qi+1, for i=0,...,r
where C0  C1    Cr
Price of qi units
Vi = C0(q1 – q0) + C1(q2 – q1) Ci–1(qi – qi–1)
Ordering Q units
Suppose qi  Q  qi+1 .
marginal cost per unit C0 C1 C2
... Cr
... q0 q1 q2 q3
... qr

24. Marginal Unit Quantity Discount
Example 10-8
r = 2, D = 120,000/year
S = $100/lot, h = 0.2
Q* = 16,961 i 1 2 qi Ci Vi
$3.00 $0
5,000
$2.96
$15,000
10,000
$2.92
$29,800

25. Marginal Unit Quantity Discount
Example 10-8 (cont’d)
Sensitivity analysis
Optimal order quantity Q* with regard to ordering cost
 Higher inventory level (longer average flow time)?
(no discount)
C = $3
(discount)
(original) S = $100/lot
6,324
16,961
(reduced) S' = $4/lot
1,256
15,775

26. Why Quantity Discount?
1. Improve coordination to increase total supply chain profit
Each stage’s independent decision making for its own profit
Hard to maximize supply chain profit (i.e., hard to coordinate)
How can a manufacturer control a myopic retailer?
Quantity discounts for commodity products
Quantity discounts for products for which firm has market power
2. Extraction of surplus through price discrimination
Revenue management (Ch. 15)
 Other factors such as marketing that motivates sellers
Munson and Rosenblatt (1998)
Manufacturer
(supplier)
Retailer
customers
supply chain

27. Coordination for Total Supply Chain Profit
Quantity discounts for commodity products
Assumption
Fixed price and stable demand  fixed total revenue
 Max. profit  min. total cost
Example case
Two stages with a manufacture (supplier) and a retailer
Manufacturer
(supplier)
Retailer
customers
SS = 250
hS = 0.2
CS = 2
SR = 100
hR = 0.2
CR = 3
D = 120,000

28. Coordination for Total Supply Chain Profit
Quantity discounts for commodity products (cont’d)
No discount
Retailer’s (local) optimal order quantity ( supply chain’s decision)
Q1 = (212,000100/0.23)1/2 = 6,325
Total cost (without material cost)
TC1 = TC1S + TC1R = $6,008 + $3,795 = $9,803
Minimum total cost, TC*, regarding supply chain (coordination)
Q* = 9,165
TC* = TC*S + TC*R = $5,106 + $4,059 = $9,165
Dilemma?
Manufacturer saving by $902, but retailer cost increase by $264
How to coordinate (decision maker is the retailer)?

29. Coordination for Total Supply Chain Profit
Quantity discounts for commodity products (cont’d)
Lot size-based quantity discount offering by manufacturer
q1 = 9,165, C0 = $3, C1 = $2.9978
Retailer’s (local) optimal order quantity (considering material cost)
Q2 = 9,165
Total cost (without material cost)
TC2 = TC2S + TC2R = $5,106 + $4,057 = $9,163
Savings (compared to no discount)
Manufacturer: $902
Retailer: $264 (material cost) – $262 (inventory cost) = $2
KEY POINT
For commodity products for which price is set by the market, manufacturers with large fixed cost per lot can use lot size-based quantity discounts to maximize total supply chain profit.
Lot size-based discount, however, increase cycle inventory in the supply chain.

30. Coordination for Total Supply Chain Profit
Quantity discounts for commodity products (cont’d)
Other approach: setup cost reduction by manufacturer
Retailer’s (local) optimal order quantity
Q3 = Q1 = 6,325
Total cost (without material cost): no need to discount!
TC3 = TC3S + TC3R = $3,162 + $3,795 = $6,957
 Same with optimal supply chain cost when material cost is considered
 Expanding scope of strategic fit
Operations and marketing departments should be cooperate!
Manufacturer
(supplier)
Retailer
customers
S'S = 100
hS = 0.2
CS = 2
SR = 100
hR = 0.2
CR = 3
D = 120,000

31. Coordination for Total Supply Chain Profit
Quantity discounts for products with market power
Assumption
Manufacturer’s cost, CS = $2
Customer demand depending on price, p, set by retailer
D = 360,000 – 60,000p
 Profit depends on price.
D = 360,000 – 60,000p
Manufacturer
(supplier)
Retailer
customers
CS = 2
CR = ?
p = ?

