Inventory Management
Chapter 13
Learning Objectives
•
•
•
•
•
•
•
•
•
Define the term inventory
List the different types of inventory
Describe the main functions of inventory
Discuss the main requirements for effective inventory
management
Describe the A-B-C approach and explain how it is useful
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.
Inventory Management at Cox
Hardware
• From the video, what are the elements relevant to inventory management?
Inventory
• Definition
– A stock or store of goods
– Dependent demand items
• Items that are subassemblies or component parts to be
used in the production of finished goods.
– Independent demand items
• Items that are ready to be sold or used
• Inventories are a vital part of business:
– necessary for operations
– contribute to customer satisfaction
Types of Inventory
•
•
•
•
•
•
Raw materials and purchased parts
Work-in-process (WIP)
Finished goods inventories or merchandise
Tools and supplies
Maintenance and repairs (MRO) inventory
Goods-in-transit to warehouses or customers
(pipeline inventory)
Discussion
• Which of them are dependent demand items
and independent demand items?
– Raw materials and purchased parts
– Work-in-process (WIP)
– Finished goods inventories or merchandise
– Tools and supplies
– Maintenance and repairs (MRO) inventory
– Goods-in-transit to warehouses or customers
(pipeline inventory)
Inventory Functions
• Inventories serve a number of functions such as:
1. To meet anticipated customer demand
2. To smooth production requirements
•
Firms that experience seasonal patterns in demand often
build up inventories during preseason periods to meet
overly high requirements during seasonal periods.
3. To decouple operations
•
Buffers between operations
4. To protect against stockouts
•
Delayed deliveries and unexpected increases in demand
increase the risk of shortages.
Inventory Functions
• Inventories serve a number of functions such as:
5. To take advantage of order cycles
•
It is usually economical to produce in large rather than small
quantities. The excess output must be stored for later use.
6. To hedge against price increases
•
Occasionally a firm will suspect that a substantial price increase is
about to occur and purchase larger-than-normal amounts to beat
the increase
7. To permit operations
•
The fact that production operations take a certain amount of
time (i.e., they are not instantaneous) means that there will
generally be some work-in-process inventory.
8. To take advantage of quantity discounts
•
Suppliers may give discounts on large orders.
Objectives of Inventory Control
• Inventory management has two main
concerns:
1. Satisfactory Level of customer service
• Having the right goods available in the right quantity in
the right place at the right time
2. Costs of ordering and carrying inventories
• The overall objective of inventory management is to
achieve satisfactory levels of customer service while
keeping inventory costs within reasonable bounds
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
MIS 373: Basic Operations Management
10
Effective Inventory Management
• Requires:
1.
2.
3.
4.
A system keep track of inventory
A reliable forecast of demand
Knowledge of lead time and lead time variability
Reasonable estimates of
• holding costs
• ordering costs
• shortage costs
5. A classification system for inventory items
MIS 373: Basic Operations Management
11
Measures of performance
• Measures of performance:
– Customer satisfaction
• Number and quantity of backorders
• Customer complaints
– Inventory turnover = a ratio of
(average) cost of goods sold
(average) inventory investment
during a period
MIS 373: Basic Operations Management
12
Inventory Counting Systems
• Periodic System
• Physical count of items in inventory made at periodic intervals
• Perpetual Inventory System
• System that keeps track of removals from inventory
continuously, thus monitoring current levels of each item
– Point-of-sale (POS) systems
• A system that electronically records actual sales
– Radio Frequency Identification (RFID)
MIS 373: Basic Operations Management
13
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
• Interest, insurance, taxes (in some states), depreciation, obsolescence,
deterioration, spoilage, pilferage, breakage, tracking, picking, and
warehousing costs (heat, light, rent, workers, equipment, security).
• Ordering costs
– Costs of ordering and receiving inventory
• determining how much is needed, preparing invoices, inspecting goods
upon arrival for quality and quantity, and moving the goods to temporary
storage.
• Shortage costs
– Costs resulting when demand exceeds the supply of inventory; often
unrealized profit per unit
MIS 373: Basic Operations Management
14
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 and 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
MIS 373: Basic Operations Management
15
ABC Classification System
• How to classify?
1.
2.
3.
#
Annual
demand
Unit
price
8
1,000
4,000
4,000,000
A
3
2,400
500
1,200,000
B
6
1,000
1,000
1,000,000
B
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
C items: the most with
the lowest dollar value.
2
1,000
70
70,000
C
5
700
70
49,000
C
B items: those in
between
7
200
210
42,000
C
For each item, multiply
annual volume by unit
price to get the annual
dollar value.
Arrange annual values in
descending order.
A items: the few with the
highest annual dollar
value
18,800
MIS 373: Basic Operations Management
Annual value Class
7,591,000
% of
items
% of
value
5.3
52.7
31.4
40.8
63.3
6.5
100
100
16
ABC Classification: Cycle Counting
• Cycle counting
– A physical count of items in inventory
• Cycle counting management
– How much accuracy is needed?
• A items: ± 0.2 percent
• B items: ± 1 percent
• C items: ± 5 percent
– When should cycle counting be performed?
– Who should do it?
MIS 373: Basic Operations Management
17
Inventory Models
• Economic Order Quantity models:
– identify the optimal order quantity
• by minimizing total annual costs that vary with order size and
frequency
1.
2.
3.
4.
5.
6.
The basic Economic Order Quantity model (EOQ)
The Quantity Discount model
The Economic Production Quantity model (EPQ)
Reorder Point Ordering (uncertainty, when to order)
Fixed-Order-Interval model
Single Period model (perishable items)
Basic EOQ Model
• The basic EOQ model:
– used to find a fixed order quantity that will minimize total
annual inventory costs
– continuous monitoring system
• Assumptions:
1. Only one product is involved
2. Annual demand requirements are known
3. Demand is even throughout the year
4. Lead time does not vary
5. Each order is received in a single delivery
6. There are no quantity discounts
MIS 373: Basic Operations Management
19
The Inventory Cycle
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place
order
Receive
order
Place
order
Receive
order
Time
Lead time
MIS 373: Basic Operations Management
20
Discussion
• What costs are associated with this model?
– (Hint: Recall the 5 types of costs)





