Inventory Management
Learning Objectives
• You should be able to:
1. Define the term inventory, list the major reasons for
holding inventories, and list the main requirements for
effective inventory management
2. Explain periodic and perpetual review systems
3. Explain the objectives of inventory management
4. Describe the A-B-C approach and explain how it is
useful
5. Describe the basic EOQ model and its assumptions and
solve typical problems
6. Describe the economic production quantity model and
solve typical problems
7. Describe the quantity discount model
8. Describe reorder point models
9. Describe situations in which the single-period model
would be appropriate, and solve typical problems
Instructor Slides
2
Inventory
• Inventory
– A stock or store of goods
• Independent demand items
– Items that are ready to be sold or used
• A “typical” firm (on “typical” times) has roughly 30% of its
current assets and as much as 90% of its working capital
invested in inventory
13-3
Types of Inventory
• Raw materials and purchased parts
• Work-in-process (WIP) = partially
completed goods
• Finished goods inventories
(manufacturing) or merchandise (retail)
• Maintenance and repairs (MRO) inventory
• Goods-in-transit to warehouses or
customers (pipeline inventory)
13-4
Inventory
• Inventories are a vital part of business:
– necessary for operations
– contribute to customer satisfaction
• But
– Costly
– May have limited shelf-life
– Carrier may have limited space
13-5
Functions of Inventory
• To meet anticipated demand
• “anticipation stock”
• To smooth production requirements
• seasonal demand
• To permit operations
• Work-In-Process
• To decouple operations
• buffer between successive operations in case of a
breakdown
• To protect against stock-outs
• Delayed deliveries or increase in demand
• To take advantage of order cycles
• minimize purchasing and holding costs or economies of
producing in large quantities
• To help hedge against price increases
• To take advantage of quantity discounts
12-6
Inventory Costs
 Purchase cost
 The amount paid to buy the inventory
 Holding (carrying) costs
 Cost to carry an item in inventory for a length of time,
usually a year (rent, equipment, materials, labor, insurance, security,
interest and other direct expenses).
 Ordering costs
 Costs of ordering and receiving inventory
 Setup costs (Analogous to ordering costs)
 The costs involved in preparing equipment for a job
 Shortage costs
 Costs resulting when demand exceeds the supply of
inventory; often unrealized profit per unit
• Lead time
• Time interval between ordering and receiving the order
13-7
Objectives of Inventory Control
 The overall objective of inventory
management is to achieve:
1. Satisfactory levels of customer service
•
•
Having the right goods available in the right quantity in
the right place at the right time
Focus on size and timing
2. While keeping inventory costs (ordering and
carrying) within reasonable bounds (minimize)
13-8
Objectives of Inventory Control
•
Measures of performance:
1. Customer satisfaction
– Number and quantity of backorders
– Customer complaints
2. Inventory turnover
= (average) cost of goods sold
(average) inventory investment
during a period
13-9
Inventory Management
• Management has two basic functions
concerning inventory:
1. Establish a system for tracking items in
inventory
2. Make decisions about
• When to order
• How much to order
13-10
ABC Classification System
• A-B-C approach
– Classifying inventory according to some measure
of importance, and allocating control efforts
accordingly
– A items (very important)
• 10 to 20 percent of the number of items in inventory
• about 60 to 70 percent of the annual dollar value
– B items (moderately important)
– C items (least important)
• 50 to 60 percent of the number
of items in inventory but
• only about 10 to 15 percent of
the annual dollar value
13-11
ABC Classification System
• How to classify?
1. For each item,
multiply annual
volume by unit
price to get the
annual dollar
value.
2. Arrange annual
values in
descending order.
3. A items: the few
with the highest
annual dollar
value
C items: the most
with the lowest
dollar value.
B items: those in
between
#
Annual
demand
Unit
price
Annual
value
Clss
% of
items
% of
value
8
1,000
4,000
4,000,000
A
5.3
52.7
3
2,400
500
1,200,000
B
6
1,000
1,000
1,000,000
B
31.4
40.8
1
2,500
360
900,000
B
4
1,500
100
150,000
C
10
500
200
100,000
C
9
8,000
10
80,000
C
2
1,000
70
70,000
C
63.3
6.5
5
700
70
49,000
C
7
200
210
42,000
C
100
100
18,800
7,591,000
13-12
Inventory Counting Systems
• Periodic System (e.g. small retailer)
• Physical count of items in inventory made at periodic intervals
– Many items ordered at the same time. Savings in processing and
shipping of orders.
vs.
– Lack of control between reviews. Having to keep extra stock to protects
against shortages.
• Perpetual Inventory System (e.g. bank transactions)
• System that keeps track of removals from inventory continuously,
thus monitoring current levels of each item
• An order is placed when inventory drops to a predetermined minimum
level (can optimize Q)
Vs.
• Added cost of record keeping. Usually has to be accompanied by a
periodic physical count.
• Note: RFID technology is used by inventory counting systems
13-13
Inventory Ordering Models
• Economic Order Quantity models:
– identify the optimal order quantity
• by minimizing total annual costs that vary with order size
and frequency
1. The basic Economic Order Quantity model (EOQ)
2. The Economic Production Quantity model (EPQ)
3. The Quantity Discount model*
• Other Models
1. Reorder Point Ordering* (uncertainty, when to order)
2. Fixed-Order-Interval Model*
3. Single Period model (perishable items)
13-14
Basic Economic Order Quantity Model
• The basic EOQ model:
– used to find a fixed order quantity that will minimize
total annual inventory costs
– Purchase price is not included (cost is unaffected by it)
• Assumptions:
1.
2.
3.
4.
5.
6.
Only one product is involved
Annual demand requirements are 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
13-15
The Inventory Cycle (EOQ)
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place
order
Receive
order
Time
Place
order
Receive
order
Lead time
13-16
Average number of
units in inventory
Total Annual Cost
Total Cost  Annual Holding Cost  Annual Ordering Cost
Q
D

