Inventory
Management
Learning Objectives
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Define the term inventory and list the major
reasons for holding inventories; and list the main
requirements for effective inventory management.
Discuss the nature and importance of service
inventories
Discuss periodic and perpetual review systems.
Discuss the objectives of inventory management.
Describe the A-B-C approach and explain how it
is useful.
Learning Objectives
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Describe the basic EOQ model and its
assumptions and solve typical problems.
Describe the economic production quantity
model and solve typical problems.
Describe the quantity discount model and
solve typical problems.
Describe reorder point models and solve
typical problems.
Describe situations in which the singleperiod model would be appropriate, and
solve typical problems.
Inventory
Inventory: a stock or store of goods
Dependent Demand
A
C(2)
B(4)
D(2)
Independent Demand
E(1)
D(3)
F(2)
Independent demand is uncertain.
Dependent demand is certain.
Inventory Models
 Independent demand – finished goods, items
that are ready to be sold
 E.g. a computer
 Dependent demand – components of
finished products
 E.g. parts that make up the computer
Types of Inventories
 Raw materials & purchased parts
 Partially completed goods called
work in progress
 Finished-goods inventories

(manufacturing firms)
or merchandise
(retail stores)
Types of Inventories (Cont’d)
 Replacement parts, tools, & supplies
 Goods-in-transit to warehouses or
customers
Functions of Inventory
 To meet anticipated demand
 To smooth production requirements
 To decouple operations
 To protect against stock-outs
Functions of Inventory (Cont’d)
 To take advantage of order cycles
 To help hedge against price increases
 To permit operations
 To take advantage of quantity
discounts
Objective of Inventory Control
 To achieve satisfactory levels of
customer service while keeping
inventory costs within reasonable
bounds
 Level of customer service
 Costs of ordering and carrying inventory
Inventory turnover is the ratio of
average annual cost of goods sold to
average inventory investment.
Effective Inventory Management
 A system to keep track of inventory
 A reliable forecast of demand
 Knowledge of lead times
 Reasonable estimates of
 Holding costs
 Ordering costs
 Shortage costs
 A classification system
Inventory Counting Systems
 Periodic System
Physical count of items made at periodic
intervals
 Perpetual Inventory System
System that keeps track
of removals from inventory
continuously, thus
monitoring
current levels of
each item
Perpetual Inventory System
 Two-Bin System - Two containers of
inventory; reorder when the first is
empty
 Universal Bar Code - Bar code
printed on a label that has
information about the item
to which it is attached
0
214800 232087768
Key Inventory Terms
 Lead time: time interval between
ordering and receiving the order
 Holding (carrying) costs: cost to carry
an item in inventory for a length of time,
usually a year
 Ordering costs: costs of ordering and
receiving inventory
 Shortage costs: costs when demand
exceeds supply
ABC Classification System
Classifying inventory according to some
measure of importance and allocating control
efforts accordingly.
A – 10-20% items, 60-70% value
B – 30-40% items, 20-30% value
C – 50-60% items,
High
A
10-15% value
Annual
Value
of items
B
C
Low
Low
High
Percentage of Items
Cycle Counting
 A physical count of items in inventory
 Cycle counting management
 How much accuracy is needed?
 When should cycle counting be performed?
 Who should do it?
Cycle Counting Example
5,000 items in inventory, 500 A items, 1,750 B items,
2,750 C items
Policy is to count
A items every month (20 working days),
B items every quarter (60 days), and
C items every six months (120 days)
Item
Class
Quantity
A
500
Each month
B
1,750
Each quarter
C
2,750
Every 6 months
Cycle Counting Policy
Number of Items
Counted per Day
500/20 = 25/day
1,750/60 = 29/day
2,750/120 = 23/day
77/day
Economic Order Quantity Models
 Economic order quantity (EOQ) model
 The order size that minimizes total annual
cost
 Economic production model
 Quantity discount model
Assumptions of EOQ Model
 Only one product is involved
 Annual demand requirements known
 Demand is even throughout the year
 Lead time does not vary
 Each order is received in a single
delivery
 There are no quantity discounts
The Inventory Cycle
Profile of Inventory Level Over Time
Q
Quantity
on hand
Usage
rate
Reorder
point
Receive
order
Place Receive
order order
Lead time
Place Receive
order order
Time
Total Cost
Annual
Annual
Total cost = carrying + ordering
cost
cost
TC =
Q
H
2
+
DS
Q
Q = Order quantity
H = Holding cost
D = Annual demand
S = Ordering cost or Setup cost
Cost Minimization Goal
Annual Cost
The Total-Cost Curve is U-Shaped
Q
D
TC  H  S
2
Q
Ordering Costs
QO (optimal order quantity)
Order Quantity
(Q)
Deriving the EOQ
Using calculus, we take the derivative of
the total cost function and set the
derivative (slope) equal to zero and solve
for Q.
