Chapter 10
Inventory
Management
BA 320 Operations Management
Inventory
 Stock of items held to meet
future demand
 Inventory management answers
two questions
 How much to order
 When to order
BA 320 Operations Management
Types of Inventory
 Raw materials
 Purchased parts and supplies
 Labor
 In-process (partially completed) products
 Component parts
 Working capital
 Tools, machinery, and equipment
BA 320 Operations Management
Reasons to Hold
Inventory
 Meet unexpected demand
 Smooth seasonal or cyclical demand
 Meet variations in customer demand
 Take advantage of
price discounts
 Hedge against price
increases
 Quantity discounts
BA 320 Operations Management
Two Forms of Demand
 Dependent
 Items used to produce final products
 Independent
 Items demanded by external customers
BA 320 Operations Management
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
BA 320 Operations Management
Inventory Control
Systems
 Continuous system (fixed-orderquantity)
 Constant amount ordered when
inventory declines to predetermined
level
 Periodic system (fixed-time-period)
 Order placed for variable amount
after fixed passage of time
BA 320 Operations Management
ABC Classification
System
 Demand volume and value of items vary
 Classify inventory into 3 categories,
typically on the basis of the dollar value
to the firm
CLASS
A
B
C
PERCENTAGE
OF UNITS
5 - 15
30
50 - 60
PERCENTAGE
OF DOLLARS
70 - 80
15
5 - 10
BA 320 Operations Management
ABC Classification
PART
1
2
3
4
5
6
7
8
9
10
UNIT COST
ANNUAL USAGE
$ 60
350
30
80
30
20
10
320
510
20
90
40
130
60
100
180
170
50
60
120
Example 10.1
BA 320 Operations Management
ABC Classification
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
3,000
7
2,400
8
1,700
9
$85,400
10
% OF TOTAL % OF TOTAL
UNIT
ANNUAL
USAGE
VALUECOSTQUANTITY
% CUMMULATIVE
35.9
$ 60
18.7
350
16.4
30
6.3
5.680
4.630
4.220
3.510
2.8
320
2.0
510
20
6.0
5.0
4.0
9.0
6.0
10.0
18.0
13.0
12.0
17.0
90
40
130
60
100
180
170
50
60
120
6.0
11.0
15.0
24.0
30.0
40.0
58.0
71.0
83.0
100.0
Example 10.1
BA 320 Operations Management
ABC Classification
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
3,000
7
2,400
8
1,700
9
$85,400
10
% OF TOTAL % OF TOTAL
UNIT
ANNUAL
USAGE
VALUECOSTQUANTITY
% CUMMULATIVE
35.9
$ 60
18.7
350
16.4
30
6.3
5.680
4.630
4.220
3.510
2.8
320
2.0
510
20
6.0
5.0
4.0
9.0
6.0
10.0
18.0
13.0
12.0
17.0
90
A
40
130
60
B
100
180
170
C
50
60
120
6.0
11.0
15.0
24.0
30.0
40.0
58.0
71.0
83.0
100.0
Example 10.1
BA 320 Operations Management
ABC Classification
PART
TOTAL
PART
VALUE
9 $30,600
1
8
16,000
2
2
14,000
3
1 CLASS
5,400
4
4
4,800
A3,900
5
3
B3,600
6
6
C3,000
5
7
10
2,400
8
7
1,700
9
$85,400
10
% OF TOTAL % OF TOTAL
UNIT
ANNUAL
USAGE
VALUECOSTQUANTITY
% CUMMULATIVE
35.9
6.0
$ 60
18.7
5.0
350
16.4 % OF TOTAL
4.0
30
ITEMS6.3
VALUE9.0
5.680
6.0
9, 8, 2 4.630
71.010.0
1, 4, 3 4.220
16.518.0
6, 5, 10,
12.513.0
3.5710
2.8
12.0
320
2.0
17.0
510
20
6.0
90
11.0
A
40
15.0
% OF TOTAL
130
24.0
QUANTITY
60
B 15.030.0
100
40.0
180 25.058.0
60.071.0
170
C
83.0
50
100.0
60
120
Example 10.1
BA 320 Operations Management
ABC Classification
C
100 –
B
% of Value
80 –
60 –
A
40 –
20 –
0 |–
0
|
20
|
40
|
60
|
80
% of Quantity
BA 320 Operations Management
|
100
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
 The order quantity is received all
at once
BA 320 Operations Management
The Inventory Order Cycle
Inventory Level
Order quantity, Q
Reorder point, R
Time
0
Figure 10.1
BA 320 Operations Management
The Inventory Order Cycle
Order quantity, Q
Inventory Level
Demand
rate
Reorder point, R
0
Figure 10.1
Lead
time
Order Order
placed receipt
Lead
time
Order Order
placed receipt
BA 320 Operations Management
Time
EOQ Cost Model
Co - cost of placing order
Cc - annual per-unit carrying cost
D - annual demand
Q - order quantity
Co D
Annual ordering cost =
Q
CcQ
Annual carrying cost =
2
CoD
CcQ
Total cost =
+
Q
2
BA 320 Operations Management
EOQ Cost Model
CDeriving
D - annual
demand
o - cost ofQplacing order
Proving
equality
of
opt
at optimal
point
Cc - annual per-unit carrying cost costs
Q - order
quantity
CoD
CcQ
TC =
+
Q
2
CoD
CcQ
Annual ordering cost =
=
Q
2
C
D
C
TC
o
c
=
+
Q2
2
Q
CcQ
2CoD
Annual carrying cost = Q2 =
2
C0D
Cc
Cc
0=
+
Q2
2
CoD
CcQ
2CoD
Total cost =
+
Q
2Q =
2CoD
opt
Cc
Qopt =
Cc
BA 320 Operations Management
