Transcript Document
Chapter 15
If I order too little, I make no profit. If I order
too much, I may go broke. Every product is
different. Help me!—A Retailer’s Plea
Inventory Decisions
with Certain Factors
1
Elements of Inventory Decisions
There are four basic inventory system costs:
Ordering costs
Procurement costs
Inventory holding or carrying costs
Inventory shortage costs
Demand is usually erratic and uncertain.
We assume it is smooth and predictable.
That makes it easier to develop mathematical
models. These can later be made more realistic.
Order quantity is the main variable.
With no uncertainty, we can schedule deliveries
to arrive exactly when we run out.
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The Economic Order Quantity
(EOQ) Model
The decision variable is
Q = Order Quantity
There are four parameters:
k = Fixed cost per order
A = Annual number of items demanded
c = Unit cost of procuring an item
h = Annual cost per dollar value of
holding items in inventory
An order quantity is to be found that
minimizes:
Total
Ordering Holding Procurement
=
Annual cost
Cost + Cost +
Cost
3
The Economic Order Quantity
(EOQ) Model
Inventory level has a cycle beginning with a
new shipment’s arrival.
T = Q/A = Duration of inventory cycle
4
The Economic Order Quantity
(EOQ) Model
The annual ordering cost is the number of
orders times the cost per order:
A
Annual ordering cost k
Q
The annual holding cost is the cost per item
held 1year times the average inventory:
Q
Annual holding cost hc
2
The annual procurement cost is the product
of annual demand and unit cost:
Procurement cost = Ac
5
The Economic Order Quantity
(EOQ) Model
The total annual inventory cost is:
Q
A
Total annual cost k hc Ac
2
Q
We drop Ac from the above, since that
amount will not vary with Q.
Ac is not a relevant cost.
That provides the function to be minimized,
the total annual relevant inventory cost:
Q
A
TC (Q) k hc
2
Q
6
The Economic Order Quantity
(EOQ) Model
It may be shown using calculus that the
level for Q minimizing the above is the
economic order quantity
2 Ak
Q*
hc
Problem. A software store sells 500 Alien
Saboteurs annually. The supplier charges
$100 per order plus $20 each. It costs $.15
per dollar value to hold inventory for a year.
How many should they order, how often,
and at what annual relevant inventory cost?
7
The Economic Order Quantity
(EOQ) Model
Solution:
The following parameters apply:
A = 500
k = 100
c = 20
h = .15
The economic order quantity is
2 Ak
2500 100
Q*
182.6 or 183
hc
.1520
The inventory cycle duration is
T = Q/A = 183/500 = .366 year or 133.6 days
The total annual relevant inventory cost is:
500
183 $273.22 274.50 $547.72
TC (183)
100
.
15
(
20
)
183
2
8
Optimal Inventory Policy
with Backordering
Retailers may not stock all demand. Orders
placed during shortages are backordered.
9
Optimal Inventory Policy
with Backordering
The new model adds the order level S, that
quantity on hand when a shipment arrives.
A shortage cost applies, based on a penalty
p for being one item short for a year.
New total annual relevant inventory cost:
hcS
A
TC (Q ,S ) k
2Q
Q
2
2
p QS
2Q
Optimal order quantity and order level:
10
2 Ak
Q*
hc
p hc
p
2 Ak
S*
hc
p
p hc
Optimal Inventory Policy
with Backordering
Shortage penalty p applies over a year, but
cost prorates to fractions of items or years.
Example: The retailer suffers lost profit on
future business equal to $.05 each day that
one Alien Saboteur is on backorder. That
translates into p = $.05×365 = $18.25.
Solution: The order quantity is computed:
2 Ak
Q*
hc
11
p hc
2500 100 18.25 .1520
p
.1520
18.25
18.25 .1520
182.6
197.04 or 197
18.25
Optimal Inventory Policy
with Backordering
Solution: The order level is computed:
p
2500 100
18.25
p hc
.1520
18.25 .1520
18.25
182.6
169.22 or 169
18.25 .1520
2 Ak
S*
hc
The relevant cost is
2
2
500
.
