Inventory Management
Outline
Basic Definitions and Ideas
•Reasons to Hold Inventory
•Inventory Costs
Inventory Control Systems
•Continuous Review Models
•Basic EOQ Model
•Quantity Discounts
•Safety Stock
•Special Case: The News Vendor Problem
•Discrete Probability Example
•Continuous Probability Example
•Periodic Review Model
What is Inventory?
• Inventory is a stock of items held to
meet future demand.
• Inventory management answers two
questions:
– How much to order
– When to order
Basic Concepts of Inventory Management
can be expanded to apply to a broad
array of types of “inventory”:
–
–
–
–
–
–
–
–
Raw materials
Purchased parts and supplies
Labor
In-process (partially completed) products
Component parts
Working capital
Tools, machinery, and equipment
Finished goods
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
Two Forms of Demand
• Dependent
– items used to produce final products
• Independent
– items demanded by external customers
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
Inventory Control Systems
• Fixed-order-quantity system
(Continuous)
– constant amount ordered when inventory
declines to predetermined level
• Fixed-time-period system (Periodic)
– order placed for variable amount after
fixed passage of time
Continuous Review Models
• Basic EOQ Model
• Quantity Discounts
• Safety Stock
The Basic EOQ Model
(Economic Order Quantity)
Assumptions of the Basic EOQ Model:
–
–
–
–
Demand is known with certainty
Demand is relatively constant over time
No shortages are allowed
Lead time for the receipt of orders is
constant
– The order quantity is received all at once
Inventory Order Cycle
Basic EOQ Model
750
Inventory
500
250
0
-250
0
50
100
150
200
250
300
350
Days
400
450
500
550
600
650
700
EOQ Model Costs
S = cost of placing order
D = annual demand
H = annual per-unit carrying cost
Q = order quantity
Annual ordering cost = SD/Q
Annual carrying cost = HQ/2
Total cost = SD/Q + HQ/2
Q* = Economic Order Quantity
EOQ Cost Curves
EOQ Example
1000
Total Annual Cost
750
Ordering
Holding
Total
500
250
0
0
100
200
300
400
500
Order Quantity
600
700
800
900
1000
EOQ Example
If D = 1,000 per year, S = $62.50 per order, and
H = $0.50 per unit per year, what is the
economic order quantity?
Q*
2DS

H
2 * 1000 * 62.5

0.5
 500
Quantity Discounts
Price per unit decreases as order quantity
increases:
Quantity
1-49
50-74
75-149
150-299
300-499
500+
Price
$35.00
$34.75
$33.55
$32.35
$31.15
$30.75
Quantity Discount Costs
DS QH
TC 

 CD
Q
2
C  per unit price
D = annual demand
Quantity Discount Cost Curves
EOQ with Quantity Discounts
$85,000
Total Annual Cost
$82,500
$80,000
0-49
50-74
75-149
150-299
300-499
500+
$77,500
$75,000
$72,500
$70,000
0
50
100
150
200
250
300
350
400
450
500
550
600
Order Quantity
650
700
750
800
850
900
950 1000
Quantity Discount Algorithm
Step 1. Calculate a value for Q*.
Step 2: For any discount, if the order
quantity is too low to qualify for
the discount, adjust Q upward to
the lowest feasible quantity.
Step 3: Calculate the total annual cost for
each Q*.
Quantity Discount Algorithm
Step 1. Calculate a value for Q*.
Q*