Purchase cost
Holding (carrying) cost
Ordering cost
Setup cost
Shortage cost (Backorder/Backlog cost)
Instructor Slides
21
Total Annual Cost
• Total Cost = Annual Holding Cost + Annual Ordering
Cost
Q
D
(TC) =
H
+
S
2
Q
Average number of
units in inventory
Number of
orders
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
MIS 373: Basic Operations Management
22
Goal: Total Cost Minimization
• The Total-Cost Curve is U-Shaped
Annual Cost
– There is a tradeoff between holding costs and
ordering costs
TC 
Q
D
H S
2
Q
Holding Costs
Ordering Costs
Q* (optimal order quantity)
MIS 373: Basic Operations Management
Order Quantity
(Q)
23
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.
• The total cost curve reaches its minimum where
the carrying and ordering costs are equal.
𝐻
−1 𝐷𝑆
𝑇𝐶 = +
=0
2
2
𝑄
′
𝑄∗
=
2𝐷𝑆
=
𝐻
→
𝐷𝑆 𝐻
=
2
𝑄
2
→
𝐷𝑆 𝐻𝑄
=
𝑄
2
2 𝑎𝑛𝑛𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑜𝑟𝑑𝑒𝑟 𝑐𝑜𝑠𝑡
𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡
MIS 373: Basic Operations Management
24
Example: Deriving EOQ
•
•
•
•
Tire distributer
D (Demand)=9,600 tires per year
H (Holding cost)=$16 per unit per year
S (Ordering cost) = $75 per order
𝑄∗
=
2𝐷𝑆
=
𝐻
2 ∗ 9,600 ∗ 75
= 300 𝑡𝑖𝑟𝑒𝑠
16
𝑄
𝐷
300
9,600
𝑇𝐶 = 𝐻 + 𝑆 =
∗ 16 +
∗ 75 = 2,400 + 2,400 = 4,800
2
𝑄
2
300
MIS 373: Basic Operations Management
25
Example: Deriving EOQ
•
•
•
•
•
D (Demand)=9,600 tires per year
H (Holding cost)=$16 per unit per year
S (Ordering cost) = $75 per order
Q*=300 tires
TCmin = 4,800
Let’s verify whether the Q*
indeed gives the lowest cost.
Q
D
250
9,600
TC 250  H  S 
16 
75  2,000  2,880  4,880
2
Q
2
250
TC 400 
Q
D
400
9,600
H S
16 
75  3,200  1,800  5,000
2
Q
2
400
MIS 373: Basic Operations Management
26
Exercise
• A local distributor for a national tire company
expects to sell approximately 9,600 tires per year.
Annual carrying cost is $16 per tire, and ordering
cost is $75 per order. The distributor operates
288 working days a year.
• a. What is the optimal order size?
• b. How many times per year the store reorder?
• c. What is the length of an order cycle?
• d. What is the total annual cost if EOQ is placed?
Instructor Slides
27
Solution
• a
2 DS
2  (9,600)  75
EOQ 

