Transcript Narrative Information Processing in Electronic Medical Report
Managing Economies of Scale in the Supply Chain: Cycle Inventory
Spring, 2014 Supply Chain Management: Strategy, Planning, and Operation Chapter 10 Byung-Hyun Ha
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
1
Introduction
Cycle inventory
inventory level time
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 2
Introduction
Analysis of cycle
Average inventory level (cycle inventory) =
Q
/2 Average flow time =
Q
/2
D
Little’s law: (arrival rate) = (avg. number in system)/(avg. flow time) Example • •
D
= 2 units/day,
Q
= 8 units Average inventory level • (7 + 5 + 3 + 1)/4 = 4 =
Q
/2 • Average flow time • (0.25 + 0.75 + 1.25 + 1.75 + 2.25 + 2.75 + 3.25 + 3.75)/8 = 2 =
Q
/2
D
3
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!
4
Introduction
Assumptions
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.
?
?
5
Introduction
Determining optimal order quantity Q*
Economy of scale vs. diseconomy of scale, or Tradeoff between
total fixed cost
and
total variable cost Q
1
D
?
Q
2
D
6
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 7
Economic Order Quantity (EOQ)
Assumption
Same price regardless of order quantity
Input
D
: demand per unit time,
S
: ordering cost,
C
: unit material 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)
TC
TO
TH
(
TM
)
D
S Q
Q
hC
2
TO
: total order cost
TH
: total holding cost
TM
: total material cost 8
Economic Order Quantity (EOQ)
Total cost by order quantity Q
TC
D
S Q
Q
hC
2
Optimal order quantity Q* that minimizes total cost
Q
*
TC
* 2
DS hC
2
hCDS TC
Opt. order frequency
n
*
D Q
DhC
2
S
Avg. flow time
Q
* 2
D
2
S DhC Q
* 9
Q
Economic Order Quantity (EOQ)
Robustness around optimal order quantity (KEY POINT)
Using order quantity
Q' T C
D Q
S
Q
2
hC
=
Q
* instead of
Q
* 1 2 1
TC
*
1/2
(
+ 1/
)
0.5
1.25
0 0.6
0.7
0.8
0.9
1.133 1.064 1.025 1.006
1.0
1.00
0 1.2
1.4
1.6
1.017 1.057 1.113
1.8
1.17
8 2.0
1.250
TC'
= 1.25
TC
*
TC
* 10 0.5
1 2
Economic Order Quantity (EOQ)
Robustness regarding input parameters
Mistake in indentifying ordering cost
S'
=
S
instead of real
S
• Misleading to
Q
2
D S
hC
2
D
β
S hC
β
Q
*
T C
1 2 β 1 β
TC
*
1/2
(
+ 1/
)
0.5
1.06
1 0.6
0.7
0.8
0.9
1.033 1.016 1.006 1.001
1.0
1.00
0 1.2
1.4
1.6
1.004 1.014 1.028
1.8
1.04
3 2.0
1.061
TC'
= 1.061
TC
*
TC
*
What does mean by robust?
11 0.5
1 2
Economic Order Quantity (EOQ)
Sensitivity regarding demand (KEY POINT)
Demand change from
D
to
D
1 =
kD Q
1 *
TC
1 * 2
D
1
hC S
2
hCD
1
S
2
kDS
k
hC
2
hCkDS
2
DS hC k
2
k Q hCDS
*
k TC
* Opt. order frequency
n
1 *
D
1
Q
1 *
D
1
hC
2
S
k n
* Avg. flow time
Q
1 * 2
D
1
S
2
D
1
hC
1
Q
*
k
2
D
12
Economic Order Quantity (EOQ)
Reducing flow time by reducing ordering cost (KEY POINT)
Efforts on reducing
S
Hoping
Q
1 * =
k
Q*
How much should
S
to
S
1 =
S
be reduced? (What is ?)
Q
1 * 2
DS
1
hC
2
D
γ
S
hC
γ 2
DS hC
γ
Q
*
k
Q
* =
k
2 (ordering cost must be reduced by a factor of
k
2 ) 13
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.
