Inventory Models
Inventory models

Help to decide

How much to order

When to order

Basic EOQ model

POQ model
Basic EOQ model

Assumptions

Known and constant demand

Known and constant lead time

Instantaneous receipt of material

No quantity discounts

Only order (setup) cost and holding cost

No stockouts
Basic EOQ model

Receive an order

Use the inventory at a constant rate

Reorder same amount

Instantaneously receive the order
On-hand inventory (units)
Economic Order
Quantity
Time
On-hand inventory (units)
Economic Order
Quantity
Time
Economic Order
Quantity
On-hand inventory (units)
Receive
order
Q
Time
Economic Order
Quantity
On-hand inventory (units)
Receive
order
Q
1 cycle
Time
Economic Order
Quantity
On-hand inventory (units)
Receive
order
Q
1 cycle
Time
Economic Order
Quantity
On-hand inventory (units)
Receive
order
Inventory depletion
(demand rate)
Q
1 cycle
Time
Economic Order
Quantity
On-hand inventory (units)
Receive
order
Inventory depletion
(demand rate)
Q
1 cycle
Time
Economic Order Quantity
On-hand inventory (units)
Receive
order
Inventory depletion
(demand rate)
Q
Average
cycle
inventory
Q
—
2
1 cycle
Time
Total Cost = Holding Cost + Order Cost
Annual cost (dollars)
Total Cost = Holding Cost +
Order Cost
Lot Size (Q)
Annual cost (dollars)
Total Cost = Holding Cost + Order Cost
Holding cost (HC)
Lot Size (Q)
Why Holding Costs Increase

More units must be stored if more are ordered
Purchase Order
Description
Qty.
Microwave
1
Order quantity
Purchase Order
Description
Qty.
Microwave
1000
Order quantity
Annual cost (dollars)
Total Cost = Holding Cost + Order Cost
Holding cost (HC)
Ordering cost (OC)
Lot Size (Q)
Why Order Costs Decrease

If we order more when we place an
order, then we order fewer times over
the year.

Example: You expect to order 10
microwave ovens over a year for a retail
Order cost $10
Purchase Order
Description
Qty.
Microwave
1
store like Sears. It cost $10 to place an
order.

If you order 1 microwave, how many
orders will you place over the year?
what is the ordering cost? What is the
ordering cost per microwave?

If you order 10 microwaves, how many
orders will you place over the year?
What is the ordering cost? What is the
ordering cost per microwave?
Order Cost $10
Purchase Order
Description
Qty.
Microwave
10
Total Cost = Holding Cost + Order Cost
Annual cost (dollars)
Total cost = HC + OC
Holding cost (HC)
Ordering cost (OC)
Lot Size (Q)
Gift Shop


A museum of natural history is having
problems managing their inventories. Low
inventory turnover is squeezing profit
margins and causing cash-flow problems.
A Class A item, a birdfeeder is also a topselling item.






Sales:
18 units/week
Purchase cost:
$60
Order cost:
$45
Annual holding cost: 25% of purchase cost
52-week year
Management has been ordering in lots of
390 units.
What is the annual cost of the
current policy?

Q – order quantity

TC – Total cost

Annual

Monthly

??
Q
Time
Holding Cost
AnnualHold ingCost 
( AveInventory )( HoldingCos t / unit / year )
Q
AnnualHold ingCost    H
2
Holding cost
Annual cost (dollars)
3000 —
2000 —
Holding cost =
Q
(H)
2
1000 —
0—
|
50
|
100
|
150
|
200
|
250
Lot Size (Q)
|
300
|
350
|
400
Ordering cost
AnnualDemand
AnnualOrde rCost 
(OrderCost )
OrderQuant ity
D
AnnualOrde rCost  S
Q
Holding & Ordering
Cost
Annual cost (dollars)
3000 —
2000 —
Holding cost =
Q
(H)
2
1000 —
Ordering cost =
0—
|
50
|
100
|
150
|
200
|
250
Lot Size (Q)
|
300
|
350
|
400
D
(S)
Q
Total Cost
Annual cost (dollars)
3000 —
Total cost =
Q
D
(H) +
(S)
2
Q
2000 —
Holding cost =
Q
(H)
2
1000 —
Ordering cost =
0—
|
50
|
100
|
150
|
200
|
250
Lot Size (Q)
|
300
|
350
|
400
D
(S)
Q
Total cost:
TotalCost  TotalOrder Cost  TotalHoldi ngCost
D
Q
TC  S  H
Q
2
What is the annual cost of the current
policy?
D – Total demand
Q – Order quantity
S – Setup/order cost
H – Holding cost
D
Q
TC  S  H
Q
2
What is the annual cost of the current
policy?
D – Total demand
936/year
Q – Order quantity
390/order
D
Q
TC  S  H
Q
2
936
390
TC 
45 
15
390
2
S – Setup/order cost
$45/order
TC  108 2925
H – Holding cost
TC  3033
= 0.25*60
= $15/unit/year
Total Cost for Q = 390
Current
cost
Annual cost (dollars)
3000 —
Total cost =
Q
D
(H) +
(S)
2
Q
2000 —
Holding cost =
Q
(H)
2
1000 —
Ordering cost =
0—
|
50
|
100
|
150
|
200
|
250
Lot Size (Q)
|
300
|
350
D
(S)
Q
|
400
Current
Q
Can the gift shop do better?
Current
cost
Annual cost (dollars)
3000 —
Total cost =
Q
D
(H) +
(S)
2
Q
2000 —
Holding cost =
Q
(H)
2
1000 —
Ordering cost =
0—
|
50
|
100
|
150
|
200
|
250
Lot Size (Q)
|
300
|
350
D
(S)
Q
|
400
Current
Q
Economic Order Quantity – Q*
Annual cost (dollars)
3000 —
2000 —
1000 —
Setup cost = Holding Cost
0—
|
50
|
100
Q*
|
150
|
200
|
250
Lot Size (Q)
|
300
|
350
|
400
Economic Order Quantity (EOQ) – Q*
2 DS
Q* 
H
2(936)( 45)
Q* 
15
 74.94  75units / order
Total Cost of Economic Order
Quantity (EOQ) – Q*
D
Q*
TC 
S
H
Q*
2
936
75
TC 
45  15
75
2
TC  1124.10
When Q = 390
TC  3033
When to order?

