Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 16
Inventory Management
Chapter 16 - Inventory Management
1
Chapter Topics
Elements of Inventory Management
Inventory Control Systems
Economic Order Quantity Models
The Basic EOQ Model
The EOQ Model with Non-Instantaneous Receipt
The EOQ Model with Shortages
EOQ Analysis with QM for Windows
EOQ Analysis with Excel and Excel QM
Quantity Discounts
Reorder Point
Determining Safety Stocks Using Service Levels
Order Quantity for a Periodic Inventory System
Chapter 16 - Inventory Management
2
Elements of Inventory Management
Role of Inventory (1 of 2)
Inventory is a stock of items kept on hand used to meet
customer demand..
A level of inventory is maintained that will meet anticipated
demand.
If demand not known with certainty, safety (buffer) stocks
are kept on hand.
Additional stocks are sometimes built up to meet seasonal
or cyclical demand.
Large amounts of inventory sometimes purchased to take
advantage of discounts.
Chapter 16 - Inventory Management
3
Elements of Inventory Management
Role of Inventory (2 of 2)
In-process inventories maintained to provide independence
between operations.
Raw materials inventory kept to avoid delays in case of
supplier problems.
Stock of finished parts kept to meet customer demand in
event of work stoppage.
Chapter 16 - Inventory Management
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Elements of Inventory Management
Demand
Inventory exists to meet the demand of customers.
Customers can be external (purchasers of products) or
internal (workers using material).
Management needs accurate forecast of demand.
Items that are used internally to produce a final product are
referred to as dependent demand items.
Items that are final products demanded by an external
customer are independent demand items.
Chapter 16 - Inventory Management
5
Elements of Inventory Management
Inventory Costs (1 of 3)
Carrying costs - Costs of holding items in storage.
Vary with level of inventory and sometimes with length
of time held.
Include facility operating costs, record keeping,
interest, etc.
Assigned on a per unit basis per time period, or as
percentage of average inventory value (usually
estimated as 10% to 40%).
Chapter 16 - Inventory Management
6
Elements of Inventory Management
Inventory Costs (2 of 3)
Ordering costs - costs of replenishing stock of inventory.
Expressed as dollar amount per order, independent of
order size.
Vary with the number of orders made.
Include purchase orders, shipping, handling,
inspection, etc.
Chapter 16 - Inventory Management
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Elements of Inventory Management
Inventory Costs (3 of 3)
Shortage, or stockout costs - Costs associated with
insufficient inventory.
Result in permanent loss of sales and profits for items
not on hand.
Sometimes penalties involved; if customer is internal,
work delays could result.
Chapter 16 - Inventory Management
8
Inventory Control Systems
An inventory control system controls the level of inventory
by determining how much (replenishment level) and when
to order.
Two basic types of systems -continuous (fixed-order
quantity) and periodic (fixed-time).
In a continuous system, an order is placed for the same
constant amount when inventory decreases to a specified
level.
In a periodic system, an order is placed for a variable
amount after a specified period of time.
Chapter 16 - Inventory Management
9
Inventory Control Systems
Continuous Inventory Systems
A continual record of inventory level is maintained.
Whenever inventory decreases to a predetermined level,
the reorder point, an order is placed for a fixed amount to
replenish the stock.
The fixed amount is termed the economic order quantity,
whose magnitude is set at a level that minimizes the total
inventory carrying, ordering, and shortage costs.
Because of continual monitoring, management is always
aware of status of inventory level and critical parts, but
system is relatively expensive to maintain.
Chapter 16 - Inventory Management
10
Inventory Control Systems
Periodic Inventory Systems
Inventory on hand is counted at specific time intervals and
an order placed that brings inventory up to a specified level.
Inventory not monitored between counts and system is
therefore less costly to track and keep account of.
Results in less direct control by management and thus
generally higher levels of inventory to guard against
stockouts.
System requires a new order quantity each time an order is
placed.
Used in smaller retail stores, drugstores, grocery stores and
