Inventory Management
Henry C. Co
Technology and Operations Management,
California Polytechnic and State University


Short-range decisions about supplies,
inventories, production levels, staffing
patterns, schedules, and distribution operations infrastructure
4 R’s




Right
Right
Right
Right
Material
Amount
Place
Time.
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Motivations
Economies of Scale
Uncertainties
Economies of Scale

Over-investment



Ties up capacity and financial resources
Inventory carrying cost, obsolescence,
etc.
Insufficient/late availability causes


Idle personnel or equipment when
components ran out
Lost sales/customer-goodwill if items are
out-of-stock.
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Uncertainties



Safety stock (demand or supply
uncertainty)
In-transit inventories (lead times)
Hedge inventories
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Drivers

Approximately 16 % of total assets are
invested in inventories (1986)


Materials’ average share in a manufacturer’s
cost of goods sold




In manufacturing firms, 25 to 35% of total assets
of typical are tied in inventories
40% in 1945
50% in 1960
> 60% Today
Spare parts (service parts) inventories of a
typical manufacturer

~ $ 5 million - $ 15 million
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

Consequently … distribution and
inventory (logistics) costs are quite
substantial
Value of inventories in U.S. ~ $ 1
trillion (1993)
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Basic definitions


An inventory is an accumulation of a
commodity that will be used to satisfy
some future demand.
Inventories of the following form:





Raw material
Components
Semi-finished goods
Spare parts
Purchased products in retailing.
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Functional classification

Cycle Inventories: Produce or buy in
larger quantities than needed.



Economies of scale
Quantity discounts
Restrictions (technological,
transportation,…)
inventory
cycle stock
time
a cycle
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
Safety Stock: Provides protection
against irregularities and
uncertainties.
inventory
place order at this time
reorder level
Safety stock
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time
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
Anticipation stock: Low demand in one
part of the year
build up stock
for the high demand season
Average
Anticipation stock
Low demand season
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

Hedge inventories : expect changes in
the conditions (price, strike, supply,
etc.)
Pipeline (or work-in-process)
inventories: goods in transit, between
levels of a supply chain, between work
stations.
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Service Operations



No tangible items to purchase or
inventory, materials management is of
minor concern
Operations that provide repair or
refurbishment services carry inventory
of replacement parts and supplies
Examples:


Automobile service centers carry
automotive parts
Hospitals carry inventory of food, linens,
medicine, and medical supplies
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Functions of Inventory



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
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To meet anticipated demand
To smooth production requirements
To decouple components of the
production-distribution
To protect against stock-outs
To take advantage of order cycles
To help hedge against price increases
or to take advantage of quantity
discounts
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Conflicting Needs

Some Excuses for Holding Excess
Inventory
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“Inaccurate sales forecast”
“Poor quality”
“Unsynchronized processes”
“Poor schedules”
“Unreliable suppliers”
“Unreliable shippers”
“Poor attitudes”
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
Pressures to Cut Inventory
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Interest/opportunity cost
Storage and handling
Property taxes
Insurance premiums
Shrinkage
INVENTORIES HIDE PROBLEMS!
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Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place
order
Receive
order
Place
order
Receive
order
Lead time
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Profile of … Frequent Orders
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The Classic EOQ Model
The Economic Order Quantity (EOQ)


The total cost curve reaches its minimum
where the carrying and ordering costs are
equal.
EOQ represents trade-off between fixed cost
associated with production or procurement
against inventory holding costs.
D = Rate of demand, units/year
S = Fixed cost of procurement, $/order
v = Variable cost of procurement.
h = Cost/unit time of holding each unit of
inventory, $/unit/year
Q = Quantity ordered, units
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The Classical EOQ Model
The total cost curve reaches its
minimum where the carrying and
ordering costs are equal.
Annual Cost