2DS
H

2 * 2 , 400 * 10
3.33
 120
Quantity Discount Algorithm
Step 2: For any discount, if the order quantity is too low
to qualify for the discount, adjust Q* upward
to the lowest feasible quantity.
Quantity
1-49
50-74
75-149
150-299
300-499
500+
Price
$35.00
$34.75
$33.55
$32.35
$31.15
$30.75
Min
1
50
75
150
300
500
Q*
120
120
120
120
120
120
Adj. Q*
120
120
120
150
300
500
EOQ with Quantity Discounts
$85,000
Total Annual Cost
$82,500
$80,000
0-49
50-74
75-149
150-299
300-499
500+
$77,500
$75,000
$72,500
$70,000
0
50
100
150
200
250
300
350
400
450
500
550
600
Order Quantity
650
700
750
800
850
900
950 1000
Quantity Discount Algorithm
Step 3: Calculate the total annual cost for each Q*.
Quantity
1-49
50-74
75-149
150-299
300-499
500+
Price Min Q* Adj. Q* Holding Cost Ordering Cost Purchasing Cost Total Cost
$35.00
1 120
120
$ 199.90
$ 199.90
$ 84,000.00
$84,399.80
$34.75 50 120
120
$ 199.90
$ 199.90
$ 83,400.00
$83,799.80
$33.55 75 120
120
$ 199.90
$ 199.90
$ 80,520.00
$80,919.80
$32.35 150 120
150
$ 249.75
$ 160.00
$ 77,640.00
$78,049.75
$31.15 300 120
300
$ 499.50
$ 80.00
$ 74,760.00
$75,339.50
$30.75 500 120
500
$ 832.50
$ 48.00
$ 73,800.00
$74,680.50
EOQ with Quantity Discounts
$85,000
Total Annual Cost
$82,500
$80,000
$77,500
$75,000
$72,500
$70,000
0
50
100
150
200
250
300
350
400
450
500
550
600
Order Quantity
650
700
750
800
850
900
950 1000
When to Order
Reorder Point = level of inventory at which to
place a new order (a.k.a. ROP, R)
R = dL
Where
d = demand rate per period
L = lead time
750
Inventory
500
250
0
-250
0
50
100
150
200
250
300
350
Days
400
450
500
550
600
650
700
Lead time for one of your fastest-moving products is 21
days. Demand during this period averages 100 units per
day. What would be an appropriate reorder point?
R
 dL
 100 * 21
 2,100
What About Random Demand?
(Or Random Lead Time?)
4000
3500
3000
Inventory
2500
2000
1500
1000
500
0
-500
0
50
100
150
200
-1000
Days
250
300
350
•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
Reorder Point with Variable Demand
(Leadtime is Constant)
R  d L  z , where
d = average daily demand
L = lead time
 d  standarddeviationof daily demand
  standarddeviationof demandduring lead time
z = number of standarddeviations for desired service level
z  safety stock
A carpet store wants a reorder point with a 95%
service level and a 5% stockout probability
during the leadtime.
d = 30 yards per day
L = 10 days
 d  5 yards per day
Determining the z-value for Service Level
Varianceof Lead Time Demand=(daily variances)x (numberof days of lead time)
= d2 L
Standarddeviation=  d2 L
= d L
d = 30 yards per day
L = 10 days
 d  5 yards per day
R  d L  z
 ( 30)(10)  ( 1.65)(5)( 10 )  300  26.1  326.1 yards
Safety stock  z  ( 1.65)(5)( 10)  26.1 yards
Determining the Safety Stock from the z-value
What If Leadtime is Random?
Random Variable
Mean
Standard Deviation
L = Leadtime
L
L
d = Demand
d
d
dL
L  d  L2
dL = Demand during Leadtime
2
d
2
Special Case: The Newsboy Problem
The News Vendor Problem is a special “single period”
version of the EOQ model, where the product drops in
value after a relatively brief selling period.
The name comes from newspapers, which are much less
valuable after the day they are originally published. This
model may be useful for any product with a short product
life cycle, such as
•Time-sensitive Materials (newspapers, magazines)
•Fashion Goods (some kinds of apparel)
•Perishable Goods (some food products)
Two new assumptions:
•There are two distinct selling periods:
•an initial period in which the product is sold at a regular
price
•a subsequent period in which the item is sold at a lower
“salvage” price.
•Two revenue values:
•a regular price P, at which the product can be sold during
the initial selling period
•a salvage value V, at which the product can be sold after
the initial selling period.
The salvage value is frequently less than the cost of
production C, and in general we wish to avoid selling
units at the salvage price.
“Damned if you do; damned if you don’t”:
• If we order too many, there will be extra units left
over to be sold at the disadvantageous salvage
price.
• If we order too few, some customer demand will
not be satisfied, and we will forego the profits that
could have been made from selling to the
customer.
Discrete Probability Example
Demand
300
400
500
600
700
800
Probability
0.05
0.10
0.40
0.30
0.10
0.05
C = $8.00
P = $20.00
V = $4.00
Newsboy Solution
In this case, it is useful to examine the marginal benefit
from each unit purchased. The expected profit from any
unit purchased is:
Expected Profit = P(Selling at Regular Price)*(Profit if Sold at Regular Price)
+ P(Selling at Salvage Price)*(Profit if Sold at Salvage Price)
= P(Selling at Regular Price)*(P - C)
+ P(Selling at Salvage Price)*(V - C)
Demand
300
400
500
600
700
800
Demand
300
400
500
600
700
800
Probability
0.05
0.10
0.40
0.30
0.10
0.05
P(Sell)
1.00
0.95
0.85
0.45
0.15
0.05
P(Not Sell)
0.00
0.05
0.15
0.55
0.85
0.95
Demand
300
400
500
600
700
800
Prob.
0.05
0.10
0.40
0.30
0.10
0.05
P(Sell)
1.00
0.95
0.85
0.45
0.15
0.05
Profit if Sold
$ 12.00
$ 12.00
$ 12.00
$ 12.00
$ 12.00
$ 12.00
P(Not Sell)
0.00
0.05
0.15
0.55
0.85
0.95
Profit if Not Sold
$ (4.00)
$ (4.00)
$ (4.00)
$ (4.00)
$ (4.00)
$ (4.00)
Weighted Average Profit
$ 12.00
$ 11.20
$ 9.60
$ 3.20
$ (1.60)
$ (3.20)
Marginal Expected Profit
Total Expected Profit
$14
$7,000
$12
$6,000
$10
$5,000
$8
$6
$4,000
$4
$3,000
$2
$2,000
$$1,000
$(2)
$(4)
$0
50
100
150
200
250
300
350
400
450
500
550
600
Quantity Ordered
650
700
750
800
850
900
950 1000
0
50
100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000
Quantity Ordered
Based on this analysis, we would order 600 units.
Continuous Probability Example
Using the same mean and standard deviation
as in the previous case (545.0 and 111.7), what
would be optimal if demand were normally
distributed?
Define CO and CU to be the “costs” of over-ordering
and under-ordering, respectively.
In this case:
CO
 V  C 
 4  8 
 $4.00
CU
 P C
 20  8
 $12.00
It can be shown that the optimal order quantity is
the value in the demand distribution that
corresponds to the “critical probability”:
Critical probability

