Chapter 12
Inventory Management
Operations Management - 5th Edition
Roberta Russell & Bernard W. Taylor, III
Copyright 2006 John Wiley & Sons, Inc.
Beni Asllani
University of Tennessee at Chattanooga
Lecture Outline
 Elements of Inventory Management
 Inventory Control Systems
 Economic Order Quantity Models
 Quantity Discounts
 Reorder Point
 Order Quantity for a Periodic Inventory
System
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What Is Inventory?
 Stock of items kept to meet future
demand
 Purpose of inventory management


how many units to order
when to order
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Types of Inventory
 Raw materials
 Purchased parts and supplies
 Work-in-process (partially completed)
products (WIP)
 Finished product
 Items being transported
 Tools and equipment
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Inventory and Supply Chain
Management
 Bullwhip effect






demand information is distorted as it moves away
from the end-use customer
higher safety stock inventories are stored to
compensate
Seasonal or cyclical demand
Inventory provides independence from vendors
Take advantage of price discounts
Inventory provides independence between
stages and avoids work stop-pages
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Two Forms of Demand
 Dependent


Demand for items used to produce
final products
Tires stored at a Goodyear plant are
an example of a dependent demand
item
 Independent


Demand for items used by external
customers
Cars, appliances, computers, and
houses are examples of independent
demand inventory
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Inventory and Quality
Management
 Customers usually perceive quality
service as availability of goods they want
when they want them
 Inventory must be sufficient to provide
high-quality customer service in TQM
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Inventory Costs
 Carrying cost
 cost of holding an item in inventory
 Ordering cost
 cost of replenishing inventory
 Shortage cost
 temporary or permanent loss of sales
when demand cannot be met
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Inventory Control Systems
 Continuous review system (fixedorder-quantity) -- Economic Order
Quantity Model

constant amount ordered when
inventory declines to predetermined
level
 Periodic review system (fixed-timeperiod)

order placed for variable amount after
fixed passage of time
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ABC Classification
 Class A


5 – 15 % of units
70 – 80 % of value
 Class B


30 % of units
15 % of value
 Class C


50 – 60 % of units
5 – 10 % of value
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ABC Classification: Example
PART
1
2
3
4
5
6
7
8
9
10
UNIT COST ANNUAL DEMAND
$ 60
350
30
80
30
20
10
320
510
20
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90
40
130
60
100
180
170
50
60
120
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ABC Classification:
Example (cont.)
PART
9
8
2
1
4
3
6
5
10
7
TOTAL
PART
VALUE
$30,600
1
16,000
2
14,000
3
5,400
4
4,800
5
3,900
3,600
6
CLASS
3,000
7
2,400
A
8
1,700
B
9
C
$85,400
10
% OF TOTAL % OF TOTAL
UNIT
ANNUAL
USAGE
VALUECOSTQUANTITY
% CUMMULATIVE
35.9
6.0
$ 60
18.7
5.0
350
16.4
4.0
30
6.3
9.0
5.680
6.0
4.630
10.0
4.220 % OF TOTAL
18.0
ITEMS
VALUE
3.510
13.0
12.0
9, 8,2.8
2
71.0
320
17.0
1, 4,2.0
3
16.5
5107
6, 5, 10,
12.5
20
6.0
90
11.0
40
A
15.0
130
24.0
30.0
B60
100
40.0
% OF TOTAL
58.0
180
QUANTITY
71.0
170
C
83.0
50 15.0
100.0
25.0
60 60.0
120
Example 10.1
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Economic Order Quantity
(EOQ) Models
 EOQ is fixed – the same with each
order

optimal order quantity that will minimize
inventory costs
 Basic EOQ model
 Production quantity EOQ model in
which the replenishment (or receipt)
rate is not instantaneous or infinite—it
is finite
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Assumptions of Basic
EOQ Model
 Demand is known with certainty and
is constant over time
 No shortages are allowed
 Lead time for the receipt of orders is
constant
 Order quantity is received all at once
-- i.e. the replenishment rate (or
receipt rate) is instantaneous or
infinite
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Inventory Order Cycle
Order quantity, Q
Inventory Level
Demand
rate
Reorder point, R
0
Lead
time
Order Order
placed receipt
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Lead
time
Order Order
placed receipt
Time
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EOQ Cost Model
Co - cost of placing order
Cc - annual per-unit carrying cost
D - annual demand
Q - order quantity
Annual ordering cost =
Co D
Q
Annual carrying cost =
CcQ
2
Total cost =
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CoD
+
Q
CcQ
2
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EOQ Cost Model
Proving equality of
costs at optimal point
Deriving Qopt
CoD
CcQ
TC =
+
Q
2
CoD
Cc
TC
=
+
-Q2
2
Q
C0D
Cc
0=
+
-Q2
2
Qopt =
2CoD
Cc
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Co D
CcQ
=
Q
2
Q2
2CoD
=
Cc
Qopt =
2CoD
Cc
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EOQ Cost Model (cont.)
Annual
cost ($)
Total Cost
Slope = 0
CcQ
Carrying Cost =
2
Minimum
total cost
CoD
Ordering Cost = Q
Optimal order
Qopt
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Order Quantity, Q
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EOQ Example
Cc = $0.75 per gallon
Qopt =
2CoD
Cc
Qopt =
2(150)(10,000)
(0.75)
Co = $150
Qopt = 2,000 gallons
D = 10,000 gallons
CoD
CcQ
TCmin =
+
Q
2
TCmin
(150)(10,000) (0.75)(2,000)
=
+
2,000
2
TCmin = $750 + $750 = $1,500
Orders per year = D/Qopt
= 10,000/2,000
= 5 orders/year
Order cycle time = 311 days/(D/Qopt)
= 311/5
= 62.2 store days
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Production Quantity
Model
 An inventory system in which an order is received
gradually, as inventory is simultaneously being
depleted—the receipt or replenishment rate is finite
 AKA non-instantaneous receipt model