32. Coordination for Total Supply Chain Profit
Quantity discounts for products with market power (cont’d)
No coordination (deciding independently)
Manufacturer’s decision on CR
Expected retailer’s profit, ProfR
ProfR = (p – CR)(360 – 60p)
Retailer’s optimal price setting (behavior) when CR is given
p1 = CR
Demand by p1 (supplier’s order quantity)
D = 360 – 60p1 = 180 – 30CR
Expected manufacturer’s profit, ProfS
ProfS = (CR – CS)(180 – 30CR)
 CR1 that maximizes ProfR (manufacturer’s decision)
CR1 = $4
Retailer’s decision on p1 with given CR1
p1 = $5 (D1 = 360,000 – 60,000p1 = 60,000)
Supply chain profit, Prof01
Prof01 = ProfR1 + ProfS1 = $120,000 + $60,000 = $180,000

33. Coordination for Total Supply Chain Profit
Quantity discounts for products with market power (cont’d)
Coordinating supply chain
Optimal supply chain profit, Prof0*
Prof0 = (p – CS)(360 – 60p)
p* = $4
D* = 120,000
Prof0* = $240,000
 Double marginalization problem (local optimization)
But how to coordinate?
i.e., ProfS* = ?, ProfR* = ?

34. Coordination for Total Supply Chain Profit
Quantity discounts for products with market power (cont’d)
Two pricing schemes that can be used by manufacturer
Two-part tariff
Up-front fee $180,000 (fixed) + material cost $2/unit (variable)
Retailer’s decision
ProfR = (p – CR)(360 – 60p)
p2 = CR = $4
Prof02 = ProfR2 + ProfS2 = $180,000 + $60,000 = $240,000
 Retailer’s side: larger volume  more discount
Volume-based quantity discount
q1 = 120,000, C0 = $4, C1 = $3.5
p3 = $4
Prof03 = ProfR3 + ProfS3 = $180,000 + $60,000 = $240,000

35. Coordination for Total Supply Chain Profit
Quantity discounts for products with market power (cont’d)
KEY POINT
For products for which the firm has market power, two-part tariffs or volume-based quantity discounts can be used to achieve coordination in the supply chain and maximizing supply chain profits.
For those products, lot size-based discounts cannot coordinate the supply chain even in the presence of inventory cost.
In such a setting, either a two-part tariff or a volume-based quantity discount, with the supplier passing on some of its fixed cost to the retailer, is needed for the supply chain to be coordinated and maximize profits.
Lot size-based vs. volume-based discount
Lot size-based: raising inventory level  suitable for supplier’s high setup cost
 Hockey stick phenomenon & rolling horizon-based discount

36. Short-term Discounting: Trade Promotion
Trade promotion by manufacturers
Induce retailers to use price discount, displays, or advertising to spur sales.
Shift inventory from manufactures to retailers and customers.
Defend a brand against competition.
Retailer’s reaction?
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 during the promotion period to exploit the temporary reduction in price.
Forward buy  demand variability increase  inventory & flow time increase  supply chain profit decrease

37. Short-term Discounting: Trade Promotion
Analysis
Determining order quantity with discount Qd
Unit cost discounted by d (C' = C – d)
Assumptions
Discount is offered only once.
Customer demand remains unchanged.
Retailer takes no action to influence customer demand. Qd Q*
Qd/D
1 – Qd/D

38. Short-term Discounting: Trade Promotion
Analysis (cont’d)
Optimal order quantity without discount Q* = (2DS/hC)1/2
Optimal total cost without discount TC* = CD + (2DShC)1/2
Total cost with Qd
Example 10-9
C = $3  Q* = 6,324
d = $0.15  Qd* = 38,236 (forward buy: 31,912  500%)
KEY POINT
Trade promotions lead to a significant increase in lot size and cycle inventory, which results in reduced supply chain profits unless the trade promotion reduces demand fluctuation.

39. Short-term Discounting: Trade Promotion
Retailer’s action of passing discount to customers
Example 10-10
Assumptions
Customer demand: D = 300,000 – 60,000p
Normal price: CR = $3
Ignoring all inventory-related cost
Analysis
Retailer’s profit, ProfR
ProfR = (p – CR)(300 – 60p)
Retailer’s optimal price setting with regard to CR
p = CR
No discount (CR1 = $3)
p1 = $4, D1 = 60,000
Discount (CR2 = $2.85)
p2 = $3.925, D2 = 64,500 (p1 – p2 = < 0.15 = CR1 – CR2)

40. Short-term Discounting: Trade Promotion
Retailer’ response to short-term discount
Insignificant efforts on trade promotion, but
High forward buying
Not only by retailers but also by end customers
Loss to total revenue because most inventory could be provided with discounted price
KEY POINT
Trade promotions often lead to increase of cycle inventory in supply chain without a significant increase in customer demand.

41. Short-term Discounting: Trade Promotion
Some implications
Motivation for every day low price (EDLP)
Suitable to
high elasticity goods with high holding cost
e.g., paper goods
strong brands than weaker brand (Blattberg & Neslin, 1990)
Competitive reasons
 Sometimes bad consequence for all competitors
Discount by not sell-in but sell-though
Scanner-based promotion

42. Managing Multiechelon Cycle Inventory
Configuration
Multiple stages and many players at each stage
General policy -- synchronization
Integer multiple order frequency or order interval
Cross-docking
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