Operations
Management
Inventory Management
Chapter 12 - Part I
12-1
Outline
 Functions of Inventory.
 ABC Analysis.
 Inventory Costs .
 Inventory Models for Independent Demand.

Economic Order Quantity (EOQ) Model.

Production Order Quantity (POQ) Model.

Quantity Discount Model.
 Probabilistic Models with Varying Demand.
 Fixed Period Systems.
12-2
Types of Inventory
Raw materials.
Work-in-progress.
Maintenance/repair/operating (MRO)
supply.
Finished goods.
12-3
The Functions of Inventory
 To ”decouple” or separate various parts of the
production process.
 To smooth production (link supply and demand).
 To provide goods for customers (quick response).
 To take advantage of quantity discounts.

Buy more to get a reduced price.
 To hedge against inflation and upward price
changes (speculation).

Buy more now if you think price will rise.
12-4
The Material Flow Cycle
12-5
Disadvantages of Inventory
 High cost - $$$$$

Money tied up in inventory could be better used
elsewhere in the organization.
 Difficult to control.

Inventories occur in many places.
 Hides production problems.

Large inventories may overcome poor quality
production or poor quality materials.
12-6
ABC Analysis
 Divide inventory into 3 classes based on annual
$ volume.

Annual $ volume = Annual demand x Unit cost.
A class - Most important.
15-20% of products.
60-80% of value.
B class -Less important.
20-40% of products.
15-30% of value.
C class - Least important.
50-60% of products.
5-10% of value.
12-7
ABC Analysis
 Sort products from largest to smallest annual $
volume.
 Divide into A, B and C classes.
 Focus on A products.

Develop class A suppliers more.

Give tighter physical control of A items.

Forecast A items more carefully.

Consider B products only after A products.
12-8
Classifying Items as ABC
Annual $ Usage (x1000)
25 products sorted by Annual $ Volume (Sales)
100
80
60
40
20
0
1
5
10
20
15
Product
12-9
25
Product
1
2
3
4
5
6
7
8
9
10
11
12
13
14-25
Total
Sales
100
92
88
60
58
53
49
41
32
26
21
18
16
66
720
%
14
13
12
8
8
7
7
6
4
4
3
2
2
9
Annual $ Usage (x1000)
Classifying Items as ABC
Class
A
B
C
100
80
60
A
40
20
0
% $ Vol
% Items
39%
12% (3/25)
52%
40% (10/25)
9%
48% (12/25)
B
0
20
C
40
80
60
% of Products
12-10
100
Inventory Accuracy
 Inventory accuracy importance:

To determine when and how much to order.

To achieve high level of service.
 Information system tracks inventory, but…

Not all items sold are entered (scanned) properly.

Some items disappear without being sold (theft,
defective, damaged, etc.)
12-11
Inventory Counting
 Count products to verify inventory records.
 Shut down facility and count everything at one
time (once per year).
 Cycle counting: count items continuously
(count some each week).

Count A items most frequently (for example,
once a month).

Count B items less frequently (twice a year).

Count C items least frequently (once a year).
12-12
Inventory for Services
 Can be large $.
 “Shrinkage” (theft) is a problem.

Often over 3%!
 Good personnel selection, training, and discipline is
key.
 Establish tight control of shipments entering and
leaving the facility.

Enforce procedures for documenting product movement.
 Information systems can monitor inventory levels and
help ensure accuracy.
12-13
Inventory Costs
Holding costs - Associated with holding or
“carrying” inventory over time.
Ordering costs - Associated with costs of
placing order and receiving goods.
Setup costs - Cost to prepare a machine or
process for manufacturing an order.
Stockout costs - Cost of not making a sale
and lost future sales.
12-14
Holding Costs
Investment costs (borrowing, interest).
Insurance.
Taxes.
Storage and handling.
Extra staffing.
Pilferage, damage, spoilage, scrap.
Obsolescence.
12-15
Inventory Holding Costs – Usually
20-30% of Total
Cost as a
% of Inventory Value
Category
Investment costs
6 - 24%
Housing costs
3 - 10%
Material handling costs
1 - 3.5%
Labor cost from extra handling
3 - 5%
Pilferage, scrap, and
obsolescence
2 - 5%
12-16
Ordering Costs
To order and receive product:
Supplies.
Forms.
Order processing.
Clerical support.
etc.
12-17
Setup Costs
To change equipment and setup for new
product:
Clean-up costs.
Re-tooling costs.
Adjustment costs.
etc.
12-18
Stockout Costs
For not making a sale and for lost future sales:
- Customer may wait for a backorder, or
- Cancel order (and acquire product elsewhere).
 Backorder costs: expediting, special orders,
rush shipments, etc.
 Lost current sale cost.
 Lost future sales (hard to estimate).
12-19
Inventory Questions
 How much to order (each time)?