assumption that Q is received all at once is relaxed
 p - daily rate at which an order is received
over time, a.k.a. production rate
 d - daily rate at which inventory is demanded (=D/250),
where it is assumed that there are 250 days/yr
 p must be strictly greater than d
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Production Quantity Model
(cont.)
Inventory
level
Q(1-d/p)
Maximum
inventory
level
Q
(1-d/p)
2
Average
inventory
level
0
Order
receipt period
Begin
End
order order
receipt receipt
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Time
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Production Quantity Model
(cont.)
p = production rate
d = demand rate
Maximum inventory level = Q - Q d
p
=Q1- d
p
Q
d
Average inventory level =
12
p
2CoD
Qopt =
d
Cc 1 p
CoD CcQ
d
TC = Q + 2 1 - p
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Production Quantity Model:
Example
Cc = $0.75 per gallon
Co = $150
d = 10,000/311 = 32.2 gallons per day
2CoD
Qopt =
Cc 1 - d
p
D = 10,000 gallons
p = 150 gallons per day
2(150)(10,000)
=
CoD CcQ
d
TC = Q + 2 1 - p
32.2
0.75 1 150
= 2,256.8 gallons
= $1,329
2,256.8
Q
Production run =
=
= 15.05 days per order
150
p
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Production Quantity Model:
Example (cont.)
Number of production runs =
10,000
D
=
= 4.43 runs/year
2,256.8
Q
Maximum inventory level = Q 1 -
d
p
= 2,256.8 1 -
32.2
150
= 1,772 gallons
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Quantity Discounts
Price per unit decreases as order
quantity increases
CoD
CcQ
TC =
+
+ PD
Q
2
where
P = per unit price of the item
D = annual demand
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Quantity Discount Model (cont.)
ORDER SIZE
0 - 99
100 – 199
200+
PRICE
$10
8 (d1)
6 (d2)
TC = ($10 )
TC (d1 = $8 )
Inventory cost ($)
TC (d2 = $6 )
Carrying cost
Ordering cost
Q(d1 ) = 100 Qopt
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Q(d2 ) = 200
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Quantity Discount: Example
QUANTITY
1 - 49
50 - 89
90+
Qopt =
PRICE
$1,400
1,100
900
2CoD
=
Cc
Co = $2,500
Cc = $190 per computer
D = 200
2(2500)(200)
= 72.5 PCs
190
For Q = 72.5
CcQopt
Co D
TC =
+
2 + PD = $233,784
Qopt
For Q = 90
CcQ
CoD
TC =
+ 2 + PD = $194,105
Q
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Reorder Point
Level of inventory at which a new order
is placed
R = dL
where
d = demand rate per period
L = lead time
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Reorder Point: Example
Demand = 10,000 yards/year
Store open 311 days/year
Daily demand = 10,000 / 311 = 32.154
yards/day
Lead time = L = 10 days
R = dL = (32.154)(10) = 321.54 yards
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Safety Stocks
 Safety stock
 buffer added to on hand inventory during lead
time
 Stockout
 an inventory shortage
 Service level
 probability that the inventory available during
lead time will meet demand
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Variable Demand with
a Reorder Point
Inventory level
Q
Reorder
point, R
0
LT
LT
Time
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Inventory level
Reorder Point with
a Safety Stock
Q
Reorder
point, R
Safety Stock
0
LT
LT
Time
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Reorder Point With
Variable Demand
R = dL + zd L
where
d = average daily demand
L = lead time
d = the standard deviation of daily demand
z = number of standard deviations
corresponding to the service level
probability
zd L = safety stock
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Reorder Point for
a Service Level
Probability of
meeting demand during
lead time = service level
Probability of
a stockout
Safety stock
zd L
dL
Demand
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R
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Reorder Point for
Variable Demand
The carpet store wants a reorder point with a 95%
service level and a 5% stockout probability
d = 30 yards per day
L = 10 days
d = 5 yards per day
For a 95% service level, z = 1.65
R = dL + z d L
Safety stock = z d L
= 30(10) + (1.65)(5)( 10)
= (1.65)(5)( 10)
= 326.1 yards
= 26.1 yards
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Order Quantity for a
Periodic Review Inventory System
Q = d(tb + L) + zd
tb + L - I
where
d
tb
L
d
zd
= average demand rate
= the fixed time between orders
= lead time
= standard deviation of demand
tb + L = safety stock
I = inventory level
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Fixed-Period Model with
Variable Demand
d
d
tb
L
I
z
= 6 bottles per day
= 1.2 bottles
= 60 days
= 5 days
= 8 bottles
= 1.65 (for a 95% service level)
Q = d(tb + L) + zd
tb + L - I
= (6)(60 + 5) + (1.65)(1.2)
60 + 5 - 8
= 397.96 bottles
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