Operations
Management
Inventory Management
Chapter 12
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Outline
 GLOBAL COMPANY PROFILE: AMAZON.COM
 FUNCTIONS OF INVENTORY

Types of Inventory
 INVENTORY MANAGEMENT
ABC Analysis
 Record Accuracy
 Cycle Counting
 Control of Service Inventories

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Outline - Continued
 INVENTORY MODELS
Independent versus Dependent Demand
 Holding, Ordering, and Setup Costs

 INVENTORY MODELS FOR INDEPENDENT
DEMAND
Basic Economic Order Quantity (EOQ) Model
 Minimizing Costs
 Reorder Points
 Production Order Quantity Model
 Quantity Discount Models

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Outline - Continued
 PROBABILISTIC MODELS WITH CONSTANT
LEAD TIME
 FIXED PERIOD (P) SYSTEMS
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Learning Objectives
When you complete this chapter, you should be
able to :
Identify or Define:
ABC analysis
 Record accuracy
 Cycle counting
 Independent and dependent demand
 Holding, Ordering, and Setup Costs

Describe or Explain:

The functions of inventory and basic inventory
models
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AMAZON.com
 Jeff Bezos, in 1995, started AMAZON.com as a
“virtual” retailer – no inventory, no warehouses, no
overhead; just a bunch of computers.
 Growth forced AMAZON.com to excel in inventory
management!
 AMAZON is now a worldwide leader in warehouse
management and automation.
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Order Fulfillment at AMAZON
1. You order items;, computer assigns your order to
distribution center [closest facility that has the
product(s)]
2. Lights indicate products ordered to workers who
retrieve product and reset light.
3. Items placed in crate with items from other
orders, and crate is placed on conveyor. Bar
code on item is scanned 15 times – virtually
eliminating error.
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Order Fulfillment at AMAZONContinued
4. Crates arrive at central point where items are
boxed and labeled with new bar code.
5. Gift wrapping done by hand (30 packages per
hour)
6. Box is packed, taped, weighed and labeled
before leaving warehouse in a truck.
7. Order appears on your doorstep within a week
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What is Inventory?
 Stock of materials
 Stored capacity
 Examples
© 1995
Corel Corp.
© 1984-1994 T/Maker Co.
© 1984-1994 T/Maker Co.
© 1995 Corel Corp.
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The Functions of Inventory
 To ”decouple” or separate various parts of the
production process
 To provide a stock of goods that will provide a
“selection” for customers
 To take advantage of quantity discounts
 To hedge against inflation and upward price
changes
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Types of Inventory
 Raw material
 Work-in-progress
 Maintenance/repair/operating supply
 Finished goods
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The Material Flow Cycle
Note the proportion of time material spends as inventory as
opposed to being actually worked on.
Effective inventory management and materials movement can
reduce overall cycle time significantly!
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Disadvantages of Inventory
 Higher costs
Item cost (if purchased)
 Ordering (or setup) cost



Costs of forms, clerks’ wages etc.
Holding (or carrying) cost

Building lease, insurance, taxes etc.
 Difficult to control
 Hides production problems
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ABC Analysis
 Divides on-hand inventory into 3 classes

A class, B class, C class
 Basis is usually annual $ volume

$ volume = Annual demand x Unit cost
 Policies based on ABC analysis
Develop class A suppliers more
 Give tighter physical control of A items
 Forecast A items more carefully

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Classifying Items as ABC
Class
A
B
C
% Annual $ Usage
100
80
60
% $ Vol
80
15
5
% Items
15
30
55
A
40
B
20
C
0
0
50
100
% of Inventory Items
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Cycle Counting
 Physically counting a sample of total inventory on
a regular basis
 Used often with ABC classification

A items counted most often (e.g., daily)
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Advantages of Cycle Counting
 Eliminates shutdown and interruption of
production necessary for annual physical
inventories
 Eliminates annual inventory adjustments
 Provides trained personnel to audit the accuracy
of inventory
 Allows the cause of errors to be identified and
remedial action to be taken
 Maintains accurate inventory records
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Why do we control inventory?
 Inventories represent a vast segment of total
economic activity.
 Even minor improvements can create large
savings.
How do we control inventory?
 Application of optimization techniques
 Information processing and retrieval techniques
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Decisions of an inventory policy
 If there is no production, i.e., pure inventory system
How much to order? Order quantity
 When to order? Reorder quantity
Ex:Order Q=100 units when the inventory level drops to
ROP=15 units.

