Transcript Narrative Information Processing in Electronic Medical Report

Managing Economies of Scale in the
Supply Chain: Cycle Inventory
Fall, 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 optimal order quantity Q* that minimizes total
inventory cost, with demand D given
2
Introduction
 Basic 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
• (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/2D
3
Introduction
 Costs that influence total cost 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
Q1
D
?
Q2
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)
 Assumptions
 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
Q
 S   hC
Q
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
Q
 S   hC
Q
2
 Optimal order quantity Q* that minimizes total cost
Q* 
2 DS
hC
TC
TC *  2hCDS
 Opt. order frequency
n* 
D

Q
DhC
2S
 Avg. flow time
Q*
S

2D
2DhC
Q*
Q
9
Economic Order Quantity (EOQ)
 Robustness around optimal order quantity (KEY POINT)
 Using order quantity Q' = Q* instead of Q*
D
Q
1
1
TC    S   hC      TC *
Q
2
2

 0.5
1/2( + 1/)
0.6
0.7
0.8
0.9
1.25 1.133 1.064 1.025 1.006
0
1.0
1.2
1.4
1.6
1.8
1.00 1.017 1.057 1.113
0
2.0
1.17 1.250
8
TC' = 1.25TC*
TC*
0.5
1
2

10
Economic Order Quantity (EOQ)
 Robustness regarding input parameters
 Mistake by indentifying ordering cost S' = S instead of real S
• Misleading to
2DS 
2DβS
Q 

 βQ*
hC
hC
 0.5
1/2( + 1/)
0.6
0.7
0.8
0.9
1.06 1.033 1.016 1.006 1.001
1
1.0
1.2
1 
1 
TC  
β
 TC *
2 
β 
1.4
1.6
1.00 1.004 1.014 1.028
0
1.8
2.0
1.04 1.061
3
TC' = 1.061TC*
TC*
 Robustness?
0.5
1
2

11
Economic Order Quantity (EOQ)
 Sensitivity regarding demand (KEY POINT)
 Demand change from D to D1 = kD
2 D1S

hC
Q 
*
1
2kDS
2 DS
 k
 k Q*
hC
hC
TC1*  2hCD1S  2hCkDS  k 2hCDS  k TC *
 Opt. order frequency
n1* 
D1
D1hC
*


k
n
Q1*
2S
 Avg. flow time
Q1*

2 D1
S
1 Q*

2 D1hC
k 2D
12
Economic Order Quantity (EOQ)
 Effect of reducing order quantity
 Using order quantity Q' = Q* instead of Q* (revisited)
TC  
D
Q
1
1
 S   hC      TC *
Q
2
2

 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 ?)
Q1* 
2DS1
2DγS
2DS

 γ
 γ  Q*  k  Q*
hC
hC
hC
  = k2 (ordering cost must be reduced by a factor of k2)
13
Economic Production Quantity (EPQ)
 Production of lot instead of ordering
 P: production per unit time
 Total cost by production lot size Q
TC 
D
 DQ
 S  1    hC
Q
 P 2
 Optimal production quantity Q*
 When P goes to infinite, Q* goes to EOQ.
2 DS
Q 
1  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
2 DS
hC
TCi*  2hCDS
 All-aggregate ordering
ASQ* 
2DS
SSQ  m 
hC
*
SSTC*  m  2hCDS
2m DS
2DS
 m
hC
hC
ASTC*  2hCm DS  m  2hCDS
 Impractical supposition (for analysis purpose)
15
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
m
m


TC   S   si   n   Di hi Ci 2n
i 1
i 1


n 
*
m
DhC
i 1
i i
i
m


2 S   si 
i 1


16
Lot Sizing with Multiple Products
 Example 10-3 and 10-4
i LE22B
 Input
• Common transportation cost, S = $4,000
• Holding cost fraction, h = 0.2
 Ordering each products independently
• ITC* = $155,140
i
LE22B
LE19B
Di
Ci
si
12,000
$500
$1,000
LE19B
LE19A
1,200
$500
$1,000
120
$500
$1,000
200,000
LE19A
180,000
160,000
Qi*
ni *
TCi*
1,095
11.0
$109,544
346
3.5
$34,642
110
1.1
$10,954
LE19B
100,000
60,000
• n* = 9.75
• JTC* = $136,528
LE22B
120,000
80,000
 Ordering jointly
i
140,000
40,000
20,000
0
LE19A
LE22B
Qi*
1,230
123
LE19B
LE19A
12.3
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)
 Number of all possible ways
• http://en.wikipedia.org/wiki/Bell_number
 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
D
Q
TC  TO  TH  TM   S   hC  DC
Q
2
TO: total ordering cost
TH: total holding cost
TM: total material cost
 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
19
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
C0
C1
C2
...
C0 , 0  q0  Q  q1
C ,
q1  Q  q2
 1
C  C2 ,
q2  Q  q3