offices.
Chapter 16 - Inventory Management
11
Economic Order Quantity Models
Economic order quantity, or economic lot size, is the
quantity ordered when inventory decreases to the reorder
point.
Amount is determined using the economic order quantity
(EOQ) model.
Purpose of the EOQ model is to determine the optimal
order size that will minimize total inventory costs.
Three model versions to be discussed:
Basic EOQ model
EOQ model without instantaneous receipt
EOQ model with shortages
Chapter 16 - Inventory Management
12
Economic Order Quantity Models
Basic EOQ Model (1 of 2)
A formula for determining the optimal order size that
minimizes the sum of carrying costs and ordering costs.
Simplifying assumptions and restrictions:
Demand is known with certainty and is relatively
constant over time.
No shortages are allowed.
Lead time for the receipt of orders is constant.
The order quantity is received all at once and
instantaneously.
Chapter 16 - Inventory Management
13
Economic Order Quantity Models
Basic EOQ Model (2 of 2)
Figure 16.1
The Inventory Order Cycle
Chapter 16 - Inventory Management
14
Basic EOQ Model
Carrying Cost (1 of 2)
Carrying cost usually expressed on a per unit basis of time,
traditionally one year.
Annual carrying cost equals carrying cost per unit per year
times average inventory level:
Carrying cost per unit per year = Cc
Average inventory = Q/2
Annual carrying cost = CcQ/2.
Chapter 16 - Inventory Management
15
Basic EOQ Model
Carrying Cost (2 of 2)
Figure 16.4
Average Inventory
Chapter 16 - Inventory Management
16
Basic EOQ Model
Ordering Cost
Total annual ordering cost equals cost per order (Co) times
number of orders per year.
Number of orders per year, with known and constant
demand, D, is D/Q, where Q is the order size:
Annual ordering cost = CoD/Q
Only variable is Q, Co and D are constant parameters.
Relative magnitude of the ordering cost is dependent on
order size.
Chapter 16 - Inventory Management
17
Basic EOQ Model
Total Inventory Cost (1 of 2)
Total annual inventory cost is sum of ordering and carrying
cost:
TC  Co D  Cc Q
Q
2
Chapter 16 - Inventory Management
18
Basic EOQ Model
Total Inventory Cost (2 of 2)
Figure 16.5
The EOQ Cost Model
Chapter 16 - Inventory Management
19
Basic EOQ Model
EOQ and Minimum Total Cost
EOQ occurs where total cost curve is at minimum value
and carrying cost equals ordering cost:
TC min  CoD  Cc Qopt
Qopt
2
Qopt  2CoD
Cc
The EOQ model is robust because Q is a square root and
errors in the estimation of D, Cc and Co are dampened.
Chapter 16 - Inventory Management
20
Basic EOQ Model
Example (1 of 2)
I-75 Carpet Discount Store, Super Shag carpet sales.
Given following data, determine number of orders to be
made annually and time between orders given store is open
every day except Sunday, Thanksgiving Day, and
Christmas Day.
Model parameters :
Cc  $0.75, Co  $150, D  10,000yd
Optimal order size :
Qopt  2CoD  2(150)(10,000)  2,000 yd
Cc
(0.75)
Chapter 16 - Inventory Management
21
Basic EOQ Model
Example (2 of 2)
Total annual inventory cost :
TC min  Co D  Cc Qopt  (150)10,000  (0.75) (2,000)  $1,500
Qopt
2
2,000
2
Number of orders per year :
D  10,000  5
Qopt 2,000
Order cycle time  311 days  311 62.2 store days
D / Qopt
5
Chapter 16 - Inventory Management
22
Basic EOQ Model
EOQ Analysis Over Time (1 of 2)
For any time period unit of analysis, EOQ is the same.
Shag Carpet example on monthly basis:
Model parameters :
Cc  $0.0625 per yd per month
Co  $150 per order
D  833.3 yd per month
Optimal order size :
Qopt  2CoD  2(150)(833.3)  2,000 yd
Cc
(0.0625)
Chapter 16 - Inventory Management
23
Basic EOQ Model
EOQ Analysis Over Time (2 of 2)
Total monthly inventory cost :
TC min  Co D  Cc Qopt  (150) (833.3)  (0.0625) (2,000)
Qopt
2
2,000
2
 $125 per month
Total annual inventory cost  ($125)(12)  $1,500
Chapter 16 - Inventory Management
24
EOQ Model
Non-Instantaneous Receipt Description (1 of 2)
In the non-instantaneous receipt model the assumption that
orders are received all at once is relaxed. (Also known as
gradual usage or production lot size model.)
The order quantity is received gradually over time and
inventory is drawn on at the same time it is being
replenished.
Chapter 16 - Inventory Management
25
EOQ Model
Non-Instantaneous Receipt Description (2 of 2)
Figure 16.6
The EOQ Model with Non-Instantaneous Order Receipt
Chapter 16 - Inventory Management
26
Non-Instantaneous Receipt Model
Model Formulation (1 of 2)
p  daily rate at which the order is received over time
d  daily rate at which inventory is demanded