 If there is also production

When to start/stop production?
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An inventory system
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Elements of Inventory Decisions
 Costs:



Ordering and Procurement costs (or Setup costs in manufacturing
systems)
Inventory holding or carrying costs
Inventory shortage costs
 Demand structure

How does it vary? Certain, uncertain?
 Supply structure

Any capacity limitations, defectives, number of suppliers?
 Lead times:

Certain, uncertain?
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Ordering and Procurement Costs
 Represent all expenses incurred in ordering or manufacturing items related
to
Acquisition
Transportation
 Collecting, sorting, placing the items in the storage
 Managerial and clerical costs associated with order placement.


 Ordering costs are fixed, independent of the order size.




Supplies
Forms
Order processing
Clerical support
 Procurement costs depend on the order size.
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Set-up Costs
In a manufacturing system order costs are
realized in the form of “set-up” costs. These
include:
 Clean-up costs
 Re-tooling costs
 Adjustment costs
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Inventory Holding or Carrying Costs
 Expenses incurred during the storage of items.
Physical Costs: Warehouse operation costs, insurence,
property taxes.
 Pilferage, spoilage, obsolescence
 Opportunity cost of investing in inventory rather than
investing somewhere else, ex. in a bank.

 Inventory costs are variable costs that depend on
the order size.
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Inventory Holding Costs
(Approximate Ranges)
Category
Cost as a
% of Inventory Value
Housing costs (building rent, depreciation,
operating cost, taxes, insurance)
6%
(3 - 10%)
Material handling costs (equipment, lease
or depreciation, power, operating cost)
3%
(1 - 3.5%)
Labor cost from extra handling
3%
(3 - 5%)
Investment costs (borrowing costs, taxes,
and insurance on inventory)
11%
(6 - 24%)
Pilferage, scrap, and obsolescence
Overall carrying cost
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3%
(2 - 5%)
26%
25
Shortage Costs
 Occur whenever the demand is not satisfied. Order is
either “backordered” or “lost”.
 Backordering Costs:
Fixed cost of extra managerial work.
 Loss of customer goodwill: Variable cost that depends on
duration of backorder.

 Lost Sales Costs:
Marginal profit that the item would have earned.
 Loss of customer goodwill.

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Demand Structure
 Continuous versus discrete demand
Ex: Natural gas consumption in houses
Detergent consumption in houses
 Deterministic (certain) versus stochastic (uncertain)
demand
Ex: Order quantities for the next months are 20,30,10,50.
Order quantities in a month are normally distributed with mean 25
and variance 4.
 Constant versus dynamic demand
Ex: Demand quantities for the next months are 20, 21, 20, 19
Demand quantities for the next months are 20, 50, 10, 2
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Supply Structure
 Any defectives?
If the received lot includes defective items this
brings uncertainty
 Any capacity limitations?
Do we fully receive what we order?
 Number of suppliers, fixed or variable?
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Lead time
 Time elapsed between the order delivery and
order receipt.
 Can be constant or stochastic.
Ex: Lead time is 10 days.
Lead time is between 8-12 days.
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Inventory Models
 Fixed order-quantity models
Help answer the
inventory planning
questions!
Economic order quantity
 Production order quantity
 Quantity discount