TC 
Q
D
H 
S
2
Q
Ordering Costs
Order Quantity (Q)
(optimal order quantity)
QO
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
Using calculus, we take the derivative
of the total cost function (TC) and set
the derivative (slope) equal to zero
and solve for Q.
Q OPT =
2DS
=
H
2(Annual Demand)(Or der or Setup Cost)
Annual Holding Cost
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Example
A grocery store pays $100 for each delivery of milk from
the dairy. They sell 200 gallons per week. Each gallon
costs the store $2, and they sell it for $3. They earn a 15%
return on cash that is invested in milk. They have a large
refrigerator that holds up to 2,000 gallons. It costs them
$10,000 per year to maintain.
D ~ 200(52) = 10,400 gallons per year.
h ~ 15%($2) = $.30 per gallon per year
S ~ $100
Q* = 2,633  Roughly one replenishment every 9 weeks?!?!
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Suppose that the refrigerator held only 100 gallons. How
much would it be worth to expand capacity to 200 gallons?
TC (100)  $100
TC (200)  $100
10,400
100
 $.3
 $10,415
100
2
10,400
200
 $.3
 $5,230
200
2
$5,185 per year in operational savings from doubling
freezer size.
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Example
A bank has determined that it costs $30 to replenish the
cash in one of its suburban ATMs. Customers take cash
out of the ATM at a rate of $2000 per day (365 days per
year). The bank earns a 10% return on cash that is not
sitting in an ATM. Let us define $1000 to be a basic unit
of cash.
D ~ $2K(365) = $730K per year.
h ~ 10%($1K) = $100 per $K per year
S ~ $30
Q* = $20.9K  Roughly one replenishment every 10 days.
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Potential Analytical Errors



Using different time units for holding
cost versus the rate of demand.
Determining the true opportunity cost
associated with holding inventory.
(Physical costs as well as financial.)
Dealing with operational constraints.
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When to Reorder?



Reorder Point R – When the quantity
on hand of an item drops to this
amount, the item is reordered
Safety Stock SS – Stock that is held in
excess of expected demand due to
variable demand rate and/or lead
time.
Service Level - Probability that
demand will not exceed supply during
lead time.
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Continuous Review
Inventory
slope = -D
R  U  SS
Cycle Stock
Reorder Point
(R)
Lead Time Demand
(U = D * LT )
Reorder Quantity
(Q)
}
SS
LT
Safety Stock
time
U = Average Lead Time Demand
SS = Safety Stock; R = U + SS
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Quantity
Maximum probable
demand during lead
time
Expected demand
during lead time
Reorder point R
Safety stock
LT
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Reorder Point
Service level =
Risk of a stock-out
probability of no stock-out
Expected
demand
Quantity
Safety-stock
0
z
U
R
Inventory Management (Henry C. Co)
z-scale
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Determining Reorder Point R



The Average lead time demand is equal to
the average rate of demand (D) multiplied
by the length of the lead time (L).
U  DL
The amount of safety stock is influenced by
the variability of demand, and our
preference for avoiding inventory versus
satisfying all of the demand.
If distribution of demand is stable over time,
and the demand in one interval of time is
statistically independent from that in another
interval, then the standard deviation of lead
time demand is proportional to the square
root of the length (L) of the lead time:
σ L  Lσ
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Standard Deviation of Lead-time Demand


If daily demand has mean 100, and standard
deviation 40, and the lead time L= 9 days:
Then Lead Time Demand has mean = 900,
and the standard deviation of lead time
demand =120 (=SQRT[9] * 40).
If annual demand has mean 1040, and
standard deviation 600, and the lead time L
= 2 weeks: Then Lead Time Demand has
mean = 40 (= 1040*(2/52) and the
standard deviation of lead time demand =
117.67 (=SQRT[2/52]*600.
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
If weekly demand has mean 20, and
standard deviation 83.2, and the lead time L
= 2 weeks: Then Lead Time Demand has
mean = 40 and the standard deviation of
lead time demand = 117.67. (Note that this
example is identical to the previous one, but
the demand was specified in terms of weekly
demand instead of annual.)
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Safety Stock


