Managing Facilitating Goods
Replenishment
order
Factory
Production
Delay
Replenishment Replenishment
order
order
Wholesaler
Distributor
Shipping
Delay
Wholesaler
Inventory
Retailer
Shipping
Delay
Distributor
Inventory
Customer
order
Customer
Item Withdrawn
Retailer
Inventory
Learning Objectives
• Discuss the role of information technology in managing
inventories.
• Describe the functions and costs of an inventory system.
• Determine the order quantity.
• Determine the reorder point and safety stock for inventory
systems with uncertain demand.
• Design a continuous or periodic review inventory-control
system.
• Conduct an ABC analysis of inventory items.
• Determine the order quantity for the single-period inventory
case.
• Describe the rationale behind the retail discounting model.
Role of Inventory in Services
•
•
•
•
•
•
Decoupling inventories
Seasonal inventories
Speculative inventories
Cyclical inventories
In-transit inventories
Safety stocks
Considerations in Inventory Systems
• Type of customer demand
• Planning time horizon
• Replenishment lead time
• Constraints and relevant costs
Relevant Inventory Costs
• Ordering costs
• Receiving and inspections costs
• Holding or carrying costs
• Shortage costs
Inventory Management Questions
• What should be the order quantity
(Q)?
• When should an order be placed,
called a reorder point (ROP)?
• How much safety stock (SS) should
be maintained?
Inventory Models
• Economic Order Quantity (EOQ)
• Special Inventory Models
With Quantity Discounts
Planned Shortages
• Demand Uncertainty - Safety Stocks
• Inventory Control Systems
Continuous-Review (Q,r)
Periodic-Review (order-up-to)
• Single Period Inventory Model
Units on Hand
Inventory Levels For EOQ Model
Q
0
Q
D
Time
Annual Costs For EOQ Model
900
700
600
Holding Cost
Ordering Cost
Total Cost
500
400
300
200
100
Order Quantity, Q
140
120
100
80
60
40
20
0
0
Annual Cost, $
800
EOQ Formula
• Notation
D = demand in units per year
H = holding cost in dollars/unit/year
S = cost of placing an order in dollars
Q = order quantity in units
• Total Annual Cost for Purchase Lots
TCp  S ( D / Q)  H (Q / 2)
• EOQ
2 DS
EOQ 
H
Annual Costs for Quantity
Discount Model
22,000
C = $20.00
C = $19.50
C = $18.75
21000
20000
2000
1000
0
100
200
300
400
Order quantity, Q
500
600
700
Inventory Levels For Planned
Shortages Model
Q-K
Q
TIME
0
-K
T1
T2
T
Formulas for Special Models
• Quantity Discount Total Cost Model
TCqd  CD  S ( D / Q)  I (CQ / 2)
• Model with Planned Shortages
D
(Q  K ) 2
K2
TCb  S  H
B
Q
2Q
2Q
Q 
*
2 DS  H  B 



H
B 
 H 
K Q 

 H  B
*
*
Values for Q* and K* as A
Function of Backorder Cost
B
Q*
B 
2DS
H
0 B 
2DS  H  B 


H  B 
B 0
undefined
K*
0
 H 
Q*
 H  B 
Q*
Inventory Levels
0
0
0
Demand During Lead Time
Example
L  3
  15
.
+
u=3
=
+
+
u=3
  15
.
  15
.
  15
.
u=3
u=3