d
Maximum inventory level  Q 1 p 



Q

d
Average inventory level  1 p 
2



Total carrying cost  Cc Q 1 dp 
2



Q

D
d
Total annual inventory cost  Co  Cc 1 p 
Q
2

Chapter 16 - Inventory Management
27
Non-Instantaneous Receipt Model
Model Formulation (2 of 2)


Cc Q 1 dp   Co D at lowest point of total cost curve
Q
2

Optimal order size : Qopt 
Chapter 16 - Inventory Management
2CoD
Cc(1 d / p)
28
Non-Instantaneous Receipt Model
Example (1 of 2)
Super Shag carpet manufacturing facility:
Co  $150
Cc  $0.75 per unit
D  10,000 yd per year  10,000/311  32.2 yd per day
p  150 yd per day
Optimal order size : Qopt 
2CoD  2(150)(10,000)








d
32
.
2
Cc1 p 
0.751



150




 2,256.8 yd
Chapter 16 - Inventory Management
29
Non-Instantaneous Receipt Model
Example (2 of 2)


Q

D
d
Total minimum annual inventory cost  Co  Cc 1 p 
Q
2



(
10
,
000
)
(
2
,
256
.
8
)


32
.
2
 (150)
 (.075)
1
  $1,329

(2,256.8)
2
150 

2,256.8 15.05 days
Production run length  Q

p
150
Number of orders per year (productio n runs)  D
Q
 10,000  4.43 runs
2,256.8






d
32
.
2
Maximum inventory level  Q 1 p   2,256.81
 1,772 yd

150 







Chapter 16 - Inventory Management
30
EOQ Model with Shortages
Description (1 of 2)
In the EOQ model with shortages, the assumption that
shortages cannot exist is relaxed.
Assumed that unmet demand can be backordered with all
demand eventually satisfied.
Chapter 16 - Inventory Management
31
EOQ Model with Shortages
Description (2 of 2)
Figure 16.7
The EOQ Model with Shortages
Chapter 16 - Inventory Management
32
EOQ Model with Shortages
Model Formulation (1 of 2)
2
S
Total shortage costs  Cs
2Q
2
(
Q

S
)
Total carrying costs  Cc
2Q
Total ordering cost  C0 D
Q
2
2
(
Q

S
)
S
Total inventory cost  Cs  Cc
 Co D
Q
2Q
2Q


2
CoD
 Cs  Cc 
Optimal order quantity  Qopt 
Cc  Cs 






Cc

Shortage level  Sopt  Qopt
Cc  Cs 
Chapter 16 - Inventory Management
33
EOQ Model with Shortages
Model Formulation (2 of 2)
Figure 16.8
Cost Model with Shortages
Chapter 16 - Inventory Management
34
EOQ Model with Shortages
Model Formulation (1 of 3)
I-75 Carpet Discount Store allows shortages; shortage cost
Cs, is $2/yard per year.
Co  $150
Cc  $0.75 per yd
Cs  $2 per yd
D  10,000 yd
Optimal order quantity :




2
(
150
)(
10
,
000
)
 2  0.75 
2
CoD
 Cs  Cc 
Qopt 

  2,345.2 yd

 



Cc  Cs 
0.75
2 

Chapter 16 - Inventory Management
35
EOQ Model with Shortages
Model Formulation (2 of 3)
Shortage level:



 0.75 
Cc

Sopt  Qopt
  639.6 yd
  2,345.2



Cc  Cs 
 2  0.75 





Total inventory cost :
2
2
(
Q

S
)
S
TC  Cs  Cc
 Co D
Q
2Q
2Q
2 (0.75)(1,705.6)2 (150)(10,000)
(
2
)(
639
.
6
)