Safety stock determines the expected
amount of unsatisfied demand
Whenever the amount of demand
during the lead time exceeds R, the
excess represents the amount of
unsatisfied demand. The expected
amount of unsatisfied demand is:

  x  R g x dx
R
where g(x) represents the distribution of
Lead Time Demand.
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A Retail Stocking Problem
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Daily demand (7 days/wk) is normally
distributed with mean = 60, standard
deviation = 30.
Orders can be placed at any time, and will
be filled in exactly 6 days.
It costs $10 to place and order, and each
unit costs $5.
Annual holding costs are 10% of the value of
the item.
We want to determine an efficient inventory
policy that allows us to satisfy 99% of
demand from inventory.
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
First, we need to determine how much to
order at a time:
2 DS
260365$10
Q

 936
h


$.50
Thus, our average cycle stock is 936/2
=468.
In order to determine the re-order point, we
need to know the length of the lead time (6
days), and the standard deviation of lead
time demand.
 L   L  30 6  73.48
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Now, to satisfy 99% of demand from inventory, we need:
E z  
Q1  P 
L

936.01
 0.127
73.48
From the table in the lecture note, we can see that
E(z) = .127 implies that z = .77 (or so).
We can now calculate the re-order point, which consists
of expected lead time demand plus safety stock.
R  dL  z L  606  .77(73.48)  417
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Batch Production
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Suppose a machine produces a
product at a production rate = p; e.g.,
p = 200 units/day.
Suppose the demand rate of this
product is d; e.g., d = 80 units/day.
Since p > d, inventory will increase at
(p-d) or (200-80 = 120) units/day.
Suppose the current inventory is 0.



In 10 days, the inventory level would be
10 days * 120 unit/day = 1,200 units.
In 20 days, the inventory level would be
20 days * 60 unit/day = 2,400 units.
etc.
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

Suppose the machine produces a batch of this
product, then stop, then resumes production at some
later time when the inventory of this item is low. This
is call batch production.
Batch production is very common in industry.



When a machine is used to produce two or more
products, one product at a time.
One decision the production manager has to make is
when to start producing each product, and when to
stop.
The run time is the amount of time the machine is
producing a batch.

For example, producing at 200 units/day, if we want to
produce 2,000 units per batch, the run time is
2,000/200 = 10 days. Here, batch size Q = 2,000 units,
the run time t = 10 days.
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Maximum Inventory Level

If the current inventory level is 0, what is the
inventory at the end of the run time?



Why is it that the machine produced 2,000 units in
10 days, and the maximum inventory level is only
1,200?


Since inventory will be rising at (200- 80 =120)
units/day, in 10 days, the inventory level will be 10
days * 120 unit/day = 1,200 units.
The inventory at the end of the run time is the
maximum inventory. It is equal to (p-d)*t = (200
units/day - 80 units/day)*10 days = 1,200 units.
Answer: We consumed d*t = 80 units/day * 10 days =
800 units.
After completing a batch, how long will it take to
deplete the inventory?

Answer: It will take (p-d)*t/d = 1,200/80 = 15 days to
deplete the inventory. This is the off-time.
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Number of Runs Per Year

If the annual demand of this product
is D = 24,000 units, how many runs of
this item do we produce each year?


Answer: Since we are producing Q =
2,000 units per batch, there will be D/Q =
24,000/2,000 = 12 batches per year.
In other words, there are D/Q = 12 cycles
per year. In each cycle, there is a period
of time the machine is producing the
product (the run time), and a period to
allow the inventory to deplete (the off
time).
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Average Inventory

What is the average inventory level?