TC
D
S Q
1
Q
*
D P Q
hC
2 1 2
DS D P
hC Q x D
(
P
–
D
)
Q
/
P Q
/
D
–
Q
/
P = Q
(
1
/
D
–
1
/
P
) 1/(
D
/
Q
) =
Q
/
D
14
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
Q i
*
TC i
* 2
DS hC
2
hCDS SSQ
*
m
SSTC
*
m
2
DS hC
2
hCDS
All-aggregate ordering
ASQ
*
ASTC
* 2
mDS
hC
2
hCmDS
m
m
2
DS hC
2
hCDS
Impractical supposition for analysis purpose
15
Lot Sizing with Multiple Products
Multiple products with different parameters
m
products
D i
,
C i
,
h i
: demand, price, holding cost fraction of product
i S
: ordering cost each time an order is placed • Independent of the variety of products
s i
: 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 •
Q i
=
D i
/
n
: order quantity of item
i
Total cost and optimal number of orders
TC
S
i m
1
s i
n
i m
1
D i h i C i
2
n
n
*
i m
1
D i h i C i
2
S
i m
1
s i
16
Lot Sizing with Multiple Products
Q i
*
n i
*
TC i
*
i
Example 10-3 and 10-4
Input • • Common transportation cost,
S
= $4,000 Holding cost fraction,
h
= 0.2
Ordering each products independently •
ITC
* = $155,140 200,000 180,000
LE22B LE19B LE19A
1,095 11.0
$109,544 346 3.5
$34,642 110 1.1
$10,954 160,000 140,000 120,000 Ordering jointly • •
n
* = 9.75
JTC
* = $155,140 100,000 80,000 60,000 40,000 20,000 0
Q i
*
i
LE22B
1,230
LE19B
123
LE19A
12.3
LE22B
D i C i s i
i
LE22B
12,000 $500 $1,000
LE19B
1,200 $500 $1,000
LE19A
120 $500 $1,000 LE19B LE19A 17
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!
18
Exploiting Quantity Discount
Total cost with quantity discount
TC
TO
TH
TM
D
S Q
Q
hC
2
DC
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 19
All Unit Quantity Discount
Pricing schedule
Quantity break points:
q
0 ,
q
1 , ...,
q r
,
q r
+1 • where
q
0 = 0 and
q r
+1 = Unit cost •
C i
where
C
0 when
q i
C
1 It is possible that
q i
C i Q C r
(
q i q i
+1 , for + 1)
C i i
=0,...,
r
Solution procedure
1. Evaluate the optimal lot size for each
C i
.
Q i
* 2
DS hC i
average cost per unit
C
0
C
1
C
2
C r
...
q
0
q
1
q
2
q
3 ...
q r
2. Determine lot size that minimizes the overall cost by the total cost of the following cases for each
i
.
• Case 1:
q i
Q i
*
q i
+1 , Case 2:
Q i
*
q i
, Case 3:
q i
+1
Q i
* 20
All Unit Quantity Discount
Example 10-7
r
= 2,
D
= 120,000/year
S
= $100/lot,
h
= 0.2
Q
* = 10,000
q i C i
i
0
0 $3.00
1
5,000 $2.96
2
10,000 $2.92
390,000 385,000 380,000 375,000 370,000 365,000 360,000 355,000 350,000 345,000 340,000 335,000 0 2000 4000 6000 8000 10000 12000 14000 16000 21
All Unit Quantity Discount
Example 10 7 (cont’d)
Sensitivity analysis • Optimal order quantity
Q
* with regard to ordering cost (original)
S
= $100/lot (reduced)
S'
= $4/lot (no discount)
C
= $3 6,324 1,256 (discount) 10,000 10,000 22
Marginal Unit Quantity Discount
Pricing schedule
Quantity break points:
q
0 ,
q
1 , ...,
q r
,
q r
+1 • where
q
0 = 0 and
q r
+1 =
Marginal
• unit cost
C i
where
C
0
C
1 when
C r q i
Q
q i
+1 , for
i
=0,...,
r
Price of q
i
V i
=
C
0 (
q
1
units
–
q
0 ) +
C
1 (
q
2 –
q
1 ) + ... +
C
i –1 (
q i
–
q i
–1 )
Ordering Q units
Suppose
q i
Q
q i
+1 .
