Graduate Program in
Business Information Systems
Inventory Decisions with Certain Factors
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A Retailer’s Plea
If I order too little, I make no profit. If I
order too much, I may go broke. Every
product is different. Help me!
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Why do we control inventory?


Inventories represent a vast segment of total
economic activity.
Even minor improvements can create large
savings.
How do we control inventory?


Application of optimization techniques
Information processing and retrieval
techniques
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Decisions of an inventory policy

If there is no production, i.e., pure inventory
system



How much to order? Order quantity
When to order? Reorder quantity
Ex:Order Q=100 units when the inventory level
drops to r=15 units.
If there is also production

When to start/stop production?
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An inventory system
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Elements of Inventory Decisions

Costs:




Demand structure


How does it vary? Certain, uncertain?
Supply structure


Ordering and Procurement costs
Inventory holding or carrying costs
Inventory shortage costs
Any capacity limitations, defectives, number of suppliers?
Lead times:

Certain, uncertain?
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Ordering and Procurement Costs

Represent all expenses incurred in ordering or
manufacturing items related to






Acquisition
Transportation
Collecting, sorting, placing the items in the storage
Managerial and clerical costs associated with order
placement.
Ordering costs are fixed, independent of the order size.
Procurement costs depend on the order size.
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Holding or Carrying Costs

Expenses incurred during the storage of items.




Physical Costs: Warehouse operation costs,
insurence, property taxes.
Pilferage, spoilage, obsolescence
Opportunity cost of investing in inventory rather
than investing somewhere else, ex. in a bank.
Inventory costs are variable costs that depend
on the order size.
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Shortage Costs


Occur whenever the demand is not satisfied.
Order is either “backordered” or “lost”.
Backordering Costs:



Fixed cost of extra managerial work.
Loss of customer goodwill: Variable cost that
depends on duration of backorder.
Lost Sales Costs:


Marginal profit that the item would have earned.
Loss of customer goodwill.
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Demand Structure

Continuous versus discrete demand
Ex: Natural gas consumption in houses
Detergent consumption in houses

Deterministic (certain) versus stochastic (uncertain)
demand
Ex: Order quantities for the next months are 20,30,10,50.
Order quantities in a month are normally distributed with
mean 25 and variance 4.

Constant versus dynamic demand
Ex: Demand quantities for the next months are 20, 21, 20, 19
Demand quantities for the next months are 20, 50, 10, 2
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Supply Structure



Any defectives?
If the received lot includes defective items
this brings uncertainty
Any capacity limitations?
Do we fully receive what we order?
Number of suppliers, fixed or variable?
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Lead time


Time elapsed between the order delivery and
order receipt.
Can be constant or stochastic.
Ex: Lead time is 10 days.
Lead time is between 8-12 days.
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The Economic Order Quantity
EOQ-Model



Decision variable: Q = Order Quantity
Parameters:
k = Fixed cost per order ($/order)
A = Annual number of items demanded (unit/year)
c = Unit cost of procuring an item ($/unit)
h = Annual cost of holding a dollar in inventory ($/$/year)
Objective is to “minimize total annual cost”.
Total
Ordering
Holding Procurement
=
+
Annual cost
Cost
Cost +
Cost
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EOQ Inventory Policy
Average Inv. Level
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Assumptions of Classical EOQ Model





Demand rate is constant or stable.
There is infinite supply availability.
Lead time is constant or zero.
No quantity discounts are made.
All demand is met on time, no backordering, no
stockout.
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Costs of EOQ Model



Total ordering cost is the number of orders times
the cost per order:
 A
Annual ordering cost   k
Q
Total holding cost is the cost per item held 1 year
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
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Annual Cost of EOQ-Model

Q
 A

Total annual cost   k  hc   Ac
2
Q
Here Ac is not a relevant cost and thus dropped.

Minimize Total Annual Inventory Cost:
Q
 A

TC (Q)   k  hc 
2
Q
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Optimal Solution of EOQ

Optimal solution is the economic order quantity
Q* 

2 Ak
hc
Optimal Total Cost
TC  2 Akhc
*
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Example:The House of Wines and
Liquors




Allex Mullen decides that the first task in utilizing
inventory models is to determine the value of model
parameters:
Annual demand 5200 cases of beer
$10 telephone charge for ordering
Purchase cost is $1.5/case beer +shipping cost
$0.5/case
10%bank interest, 5%state franchise tax, 5% theft
insurance rate
How many should he order, how often, and at what
annual relevant inventory cost?
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Solution:
The economic order quantity is
2 Ak
2520010
Q* 

 509.9 or 510
hc
.202

The inventory cycle duration is
T = Q/A = 510/5200 = 0.098 year or 36 days

The total annual relevant inventory cost is:
 5200
 510
TC (510) 
10

.
20
(
2
)


  $101.96  102.00  $203.96 / year
 510 
 2 
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Robustness of EOQ Model


EOQ is a robust model with respect to the
estimation errors in A, k, c or h.
Let Aactual=4 Aestimated
Then EOQactual=2 Aestimated
Since
EOQactual
2 Aactual k
2 Aestimated k

2
 2EOQestimated
hc
hc
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Ex: The House of Wines and Liquors

