Chapter 13
Inventory Management
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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Supply Chain Management
• Bullwhip effect
• demand information is distorted as it moves away
from the end-use customer
• higher safety stock inventories to 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 stoppages
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Quality Management in the Supply Chain
• Customers usually perceive quality service as
availability of goods they want when they want
them
• Inventory must be sufficient to provide highquality customer service in QM
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Types of Inventory
• Raw materials
• Purchased parts and supplies
• Work-in-process (partially completed) products
(WIP)
• Items being transported
• Tools and equipment
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Two Forms of Demand
• Dependent
• Demand for items used to produce final
products
• Tires for autos are 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 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 system (fixed-order-quantity)
• constant amount ordered when inventory declines to
predetermined level
• Periodic system (fixed-time-period)
• 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
PART
UNIT COST
ANNUAL USAGE
1
2
3
4
5
6
7
8
9
10
$ 60
350
30
80
30
20
10
320
510
20
90
40
130
60
100
180
170
50
60
120
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ABC Classification
PART
9
8
2
1
4
3
6
5
10
7
TOTAL
VALUE
% OF TOTAL
VALUE
$30,600
16,000
14,000
5,400
4,800
3,900
3,600
3,000
2,400
1,700
$85,400
35.9
18.7
16.4
6.3
5.6
4.6
4.2
3.5
2.8
2.0
% OF TOTAL
QUANTITY
6.0
5.0
4.0
9.0
6.0
10.0
18.0
13.0
12.0
17.0
% CUMMULATIVE
A
B
C
6.0
11.0
15.0
24.0
30.0
40.0
58.0
71.0
83.0
100.0
Example 10.1
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ABC Classification
CLASS
A
B
C
ITEMS
9, 8, 2
1, 4, 3
6, 5, 10, 7
% OF TOTAL
VALUE
% OF TOTAL
QUANTITY
71.0
16.5
12.5
15.0
25.0
60.0
Example 10.1
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Economic Order Quantity
(EOQ) Models
• EOQ
• optimal order quantity that will minimize
total inventory costs
• Basic EOQ model
• Production quantity model
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Assumptions of Basic EOQ Model
• Demand is known with certainty and is constant
over time
• No shortages are allowed
• Lead time for the receipt of orders is constant
• Order quantity is received all at once
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Inventory Order Cycle
Inventory Level
Order quantity, Q
Demand
rate
Average
inventory
Q
2
Reorder point, R
0
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Lead
time
Order Order
placed receipt
Lead
time
Order Order
placed receipt
Time
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EOQ Cost Model
Co - cost of placing order
Cc - annual per-unit carrying cost
D - annual demand
Q - order quantity
Annual ordering cost =
CoD
Q
Annual carrying cost =
CcQ
2
Total cost =
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CoD
+
Q
CcQ
2
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EOQ Cost Model
Proving equality of
costs at optimal point
Deriving Qopt
CoD
CcQ
TC =
+
Q
2
Co D
Cc
TC
= – Q2 +
2
Q
C0D
Cc
0 = – Q2 +
2
Qopt =
2CoD
Cc
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CoD
CcQ
=
Q
2
Q2
2CoD
=
Cc
Qopt =
2CoD
Cc
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EOQ Cost Model
Annual
cost ($)
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
Orders per year = D/Qopt
= 10,000/2,000
= 5 orders/year
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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
Order cycle time = 311 days/(D/Qopt)
= 311/5
= 62.2 store days
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Production Quantity Model
• Order is received gradually, as inventory is
simultaneously being depleted
• 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
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Production Quantity Model
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
p = production rate
d = demand rate
Maximum inventory level = Q - Q d
p
=Q1- d
p
Average inventory level =
Q
12
d
p
2CoD
Qopt =
d
Cc 1 p
CoD CcQ
d
TC = Q + 2 1 - p
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Production Quantity Model
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
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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Solution of EOQ Models With Excel
The optimal order
size, Q, in cell D8
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Solution of EOQ Models With Excel
The formula for Q
in cell D10
=(D4*D5/D10)+(D3*D10/2)*(1-(D7/D8))
=D10*(1-(D7/D8))
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Solution of EOQ Models With OM Tools
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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
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
QUANTITY
1 - 49
50 - 89
90+
Qopt =
PRICE
$1,400
1,100
900
2CoD
=
Cc
Co = $2,500
Cc = $190 per TV
D = 200 TVs per year
2(2500)(200)
= 72.5 TVs
190
For Q = 72.5
CcQopt
CoD
TC =
+
2 + PD = $233,784
Qopt
For Q = 90
CcQ
CoD
TC =
+ 2 + PD = $194,105
Q
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Quantity Discount Model With Excel
=IF(D10>B10,D10,B10)
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=(D4*D5/E10)+(D3*E10/2)+C10*D5
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Reorder Point
•
Inventory level at which a new order is placed
R = dL
where
d = demand rate per period
L = lead time
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Reorder Point
Demand = 10,000 gallons/year
Store open 311 days/year
Daily demand = 10,000 / 311 = 32.154
gallons/day
Lead time = L = 10 days
R = dL = (32.154)(10) = 321.54 gallons
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Safety Stock
• 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
• P(Demand during lead time <= Reorder Point)
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Inventory level
Variable Demand With Reorder Point
Q
Reorder
point, R
0
LT
LT
Time
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Inventory level
Reorder Point With Safety Stock
Q
Reorder
point, R
Safety Stock
0
LT
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Time
LT
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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 paint store wants a reorder point with a 95%
service level and a 5% stockout probability
d = 30 gallons per day
L = 10 days
d = 5 gallons 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 gallons
= 26.1 gallons
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Determining Reorder Point with Excel
The reorder point
formula in cell E7
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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
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Periodic Inventory System
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Fixed-Period Model With
Variable Demand
d
d
tb
L
I
z
= 6 packages per day
= 1.2 packages
= 60 days
= 5 days
= 8 packages
= 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 packages
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Fixed-Period Model with Excel
Formula for order
size, Q, in cell D10
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