CU
CU  CO
12
12  4
12

16

 0.75
From the standard normal table, the z-value corresponding
to a 0.75 probability is 0.6745.
Q
   0.6745
 545.0  0.6745111.7 
 620.3
Periodic Review Models
Sometimes a continuous review system doesn’t
make sense, as when the item is not very
expensive to carry, and/or when the customers
don’t mind waiting for a backorder.
A periodic review system only checks inventory
and places orders at fixed intervals of time.
A basic periodic review system might work as
follows:
Every T time periods, check the inventory level I,
and order enough to bring inventory back up to
some predetermined level.
This “order-up-to” level should be enough to cover
expected demand during the lead time, plus the
time that will elapse before the next periodic review.
Q  dT  L  I
We might also build some safety stock in
to the “order-up-to” quantity.
Q  dT  L  z T  L  I
What is Supply-Chain Management?
Supply-chain management is a total system
approach to managing the entire flow of
information, materials, and services from
raw-material suppliers through factories
and warehouses to the end customer
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What is a Supply-Chain?
Supply-chain is a term that describes how organizations
(suppliers, manufacturers, distributors, and customers)
are linked together
Services
Suppliers
Service Support
Operations
Local Service
Providers
Customers
Supply
Networks
Inputs
Transformation
Localization
Output
Manufacturing
Suppliers
Manufacturing
Distribution
Customers
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Measures of Supply-Chain Performance
•
One of the most commonly used measures in all of
operations management is “Inventory Turnover”
Cost of goodssold
Inventoryturnover
Averageaggregateinventoryvalue
•
In situations where distribution inventory is dominant,
“Weeks of Supply” is preferred and measures how many
weeks’ worth of inventory is in the system at a particular
time
 Averageaggregateinventoryvalue
 52 weeks
Weeksof supply 
Cost of goodssold