H

S
2
Q
where
Q  Order quantity in units
H  Holding (carrying) cost per unit, usually per year
D  Demand, usually in units per year
S  Ordering cost per order
Number of
orders
13-17
Annual Cost
Goal: Total Cost Minimization
The Total-Cost Curve is U-Shaped
Q
D
TC  H  S
2
Q
Holding Costs
Ordering Costs
QO (optimal order quantity)
Order Quantity
(Q)
13-18
Deriving EOQ
• Using calculus, we take the derivative of the
total cost function and set the derivative
(slope) equal to zero and solve for Q.
H (1) DS
TC '  
0
2
2
Q
DS H

2
Q
2
Length of order cycle = Q/D
Number of orders = D/Q
DS HQ

Q
2
The total cost curve
reaches its
minimum where the
carrying and
ordering costs are
equal.
2 DS
2(annual demand)(or der cost)
QO 

H
annual per unit holding cost
13-19
Example
•
•
•
•
Tire distributer
D (Demand)=9,600 tires per year
H (Holding cost)=$16 per unit per year
S (Ordering cost) = $75 per order
2DS
2 * 9,600 * 75
Q0 

 300 tires
H
16
Q
D
300
9,600
TC  H  S 
16 
75  2,400  2,400  4,800
2
Q
2
300
12-20
Example
•
•
•
•
•
•
Tire distributer
D (Demand)=9,600 tires per year
H (Holding cost)=$16 per unit per year
S (Ordering cost) = $75 per order
Q0=300 tires
TCmin = 4,800
Q
D
250
9,600
TC 250  H  S 
16 
75  2,000  2,880  4,880
2
Q
2
250
Q
D
400
9,600
TC 400  H  S 
16 
75  3,200  1,800  5,000
2
Q
2
400
• TC curve relatively flat at optimum
12-21
Economic Production Quantity (EPQ)
• The batch mode is widely used in production. In
certain instances, the capacity to produce a part
exceeds its usage (demand rate)
– Assumptions
1. Only one item is involved
2. Annual demand requirements are known
3. Usage rate is constant
4. Usage occurs continually, but production occurs periodically
5. The production rate is constant
6. Lead time does not vary
7. There are no quantity discounts
Instructor Slides
22
EPQ: Inventory Profile
Q
Production
and usage
Usage
only
Production
and usage
Usage
only
Production
and usage
Qp
Cumulative
production
Imax
Amount
on hand
Time
Instructor Slides
23
EPQ – Total Cost
TC min  Carrying Cost  Setup Cost
D
I 
  max  H  S
Q
 2 
where
I max  Maximum inventory