100 units, 50 units, 23.624 units, etc.
 When to order?

Every 3 days, every week, every month, etc.

When only 5 items are left, when only 10 items are
left, when only 20 items are left, etc.
 Many different models can be used, depending on
nature of products and demand.
12-20
Independent vs. Dependent Demand
 Independent demand - Demand for item is
independent of demand for any other item.
 Dependent demand - Demand for item depends
upon the demand for some other item.

Example: Demand for car engines depends on
demand for new cars.
 We will consider only independent demand.
12-21
Inventory Models
 Fixed order-quantity models.
 1. Economic order quantity (EOQ).

2. Production order quantity (POQ).

3. Quantity discount.
 Probabilistic models.
 Fixed order-period models.
12-22
How much and
when to order?
How Much and When to Order?
 Given a fixed annual demand for a product.
 With many small orders:

Amount on hand is always small, so inventory is small.

Frequent orders means cost of ordering is large.
 With few large orders:

Amount on hand may be large (when order arrives), so
inventory may be large.

Infrequent orders mean cost of ordering is small.
12-23
EOQ – Economic Order Quanitity
Models
 How much to order (each time)?

Order size is a constant = Q

Q is selected to minimize total cost.
 When to order?

Order when amount remaining = ROP

ROP is selected so chance of running
out is small.
12-24
EOQ Assumptions
 Known and constant demand.
 Known and constant lead time.
 Instantaneous receipt of material.
 No quantity discounts.
 Only order cost and holding cost.
 No stockouts.
12-25
EOQ Model - How Much to Order?
Annual Cost
Order Cost Curve
Order Quantity
12-26
EOQ Model - How Much to Order?
Annual Cost
Order Cost Curve
Optimal
Order Quantity (EOQ=Q*)
12-27
Order Quantity
Why Holding Costs Increase
 For fixed annual demand, larger order
quantities means:

Larger inventory (larger amount ordered each time).

Larger inventory holding cost.
 Example: Annual demand = 1200 units

Order 600 each time.


Maximum inventory = 600
Order 50 each time.

Maximum inventory = 50
12-28
Why Order Costs Decrease
 For fixed annual demand, larger order
quantities means:

Fewer orders per year.

Smaller order cost per year.
 Example: Annual demand = 1200 units

Order 600 each time.


1200/600 = 2 orders per year.
Order 50 each time.

1200/50 = 24 orders per year.
12-29
Deriving an EOQ
 Develop an expression for total costs.

Total cost = order cost + holding cost
 Find order quantity that gives minimum total
cost (use calculus).

Minimum is when slope is flat.

Slope = Derivative.