15
20
169
18
.
25
197
169
100
TC (197 , 169)
2197
2197
197
= $253.81 + 217.47 + 36.31 = $507.59
The above is smaller than before, even though
there is a shortage penalty and shortages. Why?
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Optimal Inventory Policy
with Backordering
There is a net savings in holding costs and a
slight reduction in ordering costs. Those
outweigh increased cost due to shortages.
The number of backorders is Q – S. Here
that quantity is 197 – 169 = 28.
The annual shortage cost is only $36.31,
because durations of shortage (for last of the
28) are only 28/197 = .142 year (52 days).
The results suggest that:
Retailers will run short, if they can get away
with it!
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But backordering must make sense.
Optimal Inventory Policy
with Backordering
Nobody backorders cigarettes or gasoline.
Sales for those products are lost during
shortages. This model does not apply for them.
The shortage penalty p is not usually
known. But it may be imputed from
existing policy. The service level L is used
for that purpose:
L = proportion of time fully stocked
The imputed shortage penalty is:
hcL
p =
1L
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Economic Production-Quantity
Model
The inventory model may be extended to
finding the optimal production quantity.
15
Economic Production-Quantity
Model
The new parameter is the annual
production rate B.
Parameter k is the production setup cost.
The variable production cost per unit is c.
The total annual relevant inventory cost:
Q B A
A
TC (Q) k hc
2 B
Q
The economic production quantity:
2 Ak B A
Q*
hc
B
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Economic Production-Quantity
Model
Example: Water Wheelies have annual demand of
A =100,000 units and are made at the rate of B =
500,000. Production costs are k = $2,000 setup
and c = $5 variable. It costs h = $.40/year to tie
up a dollar.
Economic production quantity is
Q*
2 Ak B A 2 100 2 500100
8.944 thousand
hc
B
.40 5
500
Total relevant cost is
100 ,000
8,944 500 ,000 100 ,000 29 ,516.56
2,000 .405
500 ,000
2
8,944
TC(8,944)
17
More Elaborate Models
Incorporate a second one-time shortage
penalty (done in Chapter 16).
These models are for single products. Add
additional products.
Incorporate uncertainty regarding:
Demand (done in Chapter 16).
Lead-time for delivery of order (Chapter 16).
Incorporate lost sales (done in Chapter 16).
Extend to single period products (Ch. 16).
NOTE: The basic EOQ model works very
well even when its ideal conditions don’t
18
apply. It is very robust.
Inventory Spreadsheet Templates
Economic Order Quantity
Sensitivity Analysis
Backordering
Production
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Economic Order Quantity Model
(Figure 15-3)
2. Enter the
problem
information in
F6:F9.
1. Enter the
problem
name in B3.
A
B
C
D
E
F
G
H
I
INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY MODEL
1
2
3 PROBLEM: House of Fine Wines and Liquors - Tres Equis Beer
4
5 Parameter Values:
6
Fixed Cost per Order: k =
$ 10.00
7
Annual Number of Items Demanded: A =
5,200
Optimal order
8
Unit Cost of Procuring an Item: c =
$
2.00
9
Annual Holding Cost per Dollar Value: h =
$
0.20
quantity
10
11
Decision Variables:
12
Order Quantity: Q =
100
13
F
14
Results:
15
Total Annual Relevant Cost: TC =
$ 540.00 15 =(F7/F12)*F6+F9*F8*(F12/2)
16
Time Between Orders (years): T =
0.0192 16 =F12/F7
20
Optimal total annual relevant cost and
time between orders
Sensitivity Analysis
(Figure 15-6)
A sensitivity analysis shows how answers vary as
data changes. Here the fixed order cost, k, varies.
1. Enter the
problem
name in B3.
2. Enter the
problem
information
in F6:I9.