2(2,345.2)
2(2,345.2)
2,345.2
 $174.44  465.16  639.60  $1,279.20
Chapter 16 - Inventory Management
36
EOQ Model with Shortages
Model Formulation (3 of 3)
Number of orders  D  10,000  4.26 orders per year
Q 2,345.2
Maximum inventory level  Q  S  2,345.2  639.6 1,705.6 yd
Time between orders  t 
days per year  311  73.0 days
number of orders 4.26
Time during which inventory is on hand
 t1  Q  S  2,345.2-639.6  0.171 or 53.2 days
D
10,000
Time during which the re is a shortage
 t 2  S  639.6  0.064 year or 19.9 days
D 10,000
Chapter 16 - Inventory Management
37
EOQ Analysis with QM for Windows
Exhibit 16.1
Chapter 16 - Inventory Management
38
EOQ Analysis with Excel and Excel QM (1 of 2)
Exhibit 16.2
Chapter 16 - Inventory Management
39
EOQ Analysis with Excel and Excel QM (2 of 2)
Exhibit 16.3
Chapter 16 - Inventory Management
40
Quantity Discounts
Price discounts are often offered if a predetermined number of
units is ordered or when ordering materials in high volume.
Basic EOQ model used with purchase price added:
TC  Co D  Cc Q  PD
Q
2
where: P = per unit price of the item
D = annual demand
Quantity discounts are evaluated under two different
scenarios:
With constant carrying costs
With carrying costs as a percentage of purchase price
Chapter 16 - Inventory Management
41
Quantity Discounts with Constant Carrying Costs
Analysis Approach
Optimal order size is the same regardless of the discount
price.
The total cost with the optimal order size must be compared
with any lower total cost with a discount price to determine
which is the lesser.
Chapter 16 - Inventory Management
42
Quantity Discounts with Constant Carrying Costs
Example (1 of 2)
University bookstore: For following discount schedule
offered by Comptek, should bookstore buy at the discount
terms or order the basic EOQ order size?
Quantity Price
1- 49
$1,400
50 – 89
1,100
90 +
900
Determine optimal order size and total cost:
Co  $2,500
Cc  $190 per unit
D  200
Qopt  2CoD  2(2,500)(200)  72.5
190
Cc
Chapter 16 - Inventory Management
43
Quantity Discounts with Constant Carrying Costs
Example (2 of 2)
Compute total cost at eligible discount price ($1,100):
TC min  CoD  Cc Qopt  PD
Qopt
2
 (2,500)(200)  (190) (72.5)  (1,100)(200)  $233,784
(72.5)
2
Compare with total cost of with order size of $90 and price
of $900:
TC  CoD  Cc Q  PD
Q
2
 (2,500)(200)  (190)(90)  (900)(200)  $194,105
(90)
2
Because $194,105 < $233,784, maximum discount price
should be taken and 90 units ordered.
Chapter 16 - Inventory Management
44
Quantity Discounts with Carrying Costs
Percentage of Price Example (1 of 3)
University Bookstore example, but a different optimal order
size for each price discount.
Optimal order size and total cost determined using basic
EOQ model with no quantity discount.
This cost then compared with various discount quantity
order sizes to determine minimum cost order.
This must be compared with EOQ-determined order size for
specific discount price.
Data:
Co = $2,500
D = 200 computers per year
Chapter 16 - Inventory Management
45
Quantity Discounts with Carrying Costs
Percentage of Price Example (2 of 3)
Quantity
Price
Carrying Cost
0 - 49
50 - 89
90 +
$1,400
1,100
900
1,400(.15) = $210
1,100(.15) = 165
900(.15) = 135
Compute optimum order size for purchase price without
discount and Cc = $210:
Qopt  2CoD  2(2,500)(200)  69
Cc
210
Compute new order size:
Qopt  2(2,500)(200)  77.8
165
Chapter 16 - Inventory Management
46
Quantity Discounts with Carrying Costs
Percentage of Price Example (3 of 3)
Compute minimum total cost:
TC  CoD  Cc Q  PD  (2,500)(200) 165 (77.8)  (1,100)(200)
Q
2
77.8
2
 $232,845
Compare with cost, discount price of $900, order quantity of
90:
TC  (2,500)(200)  (135)(90)  (900)(200)  $191,630
90
2