TC
TO D Q
S TH
V i
TM
Q
2
q i
C i
h
D
V i Q
Q
q i
C i
marginal cost per unit
C
0
C
1
C
2
C r q
0 ...
q
1
q
2
q
3 ...
q r
23
Marginal Unit Quantity Discount
Example 10-8
r
= 2,
D
= 120,000/year
S
= $100/lot,
h
= 0.2
Q
* = 16,961
q i C i V i
i
0
0 $3.00
$0
1
5,000 $2.96
$15,000
2
10,000 $2.92
$29,800 395,000 390,000 385,000 380,000 375,000 370,000 365,000 360,000 355,000 350,000 0 4000 8000 12000 16000 20000 24000 28000 24
Marginal Unit Quantity Discount
Example 10 8 (cont’d)
Sensitivity analysis • Optimal order quantity
Q
* with regard to ordering cost (original)
S
= $100/lot (reduced)
S'
= $4/lot (no discount)
C
= $3 6,324 1,256 (discount) 16,961 15,775 Higher inventory level (longer average flow time)?
25
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
Manufacturer (supplier) Retailer customers supply chain
2. Extraction of surplus through price discrimination
Revenue management (Ch. 15)
Other factors such as marketing that motivates sellers
Munson and Rosenblatt (1998) 26
Coordination for Total Supply Chain Profit
Quantity discounts for commodity products
Assumption • Fixed price and stable demand Max. profit min. total cost fixed total revenue Example case • Two stages with a manufacture (supplier) and a retailer Manufacturer (supplier)
S
S
h
S
C
S = 250 = 0.2
= 2 Retailer
S
R
h
R
C
R = 100 = 0.2
= 3 customers
D
= 120,000 27
Coordination for Total Supply Chain Profit
Quantity discounts for commodity products (cont’d)
No discount • Retailer’s (local) optimal order quantity ( •
Q
1 = (2 12,000 100/0.2
3) 1/2 = 6,325 • Total cost (without material cost) supply chain’s decision) •
TC
1
= TC
1 S +
TC
1 R = $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)?
TC
TC
S
TC
R
D
S
S
Q
Q
2
h
S
C
S
D Q
S
R
Q
2
h
R
C
R
D
S
S
Q
S
R
Q
2
h
S
C
S
h
R
C
R
Q
* 2
D h
S
C
S
S
S
S
R
h
R
C
R 9 , 165 28
Coordination for Total Supply Chain Profit
Quantity discounts for commodity products (cont’d)
Lot size-based quantity discount offering by manufacturer • • • •
q
1 = 9,165,
C
0 = $3,
C
1 = $2.9978
Retailer’s (local) optimal order quantity (considering material cost) •
Q
2 = 9,165 Total cost (without material cost) •
TC
2
= TC
2 S +
TC
2 R = $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
supply chain.
in the 29
Coordination for Total Supply Chain Profit
Quantity discounts for commodity products (cont’d)
Other approach:
setup cost reduction by manufacturer Manufacturer (supplier) Retailer customers
S'
S
h
S =
100
= 0.2
C
S = 2
S
R
h
R
C
R = 100 = 0.2
= 3
D
= 120,000 • • Retailer’s (local) optimal order quantity •
Q
3 =
Q
1 = 6,325 Total cost (without material cost): no need to discount!
•
TC
3
= TC
3 S +
TC
3 R = $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!
30
Coordination for Total Supply Chain Profit
Quantity discounts for products with market power
Assumption • • Manufacturer’s cost,
C
S = $2 Customer demand depending on price,
p
, set by retailer •
D
= 360,000 – 60,000
p
Profit depends on price.