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Example: Supply-Chain Performance Measurement
Suppose a company’s new annual report claims their
costs of goods sold for the year is $160 million and their
total average inventory (production materials + workin-process) is worth $35 million. This company
normally has an inventory turn ratio of 10.
What is this year’s Inventory Turnover ratio?
What does it mean?
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56
Cost of goodssold
Inventoryturnover
Averageaggregateinventoryvalue
A
1 COGS
$
2 Avg Inventory $
3 Turnover
B
160,000,000
35,000,000
4.57
B01.2314 -- Operations -- Prof.
Juran
C
=B1/B2
57
Since the company’s normal inventory turnover ratio is
10, a drop to 4.57 means that the inventory is not
turning over as quickly as it had in the past.
In other words, they now have more inventory relative
to their cost of goods sold than before.
What else would you want to know about this
situation?
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58
Supply Chain Strategy
Marshall Fisher:
• Adverse effects of price promotions
• Functional vs. Innovative products
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Hau Lee’s Supply Chain Concepts
• Hau Lee’s approach to supply chains centers on
aligning the supply chain with process side
uncertainties (focus on the supply side)
• A stable supply process has mature technologies and an
evolving supply process has rapidly changing
technologies
• Types of Supply Chains
– Efficient
– Risk-Hedging
– Responsive
– Agile
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60
Demand Characteristics
Functional
Low demand Uncertainty
More predictable demand
Stable Demand
Long product life
Low inventory cost
Low profit margin
Low product variety
Higher volume
Low stockout cost
Low obsolescence
B01.2314 -Operations -- Prof.
Innovative
High demand Uncertainty
Difficult to forecast
Variable Demand
Short selling season
High inventory cost
High profit margin
High product variety
Low volume
High stockout cost
High obsolescence
Supply Characteristics
Stable
Few breakdowns
Stable and higher yields
Few quality problems
More supply sources
Reliable suppliers
Few process changes
Few capacity constraints
Easy to change over
Flexible
Dependable lead times
61
Evolving
Vulnerable to breakdowns
Variable and lower yields
Potential quality problems
Limited supply sources
Unreliable suppliers
More process changes
Potential capacity constraints
Difficult to change over
Inflexible
Variable lead times
Hau Lee’s Uncertainty Framework
Low Supply
Uncertainty
(Stable Process)
High Supply
Uncertainty
(Evolving Process)
B01.2314 -Operations -- Prof.
Low Demand Uncertainty
High Demand Uncertainty
(Functional Products)
(Innovative Products)
Efficient Supply Chain
Responsive Supply Chain
(Grocery, Basic Apparel, Food, Oil
and Gas)
(Fashion Apparel, Computers, Popular
Music)
Risk-hedging Supply Chain
Agile Supply Chain
(Hydroelectric Power, Some Food
Produce)
(Telecom, High-end Computers,
Semiconductors)
62
Value Density
• Value density is defined as the value of
an item per pound of weight
• An important measure when deciding
where items should be stocked
geographically and how they should
be shipped
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Mass Customization
• Mass customization is a term used to describe the
ability of a company to deliver highly customized
products and services to different customers
• The key to mass customization is effectively
postponing the tasks of differentiating a product
for a specific customer until the latest possible
point in the supply-chain network
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Mass Customization
• Principle 1: A product should be designed so
it consists of independent modules that can
be assembled into different forms of the
product easily and inexpensively.
• Principle 2: Manufacturing and service
processes should be designed so that they
consist of independent modules that can be
moved or rearranged easily to support
different distribution network strategies.
B01.2314 -Operations -- Prof.
65
Mass Customization
• Principle 3: The supply network — the positioning of
the inventory and the location, number, and
structure of service, manufacturing, and distribution
facilities — should be designed to provide two
capabilities. First, it must be able to supply the basic
product to the facilities performing the customization
in a cost-effective manner. Second, it must have the
flexibility and the responsiveness to take individual
customers’ orders and deliver the finished,
customized good quickly.
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Summary
• Supply-Chain Management
• Measuring Supply-Chain Performance
• Outsourcing
• Value Density
• Mass Customization
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Summary
Basic Definitions and Ideas
•Reasons to Hold Inventory
•Inventory Costs
Inventory Control Systems
•Continuous Review Models
•Basic EOQ Model
•Quantity Discounts
•Safety Stock
•Special Case: The News Vendor Problem
•Discrete Probability Example
•Continuous Probability Example
•Periodic Review Model