Qp
 p  u
p
p  Production or delivery rate
u  Usage rate
Solution:
Instructor Slides
Qp 
2 DS
H
p
p u
24
Example
• Toy manufacturer makes rubber wheels for dump truck
toys.
• D=48,000 wheels per year
• S=$45
• H=$1 per wheel per year
• P=800 wheels per day
• u=200 wheels per day
Qp 
2 DS
H
Qp 
2( 48000) 45
1
Instructor Slides
p
p u
800
 2,400 wheels
800  200
25
Quantity Discount Model
• Take into account quantity discount offered by
supplier (add purchase cost to model)
• Quantity discount
– Price reduction for larger orders offered to customers to
induce them to buy in large quantities
Total Cost  Carrying Cost  Ordering Cost  Purchasing Cost
Q
D
 H  S  PD
2
Q
where
P  Unit price
Instructor Slides
26
Quantity Discounts*
Adding PD does not change EOQ
Instructor Slides
The total-cost curve with quantity
discounts is composed of a portion
of the total-cost curve for each price
27
Reorder Point Ordering
• Reorder-Point
– When the quantity on hand of an item drops to this
amount (quantity-trigger), the item is reordered.
– Determinants of the Reorder-Point
1. The rate of demand
2. The lead time
3. The extent of demand and/or lead time variability
4. The degree of stockout risk acceptable to
management
13-28
Safety Stock*
Instructor Slides
29
Fixed-order-interval (FOI) model
• Fixed-order-interval (FOI) model
– Orders are placed at fixed time intervals
• Reasons for using the FOI model
– Supplier’s policy may encourage its use
– Grouping orders from the same supplier can produce
savings in shipping costs
– Some circumstances do not lend themselves to
continuously monitoring inventory position
Instructor Slides
30
Fixed-Quantity (ROP) vs.
Fixed-Interval Ordering*
Fixed Quantity
Fixed Interval
Instructor Slides
31
Single-Period Model
• Single-period model
– Model for ordering of perishables and other items with
limited useful lives
• Period = life of the good.
• Items are not carried over to the next period
• The goal of the single-period model is to identify the order
quantity that will minimize the long-run excess and
shortage costs
• Two categories of problems:
1. Demand can be characterized by a continuous
distribution
2. Demand can be characterized by a discrete distribution
13-32
Single-Period Model
• Shortage cost
– Generally, the unrealized profit per unit
– Cshortage = Cs = Revenue per unit – Cost per unit
• Excess cost
– Different between purchase cost and salvage
value of items left over at the end of the period
– Cexcess = Ce = Cost per unit – Salvage value per
unit
13-33
Continuous Stocking Levels
Uniform Demand
• Service Level =
probability that demand
will not exceed the
stocking level (S).
• Shortage: Demand > S0
• Excess:
Demand < S0
Cs
Service level 
C s  Ce
where
Cs  shortage cost per unit
Ce  excess cost per unit
Ce
Cs
Like a Seesaw, leverage
If demand is not uniformly
distributed, consider cumulative
probability of demand
Service level
Quantity
So
Balance Point
So=min+SL*(max-min)
So =Optimum
Stocking Quantity
13-34
Continuous Stocking Levels
Uniform Demand
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Sweet cider delivered to bar
Demand uniformly distributed
between 300 and 500 liters/week.
Cost=20 cent/liter
Revenue = 80 cent/liter.
Ce=
cost-salvage =.2 $/liter
Cs=
revenue-cost =.8-.2=.6 $/liter
SL=
Cs/(Cs+Ce) =.6/(.6+.2)=.75
So=
min+SL*(max-min)
=300+.75*(500-300)=450 liter
Stockout risk =
1-SL = 1- .75=.25
Cs
Service level 
C s  Ce
where
Cs  shortage cost per unit
Ce  excess cost per unit
Ce=.2 $/liter
Cs=.6 $/liter
Service level=75%
500
300
So =450
Balance Point
13-35
Discrete Stocking Levels
•
Cs/(Cs+Ce) may not coincide with a feasible
stocking level.
•
Solution: Stock at the next higher level so that
the desired service level is equaled or
exceeded.
Cs
Service level 
C s  Ce
where
Cs  shortage cost per unit
Ce  excess cost per unit
13-36
Discrete Stocking Levels
•
•
•
•
Discrete
p = $5.5
v= $0
c= $2
• SL= Cs/(Cs+Ce)=
(p-c)/[(p-c)+(c-v)] =
(5.5-2)/[(5.5-2)+(2-0)] =
3.5/(3.5+2)= .636
• Q=13
Q
P(R≤Q)
Expected profit
1
0.05
3.5
2
0.1
6.725
3
0.15
9.675
4
0.2
12.35
5
0.25
14.75
6
0.3
16.875
7
0.35
18.725
8
0.4
20.3
9
0.45
21.6
10
0.5
22.625
11
0.55
23.375
12
0.6
23.85
13
0.65
24.05
14
0.7
23.975
15
0.75
23.625
16
0.8
23
17
0.85
22.1
18
0.9
20.925
19
0.95
19.475
20
1
17.75
37
Discrete Stocking Levels
• Shortage (downtime) cost =
$4,200
• Excess (spare part) cost =
$200
• SL =
= 4200/(4200+200) = .84
• Next level - > 2 spare parts
90% will not run out of spare
parts
Spares
used
Relative
frequency
Cumulative
frequency
0
.2
.2
1
.4
.6
2
.3
.9
3
.1
1
4 or
more
0
1
13-38
Operations Strategy*
• Too much inventory:
– Costly to maintain
– Tends to hide problems (easier to live with
problems than to eliminate them)
• Wise strategy = find ways to
– Improve demand forecasts = reduce safety
stock
– Improve inventory management
– Reduce Ordering costs
– Reduce inventory Holding costs
– Reduce variation (e.g., lead-time)
– Lean Operations
– Supply-Chain Management
12-39