Manufacturer (supplier)
C
S = 2 Retailer
C
R = ?
D
= 360,000 – 60,000
p
customers
p
= ?
31
Coordination for Total Supply Chain Profit
Quantity discounts for products with market power (cont’d)
No coordination (deciding independently) • • Manufacturer’s decision on
C
R • Expected retailer’s profit, »
Prof
R = (
p
–
Prof
R
C
R ) (360 – 60
p
) • • Retailer’s optimal price setting (behavior) when
C
R »
p
1 = 3 + 0.5
C
R Demand by
p
1 »
D
= 360 (supplier’s order quantity) – 60
p
1 = 180 – 30
C
R • Expected manufacturer’s profit,
Prof
S »
Prof
S = (
C
R –
C
S ) (180 – 30
C
R )
C
R 1 that maximizes
Prof
R (m anufacturer’s decision) is given Retailer’s decision on •
p
1 »
C
R 1 = $5 ( = $4
D
1
p
1 with given
C
R 1 = 360,000 – 60,000
p
1 = 60,000) • Supply chain profit,
Prof
0 1 •
Prof
0 1 =
Prof
R 1 +
Prof
S 1 = $120,000 + $60,000 = $180,000 32
Coordination for Total Supply Chain Profit
Quantity discounts for products with market power (cont’d)
Coordinating supply chain • • Optimal supply chain profit,
Prof
0 * •
Prof
0 = (
p
–
C
S ) (360 – 60
p
) •
p
* = $4 •
D
* = 120,000 •
Prof
0 * = $240,000 Double marginalization problem (local optimization) But how to coordinate?
• i.e.,
Prof
S * = ?,
Prof
R * = ?
33
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 »
Prof
R = (
p
–
C
R ) (360 – 60
p
) »
p
2 = 3 + 0.5
C
R = $4 •
Prof
0 2 =
Prof
R 2 +
Prof
S 2 = $180,000 + $60,000 = $240,000 Retailer’s side: larger volume more discount • Volume-based quantity discount • •
q
1 = 120,000,
p
3 = $4
C
0 = $4,
C
1 = $3.5
•
Prof
0 3 =
Prof
R 3 +
Prof
S 3 = $180,000 + $60,000 = $240,000 34
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.
KEY POINT
• 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 high setup cost suitable for supplier’s Hockey stick phenomenon & rolling horizon-based discount 35
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
increase demand variability increase supply chain profit decrease inventory & flow time 36
Short-term Discounting: Trade Promotion
Analysis
Determining order quantity with discount
Q d
• 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.
Q d Q
*
Q d
/
D
1 –
Q d
/
D
37
Short-term Discounting: Trade Promotion
Analysis (cont’d)
Optimal order quantity without discount
Q
* = (2
DS
/
hC
) 1/2 Optimal total cost without discount
TC
* =
CD
+ (2
DShC
) 1/2 Total cost with
Q d TC
C
d
Q d
S
Q d
2
h
C
d
Q d D
TC
* 1
Q d D
Q d
*
h
C dD
d
CQ
*
C
d
Example 10-9
C d
= $3 = $0.15
Q
* = 6,324
Q d
* = 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.
38
Short-term Discounting: Trade Promotion
Retailer’s action of passing discount to customers
Example 10-10 Assumptions • • • Customer demand:
D
= 300,000 – 60,000
p
Normal price:
C
R = $3 Ignoring all inventory-related cost Analysis • • Retailer’s profit,
Prof
R •
Prof
R = (
p
–
C
R ) (300 – 60
p
) Retailer’s optimal price setting with regard to
C
R •
p
= 2.5 + 0.5
C
R No discount (
C
R 1 •
p
1 = $4,
D
1 = $3) = 60,000 Discount (
C
R 2 •
p
2 = $2.85) = $3.925,
D
2 = 64,500 (
p
1 –
p
2 = 0.075 < 0.15 =
C
R 1 –
C
R 2 ) 39
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.
40
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 41
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
(Skip!)
42