Q OPT =
2DS
=
H
2( Annual Demand )(Order or Setup Cost )
Annual Holding Cost
Minimum Total Cost
The total cost curve reaches its
minimum where the carrying and
ordering costs are equal.
Q
H
2
=
DS
Q
Reorder Points
 EOQ answers the “how much” question
 The reorder point (ROP) tells when to
order
ROP =
Lead time for a
Demand
per day new order in days
=dxL
D
d = Number of working days in a year
Inventory level (units)
Reorder Point Curve
Q*
Slope = units/day = d
ROP
(units)
Time (days)
Lead time = L
Economic Production Quantity (EPQ)
 Production done in batches or lots
 Capacity to produce a part exceeds the
part’s usage or demand rate
 Assumptions of EPQ are similar to EOQ
except orders are received
incrementally during production
Economic Production Quantity
Assumptions
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Only one item is involved
Annual demand is known
Usage rate is constant
Usage occurs continually
Production rate is constant
Lead time does not vary
No quantity discounts
Inventory level
Production Order Quantity
Model
Part of inventory cycle
during which production
(and usage) is taking place
Demand part of
cycle with no
production
Maximum
inventory
t
Time
Economic Run Size
Q0 
2DS
p
H p u
Q = Order quantity
H = Holding cost
D = Annual demand
S = Ordering cost or Setup cost
p = Production rate or Delivery rate
u = Usage rate
Quantity discount model
Annual
Annual
Purchasing
+
TC = carrying + ordering cost
cost
cost
Q
H
TC =
2
+
DS
Q
+
PD
Quantity discount model
Cost
TC1
TC2
TC3
Unit price 1
Unit price 2
Unit price 3
Quantity
When to Reorder with EOQ
Ordering
 Reorder Point - When the quantity on
hand of an item drops to this amount,
the item is reordered
 Safety Stock - 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.
Determinants of the Reorder
Point
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The rate of demand
The lead time
Demand and/or lead time variability
Stockout risk (safety stock)
Stock-out Cost can be
determined
 Used when demand is not constant or
certain
 Use safety stock to achieve a desired
service level and avoid stockouts
ROP = d x L + ss
Annual stockout costs = the sum of the
{units short x the probability
x the stockout cost/unit
x the number of orders per year}
Safety Stock Example
ROP = 50 units
Orders per year = 6
Stockout cost = $40 per units
Carrying cost = $5 per units per year
30
40
50
Probability of
demand level
.2
.2
.3
60
.2
70
.1
1.0
Demand (Units)
ROP 
Safety Stock Example
ROP = 50 units
Orders per year = 6
Safety
Stock
Additional
Holding Cost
20
(20)($5) = $100
10
0
Stockout cost = $40 per frame
Carrying cost = $5 per frame per year
Total
Cost
Stockout Cost
$0
$100
= $240
$290
$0 (10)(.2)($40)(6) + (20)(.1)($40)(6) = $960
$960
(10)($5) = $50 (10)(.1)($40)(6)
A safety stock of 20 frames gives the lowest total cost
ROP = 50 + 20 = 70 frames
Inventory level
Probabilistic Demand
Minimum demand during lead time
Maximum demand during lead time
Mean demand during lead time
ROP
ROP 
Normal distribution probability of
demand during lead time
Safety stock
0
Lead
time
Time
Risk
Place
order
Receive
order
Expected
demand
during
lead time
Probabilistic Demand
Risk of a stockout
(5% of area of
normal curve)
Probability of
no stockout
95% of the time
Mean
demand
ROP = ? units
Quantity
Safety
stock
0
z
Number of
standard deviations
Z=1.65
Probabilistic Demand
Use prescribed service levels to set safety
stock when the cost of stockouts cannot be
determined
ROP = demand during lead time + Zsdlt
where
Z = number of standard deviations
sdlt = standard deviation of demand
during lead time
Other Probabilistic Models