During the run time, the inventory level rises from
0 to the maximum level of (p-d)*t = 1,200 units
(see page 4).
During the off time, the inventory level drops from
a maximum of (p-d)*t units to 0.
The average inventory level therefore = [0 + (pd)*t ]/2 = (0 + 1,200)/2 = 600 units.
Since p*t = Q, then t = Q/p. We can rewrite
the expression for the average inventory as
(p-d)*t/2 = (p-d)*(Q/p)/2 = (1-d/p)Q/2.
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Tradeoff



Batch size = production rate * run time. Large batch
size means long run time, and high average
inventory.
On the contrary, if the batch size is small, the run
time is short, and we need to run many batches per
year.
What is the average inventory if the batch size equals
the annual demand D = 24,000 units? How many
batches do we have to run per year?


Answer: The average inventory = (1-d/p)Q/2 = (180/200) (24,000)/2 = 7,200 units. We need to run one
batch per year.
What is the average inventory if the batch size equals
the weekly demand of 480 units (assuming 50
weeks/year)? How many batches do we have to run
per year?

Answer: 144 units; Run 50 batches per year.
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Optimal Tradeoff



Suppose the cost to carry one unit of inventory for
one year is H. Since the average inventory level is
(1-d/p)Q/2, the annual inventory-carrying cost is
H*(1-d/p) Q/2.
Suppose the cost to set-up the machine to produce a
batch is S. Since we need to run D/Q batches per
year (see page 5), the annual set-up cost is S*D/Q.
Adding the two costs, we have H*(1-d/p) Q/2 +
S*D/Q. Using calculus, the optimal batch size is
Q* 
2 DS
H  1  d 
p

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Illustration
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



Annual demand D = 24,000 units.
Production rate p = 200 units/day.
Demand rate d = 80 units/day (25
days/month).
Set-up cost S = $100.
Inventory-carrying cost H =
$2/unit/year.
The optimal batch size =2,000 units.
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




The machine will produce D/Q = 12 batches
a year.
Run time t = Q/p = 2,000/200 = 10 days.
Maximum inventory = (p-d)*t = (10040)*20 = 1,200 units.
Off time = 1,200 units/ 80 units/day = 15
days.
In other words, the machine will run this
product for 10 days, stop (to do something
else) for 15 days, before running this item
again.
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The New Boy Problem
Johnson & Pike, 1999
A-B-C Classification
A specific unit of stock to be controlled
is called a Stock Keeping Unit (SKU)
% of
total
annual 80
$ usage
80
20
40
100
60 % of total number of SKUs
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
20 % of SKUs account for 80 % of
total annual usage


1.
2.
3.
Large number of cheap items
A few very expensive items
A (most important) [ First 5-10 % of
items]
B (intermediate important) [ 50 % of
items]
C (least important) [40-45 % of
items only ~ 20 % of value]
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Inventory Management:
Then and Now

Innovations in information
technology and computer networking


tracking customer demand
production ~ demand
3.
Electronic Data Interchange
Efficient Consumer Response (ECR)
Vendor Managed Inventory (VMI)

How to utilize available information ?
1.
2.
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Electronic Data Interchange



Computer to computer transmission of
data (orders, invoices, payments, etc.)
Fast and reliable tracking of inventory
levels, outstanding customer orders,
backorders.
Shorter lead times for order
processing, more reliable due-date
quotation.
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Efficient Consumer Response (ECR)

Distributors and suppliers work
together so that information and
goods can be exchanged quickly,
efficiently and reliably





Efficient
Efficient
Efficient
Efficient
store assortment
replenishment
promotion
product introduction
Wegmans, Spartan Stores, HP, IBM,
Compaq
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
Vendor Managed Inventory (VMI)
Supplier manages the inventory on it’s
customer’s shelf (when and how much
to order)
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Framework for Inventory Management

Large number of items




Large manufacturer ~ 500,000 items
Retailer ~ 100,000
Items show different characteristics
Demand can occur in many ways:

Unit by unit, in cases, by the dozen, etc.
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

Decision making in production and
inventory management involves
dealing with large number of items,
with very diverse characteristics and
with external factors.
We want to resolve:



How often the inventory status (of an
item) should be determined ?
When a replenishment order should be
placed ?
How large the replenishment order should
be ?
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