 300(tires )
H
16
• b
D
9,600

 32(times)
EOQ
300
• c
• d
# of working days 288
Length 

9
# of reorder times
32
Total Cost  (Q / 2) H  ( D / Q) S
（300/ 2）
16  (9,600 / 300)75
 $4800 Instructor Slides
28
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.
2.
3.
4.
5.
6.
7.
8.
Only one item is involved
Annual demand requirements are known
Usage rate is constant (= even demand throughout the year)
Usage occurs continually, but production occurs periodically (batch
production)
The production rate is constant
Production rate is greater than usage rater
Lead time does not vary
There are no quantity discounts
Instructor Slides
13-29
EPQ: Inventory Cycle
Q
Production
and usage
Usage
only
Production
and usage
Usage
only
Production
and usage
Qp
Cumulative
production
Imax
Amount
on hand
Time
Instructor Slides
13-30
Discussion
• What costs are associated with this model?
– (Hint: Recall the 5 types of costs)





Purchase cost
Holding (carrying) cost
Ordering cost
Setup cost
Shortage cost (Backorder/Backlog cost)
• Our decision making in EPQ model is to
determine the size of order (Q) each time.
Why?
Instructor Slides
31
EPQ Model
Q  Batch size for production
I  Maximum Inventory Level
H  Holding (carrying) cost per unit per year
D  Annual Demand
S  Setup cost per order
p  production rate (unit per day)
u  usage rate (unit per day)
Instructor Slides
32
EPQ – Total Cost
TC min  Carrying Cost  Setup Cost
I
  max
 2
D

H

S

Q

where
I max  Maximum inventory

Qp
 p  u
p
p  Production or delivery rate
u  Usage rate
Instructor Slides
13-33
EPQ
2 DS
Qp 
H
p
p u
Optimal Batch size a.k.a Economic Produciton Quantity (EPQ)
Instructor Slides
13-34
Exercise
• A toy manufacturer uses 48,000 wheels per year for
toy trucks. The firm produces its own wheels, which it
can produce at a rate of 800 per day. The toy trucks are
assembled uniformly over the entire year. Carrying cost
is $1 per wheel a year. Setup cost is $45 per production
run. The firm operates 240 days per year.
• a. What is the optimal batch size?
• b. What is the total annual cost if EPQ is produced each
run?
• c. What is cycle time (production stage + usage stage)?
• d. What is the production cycle time?
Instructor Slides
35
Solution
u  48,000 / 240  200 wheels per day
• a
• b
EPQ 
2 DS
H
p
2  (48,000)  45
800

 2,400( wheels )
p u
1
800  200
EPQ
2,400
I max 
( p  u) 
(800  200)  1,800( wheels )
p
800
Total Cost  ( I max / 2) H  ( D / Q) S
（1,800/ 2）
1  (48,000 / 2,400)45
 $1,800
Instructor Slides
36
Solution
• c
EPQ 2,400
CycleTime 