d L  12
ROP
ss
Four Days Lead Time
Demand During Lead time
Safety Stock (SS)
• Demand During Lead Time (LT) has
Normal Distribution with
- Mean(d L )   ( LT )
- Std . Dev.( L )   LT
• SS with r% service level
SS  zr LT
• Reorder Point
ROP  SS  d L
Continuous Review System (Q,r)
Amount used during first lead time
Inventory on hand
EOQ
Reorder point, ROP
d3
Average lead time usage, dL
Safety stock, SS
d1
d2 EOQ
First lead
time, LT1
LT2
LT3
Time
Order 1 placed
Order 2 placed
Shipment 1 received
Order 3 placed
Shipment 2 received
Shipment 3 received
Periodic Review System
(order-up-to)
Inventory on Hand
Review period
Target inventory level, TIL
RP
RP
RP
First order quantity, Q1
Q3
Q2
d3
d1
Amount used during
first lead time
d2
Safety stock, SS
First lead time, LT1
LT2
LT3
Time
Order 1 placed
Order 2 placed
Shipment 1 received
Order 3 placed
Shipment 2 received Shipment 3 received
Inventory Control Systems
• Continuous Review System
2 DS
H
ROP  SS   LT
EOQ 
SS  zr LT
• Periodic Review System
RP  EOQ / 
TIL  SS   ( RP  LT )
SS  zr RP  LT
Percentage of inventory items (SKUs)
100
90
80
70
60
C
50
30
B
20
10
A
40
100
90
80
70
60
50
40
30
20
10
0
0
Percentage of dollar volume
ABC Classification of Inventory
Items
Inventory Items Listed in
Descending Order of Dollar Volume
Unit cost
($)
Monthly
Sales
(units)
Dollar
Volume ($)
Computers
Entertainment center
3000
2500
50
30
150,000
75,000
Television sets
Refrigerators
Monitors
400
1000
200
60
15
50
Stereos
Cameras
Software
Computer disks
CDs
150
200
50
5
20
60
40
100
1000
200
Inventory Item
Totals
Percent of
Dollar
Volume
Percent of
SKUs
Class
74
20
A
24,000
15,000
10,000
16
30
B
9,000
8,000
5,000
5,000
4,000
10
50
C
100
100
305,000
Single Period Inventory Model
Newsvendor Problem Example
D = newspapers demanded
p(D) = probability of demand
Q = newspapers stocked
P = selling price of newspaper, $10
C = cost of newspaper, $4
S = salvage value of newspaper, $2
Cu = unit contribution: P-C = $6
Co = unit loss: C-S = $2
Single Period Inventory Model
Expected Value Analysis
p(D)
D
6
7
Stock Q
8
.028
.055
.083
.111
.139
.167
.139
.111
.083
.055
.028
2
3
4
5
6
7
8
9
10
11
12
4
12
20
28
36
36
36
36
36
36
36
2
10
18
26
34
42
42
42
42
42
42
0
8
16
24
32
40
48
48
48
48
48
-2
6
14
22
30
38
46
54
54
54
54
-4
4
12
20
28
36
44
52
60
60
60
$31.54
$34.43
$35.77
$35.99
$35.33
Expected Profit
9
10
Single Period Inventory Model
Incremental Analysis

P ( revenue) (unit revenue) 
E (revenue on last sale)
E (loss on last sale)
P (loss) (unit loss)
P( D  Q)Cu  P( D  Q)Co
1 P( D  Q)C
u
 P( D  Q)Co
Cu
P ( D  Q) 
Cu  Co
(Critical Fractile)
where:
Cu = unit contribution from newspaper sale ( opportunity cost of underestimating demand)
Co = unit loss from not selling newspaper (cost of overestimating demand)
D = demand
Q = newspaper stocked
Critical fractile for the
newsvendor problem
P(D<Q)
(Co applies)
Probability
P(D>Q)
(Cu applies)
0.722
0
2
4
6
8
10
New spaper demand, Q
12
14
Retail Discounting Model
•
•
•
•
S = current selling price
D = discount price
P = profit margin on cost (% markup as decimal)
Y = average number of years to sell entire stock of “dogs” at
current price (total years to clear stock divided by 2)
• N = inventory turns (number of times stock turns in one year)
Loss per item = Gain from revenue
S – D = D(PNY)
S
D
(1  PNY )
Topics for Discussion
• Discuss the functions of inventory for different
organizations in the supply chain.
• How would one find values for inventory costs?
• How can information technology create a competitive
advantage through inventory management?
• How valid are the assumptions for the EOQ model?
• How is a service level determined for inventory
items?
• What inventory model would apply to service capacity
such as seats on an aircraft?
Interactive Exercise
The class engages in an estimation of the
cost of a 12-ounce serving of Coke in
various situations (e.g., supermarket,
convenience store, fast-food restaurant,
sit-down restaurant, and ballpark).
What explains the differences?