EOQ Cost Model
Annual
cost ($)
Order Quantity, Q
Figure 10.2
BA 320 Operations Management
EOQ Cost Model
Annual
cost ($)
CoD
Ordering Cost = Q
Order Quantity, Q
Figure 10.2
BA 320 Operations Management
EOQ Cost Model
Annual
cost ($)
CcQ
Carrying Cost =
2
CoD
Ordering Cost = Q
Order Quantity, Q
Figure 10.2
BA 320 Operations Management
EOQ Cost Model
Annual
cost ($)
Total Cost
Slope = 0
CcQ
Carrying Cost =
2
Minimum
total cost
CoD
Ordering Cost = Q
Optimal order
Qopt
Order Quantity, Q
Figure 10.2
BA 320 Operations Management
EOQ Example
Cc = $0.75 per yard
Qopt =
2CoD
Cc
Qopt =
2(150)(10,000)
(0.75)
Qopt = 2,000 yards
Co = $150
D = 10,000 yards
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
Order cycle time = 311 days/(D/Qopt)
= 10,000/2,000
= 311/5
= 5 orders/year
= 62.2 store days
Example 10.2
BA 320 Operations Management
EOQ with
Noninstantaneous Receipt
Inventory
level
Q(1-d/p)
Maximum
inventory
level
Q
(1-d/p)
2
Average
inventory
level
0
Time
Figure 10.3
BA 320 Operations Management
EOQ with
Noninstantaneous Receipt
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
Time
Figure 10.3
BA 320 Operations Management
EOQ with
Noninstantaneous Receipt
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
BA 320 Operations Management
Production Quantity
Cc = $0.75 per yard
Co = $150
d = 10,000/311 = 32.2 yards per day
2CoD
Qopt =
Cc 1 - d
p
D = 10,000 yards
p = 150 yards per day
2(150)(10,000)
=
CoD CcQ
d
TC = Q + 2 1 - p
32.2
0.75 1 150
= 2,256.8 yards
= $1,329
2,256.8
Q
Production run =
=
= 15.05 days per order
150
p
Example 10.3
BA 320 Operations Management
Production Quantity
Cc = $0.75 per yard
Co = $150
d = 10,000/311 = 32.2 yards per day
D = 10,000 yards
p = 150 yards per day
2CoD
2(150)(10,000)
10,000 = 2,256.8 yards
D
Qopt =
=
d
32.2
Number of
production
runs
=
=
= 4.43 runs/year
Cc 1 0.75Q1 - 2,256.8
150
p
d
32.2
Maximum
inventory
level
=
Q
1
=
2,256.8
1
CoD CcQ
d
p
150
TC = Q + 2 1 - p = $1,329
= 1,772 yards
2,256.8
Q
Production run =
=
= 15.05 days per order
150
p
Example 10.3
BA 320 Operations Management
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
BA 320 Operations Management
Quantity Discounts
 Price per unit decreases as order
quantity increases
CoD
CcQ
TC =
+
+ PD
Q
2
where
ORDER SIZE
P = per unit price
0 - of
99the item
D = annual
100demand
- 199
200+
BA 320 Operations Management
PRICE
$10
8 (d1)
6 (d2)
Inventory cost ($)
Quantity Discount Model
Figure 10.4
BA 320 Operations Management
Quantity Discount Model
TC = ($10 )
TC (d1 = $8 )
Inventory cost ($)
TC (d2 = $6 )
Carrying cost
Ordering cost
Q(d1 ) = 100 Qopt
Q(d2 ) = 200
Figure 10.4
BA 320 Operations Management
Quantity Discount Model
TC = ($10 )
TC (d1 = $8 )
Inventory cost ($)
TC (d2 = $6 )
Carrying cost
Ordering cost
Q(d1 ) = 100 Qopt
Q(d2 ) = 200
Figure 10.4
BA 320 Operations Management
Quantity Discount
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
C oD
TC =
+ 2 + PD = $194,105
Q
Example 10.4
BA 320 Operations Management
When to Order
Reorder Point is the level of inventory
at which a new order is placed
R = dL
where
d = demand rate per period
L = lead time
BA 320 Operations Management
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
Example 10.5
BA 320 Operations Management
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
BA 320 Operations Management
Variable Demand with
a Reorder Point
Inventory level
Q
Reorder
point, R
0
Figure 10.5
Time
BA 320 Operations Management
Variable Demand with
a Reorder Point
Inventory level
Q
Reorder
point, R
0
LT
Figure 10.5
LT
Time
BA 320 Operations Management
Inventory level
Reorder Point with
a Safety Stock
Q
Reorder
point, R
Safety Stock
0
LT
Figure 10.6
LT
Time
BA 320 Operations Management
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
BA 320 Operations Management
Reorder Point for
a Service Level
Probability of
meeting demand during
lead time = service level
Probability of
a stockout
Safety stock
zd L
Figure 10.7
dL
Demand
R
BA 320 Operations Management
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
Example 10.6
BA 320 Operations Management
Order Quantity for a
Periodic 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
BA 320 Operations Management
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
BA 320 Operations Management