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.
2
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
2500100
Q* 

 182.6 or 183
hc
.1520 
 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:
2
2


A
hcS
p
Q

S
 
TC (Q , S )   k 

2Q
2Q
Q
 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
2500100 18.25  .1520 

p
.1520 
18.25
18.25  .1520 
 182.6
 197.04 or 197
18.25
Optimal Inventory Policy
with Backordering
 Solution: The order level is computed:
p
2500100
18.25

p  hc
.1520 
18.25  .1520 
18.25
 182.6
 169.22 or 169
18.25  .1520 
2 Ak
S* 
hc
 The relevant cost is
2
2





500
.
15
20
169
18
.
25
197

169
100 
TC (197, 169)  


2197 
2197 
 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?
12
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!
13
 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 =
1L
14
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
16
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  500100

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  .405


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
19
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