Set derivative of total cost equal to 0 and solve for
best order quantity.
12-30
EOQ Model Equations
D = Annual demand (relatively constant)
S = Order cost per order
H = Holding (carrying) cost per unit per year
d = Demand rate (units per day, units per week, etc.)
L = Lead time (constant) (in days, weeks, hours, etc.)
Determine: Q = Order size (number of items per order)
D
Expected Number of Orders per year = N =
Q
D
Order Cost per year =
S
Q
Holding Cost per year = (average inventory level)  H
12-31
Given
EOQ Model - Average Inventory Level
Maximum inventory = Q
Minimum inventory = 0
Inventory Level
Order
Quantity
(Q)
Average
Inventory
(Q/2)
0
Time
12-32
EOQ Model Equations
D = Annual demand (relatively constant)
S = Order cost per order
H = Holding (carrying) cost per unit per year
d = Demand rate (units per day, units per week, etc.)
L = Lead time (constant) (in days, weeks, hours, etc.)
Given
Determine: Q = Order size (number of items per order)
D
Expected Number of Orders per year = N =
Q
D
Order Cost per year =
S
Q
Q
H
Holding Cost per year = (average inventory level)  H =
2
12-33
EOQ Model - How Much to Order?
Annual Cost
Order Cost Curve = (D/Q)S
Optimal
Order Quantity (EOQ=Q*)
12-34
Order Quantity
EOQ Total Cost Optimization
Total Cost =
D
Q
S+
H
Q
2
Take derivative of total cost with respect to Q and set
equal to zero:
D
1
S
+
H=0
Q2
2
Solve for Q to get optimal order size:
EOQ = Q* =
2 ×D ×S
H
12-35
EOQ Model Equations
D = Annual demand
S = Order cost per order
H = Holding (carrying) cost
2 ×D ×S
= Q* =
H
D
Expected Number of Orders = N =
Q*
Optimal Order Quantity
Expected Time Between Orders = T =
12-36
Working Days / Year
N
EOQ Model - When to order?
D = Annual demand (relatively constant)
d = Demand per day
L = Lead time in days
Given
Determine: ROP = reorder point (number of pieces or
items remaining when order is to be placed)
d =
D
Working Days / Year
ROP = d × L
Suppose demand is 10 per day and
lead time is (always) 4 days.
When should you order?
When 40 are left!
12-37
EOQ Model - When To Order
Inventory Level
Lead Time = time between placing
and receiving an order
Q*
Reorder
Point
(ROP)
Time
1st order
placed
1st order
received
2nd order
12-38
3rd order
4th order
EOQ Example
Demand = 1200/year
Order cost = $50/order
Holding cost = $5 per year per item
260 working days per year
2 ×1200 ×50 = 154.92 units/order; so order 155 each time
5
1200/year
Expected Number of Orders = N =
= 7.74/year
155
260 days/year
Expected Time Between Orders = T =
= 33.6 days
7.74/year
1200
155
Total Cost =
5 = 387.10 + 387.50 = $774.60/year
50 +
2
155
Q* =
12-39
EOQ is Robust
Demand = 1200/year
Order cost = $50/order
Holding cost = $5 per year per item
260 working days per year
Total Cost =
1200
Q
50 +
Q
2
5
Q = 155 units/order
TC = $774.60/year
Q* = 154.92 units/order
TC = $774.60/year = 387.30 + 387.30
Suppose we must order in multiples of 20:
Q = 140 units/order
TC = $778.57/year (+0.5%)
Q = 160 units/order
TC = $775.00/year (+0.05%)
Cost is nearly optimal!
12-40
EOQ is Robust
Demand = 1200/year
Order cost = $50/order
Holding cost = $5 per year per item
260 working days per year
Total Cost =
1200
Q
50 +
Q
2
5
Q = 155 units/order
TC = $774.60/year
Q* = 154.92 units/order
TC = $774.60/year = 387.30 + 387.30
Suppose we wish to order 6 times per year (every 2 months):
Q = 1200/6 = 200 units/order (200/order is 29% above Q*)
TC = $800.00/year = 300.00 + 500.00
(Cost is only 3.3% above optimal: $800 vs. $774.60)
12-41
EOQ Model is Robust
Annual Cost
Small
variation
in cost
Order Quantity
154.92
Large variation
in order size
12-42
Robustness
 EOQ amount can be adjusted to facilitate business
practices.
 If order size is reasonably near optimal (+ or - 20%),
then cost will be very near optimal (within a few
percent).
 If parameters (order cost, holding cost, demand)
are not known with certainty, then EOQ is still very
useful.
12-43
EOQ Model - When to order?
Demand = 1200/year
Order cost = $50/order
Holding cost = $5 per year per item
260 working days per year
Lead time = 5 days
d =
1200/year
= 4.615/day
260 days/year
ROP = 4.615 units/day 5 days = 23.07 units
-> Place an order whenever inventory falls to (or below) 23 units
12-44