Cr ,
qr  Q  qr 1  
average
cost
per unit
Cr
q0
...
q1
q2
q3 ... qr
20
All Unit Quantity Discount
 Solution procedure
1. Evaluate the optimal lot size for each Ci.
Qi* 
2 DS
hC i
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*
21
All Unit Quantity Discount
 Example 10-7
 r = 2, D = 120,000/year
 S = $100/lot, h = 0.2
 Q* = 10,000
i
0
qi
Ci
0
$3.00
1
2
5,000 10,000
$2.96 $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
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
10,000
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 .
C2
...
TC  TO  TH  TM
V  Q  qi Ci
D
 S  i
h
Q
2
D
  Vi  Q  qi Ci 
Q
marginal
cost C0
per unit C1
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
0
1
2
qi
Ci
Vi
0
$3.00
$0
5,000
$2.96
$15,000
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
25
Marginal Unit Quantity Discount
 Example 10-8 (cont’d)
 Sensitivity analysis
• Optimal order quantity Q* with regard to ordering cost
(no discount)
C = $3
(discount)
(original) S = $100/lot
6,324
16,961
(reduced) S' = $4/lot
1,256
15,775
 Higher inventory level (longer average flow time)
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
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)
27
Coordination for Total Supply Chain Profit
 Quantity discounts for commodity products
 Assumptions
• 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)
 (a) No discount
• Retailer’s (local) optimal order quantity ( supply chain’s decision)
• Q(a) = (2120,000100/0.23)1/2 = 6,325
• Total cost (without material cost)
• TC0(a) = TCS(a) + TCR(a) = $6,008 + $3,795 = $9,803
 () Minimum total cost, TC*, regarding supply chain (coordination)
• Q* = 9,165
• TC0* = TCS* + TCR* = $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)?
D
Q
D
Q
 SS   hSCS   S R   hR CR
Q
2
Q
2
D
Q
 SS  S R   hSCS  hR CR 
Q
2
TC  TCS  TCR 
 Q* 
2 DSS  S R 
 9,165
hSCS  hR CR
29
Coordination for Total Supply Chain Profit
 Quantity discounts for commodity products (cont’d)
 (b) Lot size-based quantity discount offering by manufacturer
• Price schedule of CR
• q1 = 9,165, C0 = $3, C1 = $2.9978
• Retailer’s (local) optimal order quantity (considering material cost)
• Q(b) = 9,165
• Total cost (without material cost)
• TC0(b) = TCS(b) + TCR(b) = $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)
 (c) Other approach: setup cost reduction by manufacturer
Manufacturer
(supplier)
Retailer
customers
S'S = 100
hS = 0.2
CS = 2
SR = 100
hR = 0.2
CR = 3
D = 120,000
• Retailer’s (local) optimal order quantity
• Q(c) = Q(a) = 6,325
• Total cost (without material cost): no need to discount!
• TC0(c) = TCS(c) + TCR(c) = $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!
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)
CS = 2
Retailer
CR = ?
customers
p=?
32
Coordination for Total Supply Chain Profit
 Quantity discounts for products with market power (cont’d)
 (a) 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
» p = 3 + 0.5CR
• Demand by p (supplier’s order quantity)
» D = 360 – 60p = 180 – 30CR
• Expected manufacturer’s profit, ProfS
» ProfS = (CR – CS)(180 – 30CR)
 CR(a) that maximizes ProfS (manufacturer’s decision)
» CR(a) = $4
• Retailer’s decision on p(a) with given CR(a)
• p(a) = $5 (D(a) = 360,000 – 60,000p(a) = 60,000)
• Supply chain profit, Prof0(a)
• Prof0(a) = ProfS(a) + ProfR(a) = $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
• (b) Two-part tariff
• Up-front fee $180,000 (fixed) + material cost $2/unit (variable)
• Retailer’s decision
» ProfR = (p – CR)(360 – 60p) – 180,000
» p(b) = 3 + 0.5CR = $4
• Prof0(b) = Prof S(b) + ProfR(b) = $180,000 + $60,000 = $240,000
 Retailer’s side: larger volume  more discount
• (c) Volume-based quantity discount
• Pricing schedule of CR
» q1 = 120,000, C0 = $4, C1 = $3.5
• p(c) = $4
• Prof0(c) = ProfS(c) + ProfR(c) = $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
Q
Q
 Q 
TC  S  d  hC  d   d  C  d Qd  TC * 1  d 
2
D
D

dD
CQ*
Q 

hC  d  C  d
*
d
 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 = 2.5 + 0.5CR
 (a) No discount: CR = $3
• p(a) = $4, D(a) = 60,000
 (b) Discount: C'R = $2.85
• p(b) = $3.925, D(b) = 64,500  p(a) – p(b) = 0.075 < 0.15 = CR – C'R
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Short-term Discounting: Trade Promotion
 Retailers’ 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.
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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-through
• Scanner-based promotion
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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!)
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