21
A
B
C
D
E
F
G
H
I
1
INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY MODEL
2
3 PROBLEM: Sensitivity Analysis for House of Fine Wines and Liquors - Chilean Wines
4
5 Parameter Values:
6
Fixed Cost per Order: k =
$
50.00 $ 100.00 $
150.00 $ 200.00
7
Annual Number of Items Demanded: A =
1,000
1,000
1,000
1,000
8
Unit Cost of Procuring an Item: c =
$
20.00 $ 20.00 $
20.00 $
20.00
9
Annual Holding Cost per Dollar Value: h =
$
0.20 $
0.20 $
0.20 $
0.20
10
11
Decision Variables:
12
Order Quantity: Q =
158.1
223.6
273.9
316.2
13
14
Results:
15
Total Annual Relevant Cost: TC =
$ 632.46 $ 894.43 $ 1,095.45 $ 1,264.91
16
Time Between Orders (years): T =
0.16
0.22
0.27
0.32
The fixed order cost has a diminishing effect on the
results. For example, a 100% increase in k causes
both Q* and TC(Q)* to increase by only 41%.
Graphing the Sensitivity Analysis
(Figure 15-7)
Graphing sensitivity analysis results makes It
is easier to see relationships.
Sensitivity Analysis
Units for Q* and
Dollars for TC(Q*)
1,400
1,200
1,000
Order Quantity, Q*
800
TC(Q*)
600
400
200
0
$50
$100
$150
Fixed Cost per Order, k
22
$200
Backordering Model
(Figure 15-9)
1. Enter the
problem name
in B3.
A
B
C
2. Enter the problem
information in
F6:F10.
D
E
F
G
1
INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY
2
3 PROBLEM: House of Fine Wines and Liquors - Chilean Wine
4
5 Parameter Values:
6
Fixed Cost per Order: k =
$ 100.00
7
Annual Number of Items Demanded: A =
1,000
8
Unit Cost of Procuring an Item: c =
$ 20.00
9
Annual Holding Cost per Dollar Value: h =
$
0.20
10
Annual Cost of Being Short One Item: p =
$
3.65
11
12
Decision Variables:
13
Economic Order Quantity: Q =
324
14
Economic Order Level: S =
154
15
16
Results:
17
Total Annual Relevant Cost: TC =
$ 617.82
18
Time Between Orders (years): T =
0.32
23
Optimal total annual relevant cost and
time between orders
H
I
J
MODEL WITH BACKORDERING
F
=SQRT((2*$F$7*$F$6)/($F$9*$F$8))
13 *SQRT(($F$10+$F$9*$F$8)/$F$10)
=SQRT((2*$F$7*$F$6)/($F$9*$F$8))
14 *SQRT($F$10/($F$10+$F$9*$F$8))
F
=($F$7/$F$13)*$F$6+$F$9*$F$8*
(($F$14^2)/(2*F13))+((F10*(F1317 F14)^2/(2*F13)))
18 =F13/F7
Optimal order quantity
and order level
Production Model
(Figure 15-13)
1. Enter the problem
name in B3.
A
B
C
2. Enter the problem
information in F6:F10.
D
E
F
G
H
I
1
INVENTORY ANALYSIS - ECONOMIC PRODUCTION-QUANTITY MODEL
2
3 PROBLEM: Lambda Optics
4
5 Parameter Values:
6
Fixed Set-Up Cost per Run: k =
$ 5,000.00
7
Annual Number of Items Demanded: A =
100,000
8
Annual Production Rate: B =
200,000
9
Variable Production Cost per Unit: c =
$
10.00
F
10
Annual Holding Cost per Dollar Value: h =
$
0.20
=SQRT((2*F7*F6)/(F10*F9))*S
11
13 QRT((F8)/(F8-F7))
12
Decision Variables:
F
13
Economic Production Quantity: Q =
31,623
14
16 =F13/F7
15
Results:
17 =F13/F8
16
Time Between Production Runs (year): T =
0.32
=(F7/F13)*F6+F10*F9*(F13/2)*
17
Duration of Production Run (year): T1 =
0.16
18 ((F8-F7)/F8)
18
Total Annual Relevant Cost: TC =
$ 31,623
Optimal time between production runs,
duration of production run, and total
24
annual relevant cost.
Optimal order quantity