 12(days)
u
200
• d
EPQ 2,400
Pr oductionCycle 

 3(days)
p
800
Instructor Slides
37
Quantity Discount 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
13-38
When to Reorder
• If delivery is not instantaneous, but there is a
lead time:
Order to order?
When
Inventory
Quantity
Q
Lead Time
Place
Receive
MIS 373:
Basic Operations Management
order
order
Time
39
When to Reorder
• EOQ answers the “how much” question
• The reorder-point (ROP) tells “when” to order
• 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
MIS 373: Basic Operations Management
40
Reorder-Point
ROP
= (Demand per day) * (Lead time for a new order in days)
=
d * L
where
d = (Demand per year) / (Number of working days in a year)
Example:
– Demand = 12,000 iPads per year
– 300 working day year
– Lead time for orders is 3 working days
In other words, the
manager should place the
order when only 120 units
left in the inventory.
d = 12,000 / 300 = 40 units
ROP = d * L = 40 units per day * 3 days of leading time = 120 units
MIS 373: Basic Operations Management
41
The Inventory Cycle
Q: When shall we order?
A: When inventory = ROP
Q: How much shall we order?
A: Q = EOQ
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
ROP = LxD
Receive
order
D: demand per period
L: Lead time in periods
Place
order
Receive
order
Place
order
Receive
order
Time
Lead time
MIS 373: Basic Operations Management
42
Exercise: EOQ & ROP
• Assume a car dealer that faces demand for 5,000 cars per year, and that it costs
$15,000 to have the cars shipped to the dealership. Holding cost is estimated at
$500 per car per year. How many times should the dealer order, and what
should be the order size? (Assuming that the lead time to receive cars is 10
days and that there are 365 working days in a year)
Recall:
EOQ
ROP
=
∗
𝑄 =
2𝐷𝑆
=
𝐻
2 𝑎𝑛𝑛𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑜𝑟𝑑𝑒𝑟 𝑐𝑜𝑠𝑡
𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡
= (Demand per day) * (Lead time for a new order in days)
=
d * L
where
d = (Demand per year) / (Number of working days in a year)
Exercise: EOQ & ROP
• Assume a car dealer that faces demand for 5,000 cars per year, and that it costs
$15,000 to have the cars shipped to the dealership. Holding cost is estimated at
$500 per car per year. How many times should the dealer order, and what
should be the order size?
2(15,000)(5,000)
Q 
 548
500
*
Since d is given in years, first convert: 5000/365 =13.7 cars per working day
So, ROP = 13.7 * 10 = 137
So, when the number of cars on the lot
reaches 137, order 548 more cars.
But demand is rarely predictable!
Inventory
Level
When Actual Demand < Expected Demand
Order
Quantity
Lead Time Demand
ROP
Inventory at time of receipt
Time
Place
order
Lead Time
Receive
order
MIS 373: Basic Operations Management
45
But demand is rarely predictable!
Inventory
Level
When Actual Demand > Expected Demand
Order
Quantity
Stockout Point
ROP
Unfilled demand
Place
order
Time
Lead Time
Receive
MIS 373: Basic Operations Management
order
46
But demand is rarely predictable!
Inventory
Level
Order
Quantity
If ROP = expected demand,
inventory left 50% of the time, stock outs 50% of the time.
ROP = Expected Demand
Average
Uncertain Demand
Time
MIS 373: Basic Operations Management
47
Safety Stock
Inventory
Level
To reduce stockouts we add safety stock
Order
Quantity
ROP =
Safety
Stock +
Expected
LT
Demand
Order Quantity
Q = EOQ
Expected
LT Demand
Safety Stock
Place
order
Lead Time
Receive
MIS 373: Basic Operations Management
order
Time
48
How Much Safety Stock?
• The amount of safety stock that is appropriate for a given
situation depends upon:
1. The average demand rate and average lead time
2. Demand and lead time variability
3. The desired service level
•
Service level = probability of NOT stocking out
• Reorder point (ROP) =
Expected demand during lead time + z*σdLT
Where
z = number of standard deviations
σdLT = the standard deviation of lead time demand
MIS 373: Basic Operations Management
49
Reorder Point
• The ROP based on a normal distribution of
lead time demand
Service
level
Risk of stockout
(probability of
not stockout)
Safety
Stock
Expected
ROP
demand
MIS 373: Basic0
Operations Management
Z
Quantity
Z scale
50
Recap