Alex Mullen applies EOQ to another product,
a particular variety of Chilean wine that sells
1000 cases annually. The cost is $20 per case.
A telephone call to Chile to place an order
costs $100. The holding costs are the same as
for Tres Equis Beer.
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Ex:
Q* 
2 Ak
21000100

 223.6 or 224
hc
.2020
T = Q/A = 24/1000 = .224 year or 82 days
 1000
 224
TC (224) 
ar
100 .20(20)
  $894.43/ye
224
2




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Optimal Inventory Policy
with Backordering
Orders placed during shortages are backordered.
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Optimal Inventory Policy
with Backordering
S: Quantity on hand when a shipment arrives.
P: Cost of being one item short for a year
2
2


A
hcS
p
Q

S
 
TC (Q ,S )   k 

2Q
2Q
Q
Optimal order quantity and order level:
2 Ak
Q* 
hc
p  hc
p
2 Ak
S* 
hc
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p
p  hc
Example:The House of Wines and
Liquors-Backorders
The marketing department tells Alex that beer is a convenience product that
can not be backordered, so sale is lost! However some wine customers are
connoisseurs who are willing to order out-of-stock items. Nevertheless, the
store owner will incur some penalty cost if there is a shortage of wine.
Suppose that retailer suffers lost profit on future business equal to $0.01/unit
each day that a wine is on backorder. What should be the optimal ordering
policy if backordering is allowed?
Solution: The order quantity is computed:
p = $.01×365 = $3.65/unit/year.
2 Ak
Q* 
hc
p  hc
21000100 3.65  .2020

 324
p
.2020
3.65
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Example: Solution

The order level S is
S* 

2 Ak
hc
p

p  hc
21000100
3.65
 154
.2020
3.65  .2020
The relevant cost is
.2020154 3.65170
 1000
TC (324,154)  
100


 617.82

2324
2324
 324 
smaller than before, why?
2
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2
Is backordering better?


Fewer orders are placed when there is backordering.
Average inventory level is smaller.
Backorders/cycle= Q* – S*=324 – 154 = 170 units/cycle.
Proportion of demand not satisfied on time
=(Q*-S*)/Q*=170/324= 52.5%


The results suggest that:
Retailers will run short in each cycle.
But can they get away with it?
So backordering must make sense!
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Imputed Shortage Penalty
An alternative approach for establishing an inventory policy is
based on achieving a desired service level.
Service Level, L is the proportion of demand met on time
Q*  S *
 1  L, so
*
Q
Imputed shortage penalty
p =
hcL
1L
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LQ*  S*
As p increases EOQ is more robust
A=1000 units/yr
k=$100/order
c=$20/unit h= $0.20/$/year
L=47.5%
324
L=90%
Q*
236
EOQ with no
backordering
224
S*
212
154
P
$3.65
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$36
imputed shortage penalty
Economic Production-Quantity
Model
The inventory model may be extended to finding the
optimal production quantity.
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Economic Production-Quantity
Model




B: Annual production rate
K: Production setup cost.
c: Variable production cost per unit.
Total Annual Cost:
 A
 Q  B  A 
TC(Q )   k  hc 

 2  B 
Q

Economic Production Quantity:
2 Ak B  A
Q* 
hc
B
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Example:
Water Wheelies have annual demand of A =100,000 units and
are made at the rate of B = 500,000 units. Production costs are
k = $2,000/setup and c = $5/unit variable.
It costs h = $.40/year to tie up a dollar.

Economic production quantity is
2 Ak
hc
Q *

B A

B
2 100   2 
.40 5 
500 100
 8 , 944 units
500
Total relevant cost is
 100,000
 8,944 500,000 100,000
2,000  .405
  $29,516.56

500,000
 2 
 8,944 

TC(8,944)  
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More Elaborate Models



Incorporate a second one-time shortage penalty.
Add additional products.
Incorporate uncertainty regarding:




Demand
Lead-time for delivery of order
Incorporate lost sales
Extend to single period products
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Economic Order Quantity Model
(Figure 15-3)
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
8
Unit Cost of Procuring an Item: c =
$
2.00
9
Annual Holding Cost per Dollar Value: h =
$
0.20
10
11
Decision Variables:
12
Order Quantity: Q =
100
13
F
14
Results:
15
=(F7/F12)*F6+F9*F8*(F12/2)
15
Total Annual Relevant Cost: TC =
$ 540.00
16
Time Between Orders (years): T =
0.0192 16 =F12/F7
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Sensitivity Analysis
(Figure 15-6)
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
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Graphing the Sensitivity
Analysis (Figure 15-7)
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
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$200
Backordering Model
(Figure 15-9)
A
B
C
D
E
F
INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY
1
2
3 PROBLEM: House of Fine Wines and Liquors - Chilean Wine
4
5 Parameter Values:
$ 100.00
Fixed Cost per Order: k =
6
1,000
Annual Number of Items Demanded: A =
7
$ 20.00
Unit Cost of Procuring an Item: c =
8
$ 0.20
Annual Holding Cost per Dollar Value: h =
9
$ 3.65
Annual Cost of Being Short One Item: p =
10
11
Decision Variables:
12
324
Economic Order Quantity: Q =
13
154
Economic Order Level: S =
14
15
Results:
16
$ 617.82
Total Annual Relevant Cost: TC =
17
0.32
Time Between Orders (years): T =
18
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G
I
H
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
Production Model
(Figure 15-13)
A
B
C
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
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