 Probabilistic models
 Fixed order-period models
© 1984-1994
T/Maker Co.
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The Economic Order Quantity
EOQ-Model
 Decision variable:
Q = Order Quantity
 Parameters:
S = Fixed cost per order ($/order)
D = Annual number of items demanded (unit/year)
P = Unit cost of procuring an item ($/unit)
I = Annual cost of holding a dollar in inventory ($/$/year)
H= Annual cost of holding a unit item in inventory ($/unit/year)
H=IP
 Objective is to “minimize total annual cost”.
Total
Annual cost
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=
Ordering
Cost
+
Holding
Cost
+
Procurement
Cost
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EOQ Inventory Policy
Average Inv. Level
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Assumptions of Classical EOQ Model
 Demand rate is constant or stable.
 There is infinite supply availability.
 Lead time is constant or zero.
 No quantity discounts are made.
 All demand is met on time, no backordering, no stockout.
 Only order (or setup) cost and holding cost
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EOQ Model
How Much to Order?
Annual Cost
Minimum
total cost
Order (Setup) Cost Curve
Optimal
Order Quantity (Q*)
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Order quantity
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Costs of EOQ Model
 Total ordering cost is the number of orders times the cost
per order:
D
Annual ordering cost   S
Q
 Total holding cost is the cost per item held 1 year times the
average inventory:
Q
Annual holding cost  H  
2
 The annual procurement cost is the product of annual
demand and unit cost:
Procurement cost = PD
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Annual Cost of EOQ-Model
D
Q
Total annual cost    S  H    PD
2
Q
 Here PD is not a relevant cost and thus dropped.
 Minimize Total Annual Inventory Cost:
D
Q
TC (Q)    S  H  
2
Q
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Optimal Solution of EOQ
 Optimal solution is the economic order quantity
Q* 
2SD
H
 Optimal Total Cost
TC *  2SDH
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Example:The House of Wines and
Liquors
Allex Mullen decides that the first task in utilizing inventory
models is to determine the value of model parameters:
 Annual demand 5200 cases of beer
 $10 telephone charge for ordering
 Purchase cost is $1.5/case beer +shipping cost $0.5/case
 10%bank interest, 5%state franchise tax, 5% theft
insurance rate
How many should he order, how often, and at what annual
relevant inventory cost?
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Solution:
The economic order quantity is
Q* 
2 DS
25200 10 

 509.9 or 510
IP
.202 
 The inventory cycle duration is
T = Q/D = 510/5200 = 0.098 year or 36 days
 The total annual relevant inventory cost is:
 5200 
 510 
TC (510)  
10  .20(2)
  $101.96  102.00  $203.96 / year
510
2




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Robustness of EOQ Model
 EOQ is a robust model with respect to the estimation
errors in D, S, P or I.
 Let Dactual=4 Destimated
Then EOQactual=2 Destimated
Since
2DactualS
2DestimatedS
EOQactual 
2
 2EOQestimated
H
H
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Ex: The House of Wines and Liquors
 Alex Mullen applies EOQ to another product, a
particular variety of Chilean wine that sells 1000
cases annually. The cost is $20 per case. A
telephone call to Chile to place an order costs
$100. The holding costs are the same as for Tres
Equis Beer.
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Ex: The House of Wines and Liquors
Q* 
2 DS
21000100

 223.6 or 224
IP
.2020
T = Q/D = 24/1000 = .224 year or 82 days
 1000 
 224 
TC (224)  
100  .20(20)
  $894.43/ye ar
 224 
 2 
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The Reorder Point ROP
When to give orders?
Q*
Inventory level (units)
Slope = units/day = d
ROP
(Units)
ROP=d*LT
Time (days)
Lead time = LT
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Optimal Inventory Policy
with Backordering
Orders placed during shortages are backordered.
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Optimal Inventory Policy
with Backordering
Imax: Quantity on hand when a shipment arrives.
C: Cost of being one item short for a year
2
2
D
H I max
C Q  I max 
TC (Q, I max )    S 

2Q
2Q
Q
Optimal order quantity and order level
2DS C  H
Q* 
H
C
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2 DS
C
I max * 
H CH
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Example:The House of Wines and
Liquors-Backorders
The marketing department tells Alex that beer is a convenience product that can not be
backordered, so sale is lost! However some wine customers are connoisseurs who are
willing to order out-of-stock items. Nevertheless, the store owner will incur some penalty
cost if there is a shortage of wine.
Suppose that retailer suffers lost profit on future business equal to $0.01/unit each day
that a wine is on backorder. What should be the optimal ordering policy if backordering
is allowed?
Solution: The order quantity is computed:
p = $.01×365 = $3.65/unit/year.
Q* 
2 DS
IP
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C  IP
21000100  3.65  .2020 

 324
C
.2020
3.65
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Example: Solution
 The max. order level Imax is
I
*
max

2 DS
IP
C

C  IP
21000 100 
3.65
 154
.2020 
3.65  .2020 
 The relevant cost is
.2020154 3.65170
 1000 
TC (324,154)  