When data on demand during lead time is
not available, there are other models
available
1. When demand is variable and lead
time is constant
2. When lead time is variable and
demand is constant
3. When both demand and lead time
are variable
Other Probabilistic Models
Demand is variable and lead time is constant
ROP = (average daily demand
x lead time in days) + Zsdlt
where
sd = standard deviation of demand per day
sdlt = sd
lead time
Probabilistic Example
Average daily demand (normally distributed) = 15
Standard deviation = 5
Lead time is constant at 2 days
90% service level desired
Z for 90% = 1.28
ROP = (15 units x 2 days) + Zsdlt
= 30 + 1.28(5)( 2)
= 30 + 8.96 = 38.96 ≈ 39
Safety stock is about 9 units
Other Probabilistic Models
Lead time is variable and demand is constant
ROP = (daily demand x average lead time)
+ Zsdlt
Zsdlt = Z x (daily demand) x σlt
where
slt = standard deviation of lead time in days
Probabilistic Example
Daily demand (constant) = 10
Average lead time = 6 days
Standard deviation of lead time = slt = 3
98% service level desired
Z for 98% = 2.055
ROP = (10 units x 6 days) + 2.055(10 units)(3)
= 60 + 61.55 = 121.65
Reorder point is about 122 units
Other Probabilistic Models
Both demand and lead time are variable
ROP = (average daily demand
x average lead time) + Zsdlt
where
sd = standard deviation of demand per day
slt = standard deviation of lead time in days
sdlt = (average lead time x sd2)
+ (average daily demand) 2slt2
Probabilistic Example
Average daily demand (normally distributed) = 150
Standard deviation = sd = 16
Average lead time 5 days (normally distributed)
Standard deviation = slt = 1 days
95% service level desired
Z for 95% = 1.65
ROP = (150 packs x 5 days) + 1.65sdlt
= (150 x 5) + 1.65 (5 days x 162) + (1502 x 12)
= 750 + 1.65(154) = 1,004 packs
Fixed-Order-Interval Model
 Orders are placed at fixed time intervals
 Order quantity for next interval?
 Suppliers might encourage fixed
intervals
 May require only periodic checks of
inventory levels
 Risk of stockout
Fixed-Order-Interval Model
On-hand inventory
Target maximum (T)
Q4
Q2
P
Q1
Q3
P
P
Time
Fixed-Interval Benefits
 Tight control of inventory items
 Items from same supplier may yield
savings in:
 Ordering
 Packing
 Shipping costs
 May be practical when inventories
cannot be closely monitored
Fixed-Interval Disadvantages
 Requires a larger safety stock
 Increases carrying cost
 Costs of periodic reviews
Single Period Model
 Single period model: model for ordering of
perishables and other items with limited useful
lives
 Shortage cost: generally the unrealized profits
per unit
= revenue per unit – Cost per unit
 Excess cost: difference between purchase cost
and salvage value of items left over at the end
of a period
= Original cost per unit – Salvage value per unit
Optimal Stocking Level
Service level =
Cs
Cs + Ce
Cs = Shortage cost per unit
Ce = Excess cost per unit
Ce
Cs
Service Level
Quantity
So
Balance point
Example 15
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Ce = $0.20 per unit
Cs = $0.60 per unit
Service level = Cs/(Cs+Ce) = .6/(.6+.2)
Service level = .75
Ce
Cs
Service Level = 75%
Quantity
Stockout risk = 1.00 – 0.75 = 0.25
Supermarket
 Items used by many products are held
in a common area often called a
supermarket
 Items are withdrawn as needed
 Inventory is maintained using JIT
systems and procedures
 Common items are not planned by the
MRP system
Operations Strategy
 Too much inventory
 Tends to hide problems
 Easier to live with problems than to
eliminate them
 Costly to maintain
 Wise strategy
 Reduce lot sizes
 Reduce safety stock