Reorder point (ROP)

Lead time – amount of time from order placement
to receipt of goods

Lead time demand – the demand the occurs during
the lead time
On-hand inventory
Reorder point
Order
received
OH
On-hand inventory
Reorder point
Order
received
OH
TBO
Time between orders
On-hand inventory
Reorder point
Order
received
OH
L
TBO
Lead time
On-hand inventory
Reorder point
Order
received
OH
R
Order
placed
L
TBO
Gift shop reorder point

Demand: 18 birdfeeders/week

Lead time: 2 weeks

Lead time demand: 36 birdfeeders

ROP: 36 birdfeeders
Gift shop order policy




Place order when the on-hand inventory is 36
birdfeeders.
Order 75 birdfeeders
Order received in 2 weeks
Place next order when the on-hand inventory is
36 birdfeeders
On-hand inventory
Gift shop order policy
Order
received
75
OH
36
Order
placed
2 wks
Distribution Game

What is the EOQ for
the central warehouse
in the distribution
game?
 Order
cost:
 Holding
S=
cost: H =
 Demand:
D=
$200
$14.70/unit/year
2190
Distribution Game
2 DS
Q* 
H
2(2190)( 200)
Q* 
14.7
Q*  244.1  244units
Production Order
Quantity
On-hand inventory
Production Order
Quantity
Time
On-hand inventory
Production Order
Quantity
Q
Production quantity
Time
On-hand inventory
Production Order
Quantity
Production quantity
Q
Demand during
production interval
p–d
Time
On-hand inventory
Production Order
Quantity
Production quantity
Q
Demand during
production interval
p–d
Time
On-hand inventory
Production Order
Quantity
Production quantity
Q
Demand during
production interval
p–d
Time
Production
and demand
Demand
only
TBO
On-hand inventory
Production Order
Quantity
Production quantity
Q
Demand during
production interval
p–d
Time
Production
and demand
Demand
only
TBO
On-hand inventory
Production Order
Quantity (POQ Model)
Production quantity
Q
Demand during
production interval
Maximum inventory
p–d
Time
Production
and demand
Demand
only
POQ model

Assumptions
Known and constant demand
 Known and constant lead time


Partial receipt of material
No quantity discounts
 Only order (setup) cost and holding cost
 No stockouts

POQ model

Producing and shipping product simultaneously
until production quantity is reached.

Ship the inventory at a constant rate (no
production)

Reorder same amount

Begin production and shipping product
simultaneously
POQ Model
D – annual demand
S – Setup cost
Q 
*
p
2 DS
H (1  d / p)
H – Holding cost
d – daily demand rate
p – daily production rate
Chemical Plant
A plant manager of a chemical plant must
determine the lot size for a particular
chemical that has a steady demand of 30
barrels/day. The production rate is 190
barrels/day, annual demand is 10,500 barrels,
setup cost is $200, annual holding cost is
$0.21/barrel, and the plant operates 350
days/year. Determine the production order
quantity.
Chemical Plant
Demand:
d = 30 barrels/day
D = 10,500 barrels/year
Setup cost:
$200/setup
Holding cost:
$0.21/barrel/year
Production:
190 barrels/day
Chemical Plant
Q 
*
p
2 DS
H (1  d / p)
2(10500)( 200)
Q 
0.21(1  30 / 190)
*
p
Q  4873.4  4873barrels
*
p