 617.82
100 




324
2
324
2
324
 why?
smaller than before,
2
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2
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Is backordering better?
 Fewer orders are placed when there is backordering.
 Average inventory level is smaller.
Backorders/cycle= Q* – Imax* =324 – 154 = 170 units/cycle.
Proportion of demand not satisfied on time
=(Q*- Imax* )/Q*=170/324= 52.5%
Service level = 1- Proportion of demand not satisfied on time
= 1-52.5%=47.5%
 The results suggest that:
Retailers will run short in each cycle.
But can they get away with it?
 So backordering must make sense!
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Shortage Penalty Considerations
Shortage Cost
Shortage Cost
C
C
Time
Time
1 year
Theoretical Assumption
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1 year
Case excluded by the model
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Imputed Shortage Penalty
An alternative approach for establishing an inventory policy is based on
achieving a desired service level.
Service Level, L is the proportion of demand met on time
I *max
 L, so
*
Q
LQ*  I *max
Imputed shortage penalty
C =
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HL
1L
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As C increases EOQ is more
robust
D=1000 units/yr
S=$100/order
P=$20/unit
I= $0.20/$/year
L=47.5%
324
L=90%
Q*
236
EOQ with no
backordering
224
Imax*
212
154
C
$3.65
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$36
imputed shortage penalty
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Production Order Quantity Model
 Answers how much to order and when to order
 Allows partial receipt of material

Other EOQ assumptions apply
 Suited for production environment
Material produced, used immediately
 Provides production lot size

 Lower holding cost than EOQ model
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Reasons for Variability in Production
Most variability is caused by waste or by poor
management. Specific causes include:
employees, machines, and suppliers produce units that do
not conform to standards, are late or are not the proper
quantity
 inaccurate engineering drawings or specifications
 production personnel try to produce before
drawings or
specifications are complete
 customer demands are unknown

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Economic Production-Quantity
Model
The EOQ model may be extended to find the optimal production quantity,Q.
p: Daily production rate, d : Daily demand rate
1-d/p
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POQ Model Equations
Maximum inventory level
Setup Cost
Holding Cost
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=
D
Q
2*D*S
d
H* 1 p
= Q* =
p
Optimal Order Quantity
= Q*
(
1 -
( )
d
p
)
* S
= 0.5 * H * Q
( )
1-
d
p
D = Demand per year
S = Setup cost
H = Holding cost
d = Demand per day
p = Production per day
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Quantity Discount Model
 Answers how much to order &
when to order
 Allows quantity discounts


Reduced price when item is purchased in larger
quantities
Other EOQ assumptions apply
 Trade-off is between lower price & increased
holding cost
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Quantity Discount Schedule
Discount
Number
1
Discount
Quantity
0 to 999
Discount
(%)
No discount
Discount
Price (P)
$5.00
2
1,000 to 1,999
4
$4.80
3
2,000 and over
5
$4.75
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Quantity Discount – How Much to
Order
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Probabilistic Models
 Answer how much & when to order
 Allow demand to vary
Follows normal distribution
 Other EOQ assumptions apply

 Consider service level, L & safety stock
Service level = L =1 - Probability of stockout
 Higher service level means more safety stock


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More safety stock means higher ROP
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Probabilistic Models
When to Order?
Inventory Level
Frequency
Service
Level
P(Stockout)
Optimal
Order
Quantity
SS
X
ROP
Reorder
Point
(ROP)
Safety Stock (SS)
Place
order
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Lead Time
Receive
order
Time
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Fixed Period Model
Answers how much to order
Orders placed at fixed intervals
Inventory brought up to target amount
 Amount ordered varies

No continuous inventory count

Possibility of stockout between intervals
Useful when vendors visit routinely

Example: P&G representative calls every 2 weeks
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Inventory Level in a Fixed Period
System
Various amounts (Qi) are ordered at regular time intervals
(p) based on the quantity necessary to bring inventory up
to target maximum
On-Hand Inventory
Target maximum
Q1
Q4
Q2
Q3
p
p
p
Time
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Fixed Period Model
When to Order?
Inventory Level
Period
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Target maximum
Period
Period
Time
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