Optimal order size computed as follows:
Qopt  2(2,500)(200)  86.1
135
Since this order size is less than 90 units , it is not
feasible,thus optimal order size is 90 units.
Chapter 16 - Inventory Management
47
Quantity Discount Model
Solution with QM for Windows
Exhibit 16.4
Chapter 16 - Inventory Management
48
Reorder Point (1 of 4)
The reorder point is the inventory level at which a new
order is placed.
Order must be made while there is enough stock in place to
cover demand during lead time.
Formulation:
R = dL
where d = demand rate per time period
L = lead time
For Carpet Discount store problem:
R = dL = (10,000/311)(10) = 321.54
Chapter 16 - Inventory Management
49
Reorder Point (2 of 4)
Figure 16.9
Reorder Point and Lead Time
Chapter 16 - Inventory Management
50
Reorder Point (3 of 4)
Inventory level might be depleted at slower or faster rate
during lead time.
When demand is uncertain, safety stock is added as a
hedge against stockout.
Figure 16.10
Inventory Model with Uncertain Demand
Chapter 16 - Inventory Management
51
Reorder Point (4 of 4)
Figure 16.11
Inventory model with safety stock
Chapter 16 - Inventory Management
52
Determining Safety Stocks Using Service Levels
Service level is probability that amount of inventory on hand
is sufficient to meet demand during lead time (probability
stockout will not occur).
The higher the probability inventory will be on hand, the
more likely customer demand will be met.
Service level of 90% means there is a .90 probability that
demand will be met during lead time and .10 probability of a
stockout.
Chapter 16 - Inventory Management
53
Reorder Point with Variable Demand (1 of 2)
R  d L  Z d L
where:
R  reorder point
d  average daily demand
L  lead time
 d  the standard deviation of daily demand
Z  number of standard deviations correspond ing
to service level probability
Z d L  safety stock
Chapter 16 - Inventory Management
54
Reorder Point with Variable Demand (2 of 2)
Figure 16.12
Reorder Point for a Service Level
Chapter 16 - Inventory Management
55
Reorder Point with Variable Demand Example
I-75 Carpet Discount Store Super Shag carpet.
For following data, determine reorder point and safety stock
for service level of 95%.
d  30 yd per day
L  10 days
d  5 yd per day
For 95% service level, Z  1.65 (Table A -1, appendix A )
R  d L  Zd L  30(10)  (1.65)(5)( 10 )  300  26.1
 326.1 yd
Safety stock is second term in reorder point formula : 26.1.
Chapter 16 - Inventory Management
56
Determining Reorder Point with Excel
Determining the Reorder Point with Excel
Exhibit 16.5
Chapter 16 - Inventory Management
57
Reorder Point with Variable Lead Time
For constant demand and variable lead time:
R  d L  Zd L
where:
d  constant daily demand
L  average lead time
 L  standard deviation of lead time
d L  standard deviation of demand during lead time
Zd L  safety stock
Chapter 16 - Inventory Management
58
Reorder Point with Variable Lead Time Example
Carpet Discount Store:
d  30 yd per day
L  10 days
 L  3 days
Z  1.65 for a 95% service level
R  d L  Zd L  (30)(10)  (1.65)(30)(3)  300 148.5  448.5 yd
Chapter 16 - Inventory Management
59
Reorder Point
Variable Demand and Lead Time
When both demand and lead time are variable:
2
2 2
R  d L  Z ( d ) L  ( L) d
where:
d  average daily demand
L  average lead time
2
2 2
( d ) L  ( L) d  standard deviation of demand during lead time
2
2 2
Z ( d ) L  ( L) d  safety stock
Chapter 16 - Inventory Management
60
Reorder Point
Variable Demand and Lead Time Example
Carpet Discount Store:
d  30 yd per day
 d  5 yd per day
L  10 days
 L  3 days
Z  1.65 for 95% service level
2
2 2
R  d L  Z ( d ) L  ( L) d
 (30)(10)  (1.65) (5)(5)(10)  (3)(3)(30)(30)
 300  150.8
 450.8 yds
Chapter 16 - Inventory Management
61
Order Quantity for a Periodic Inventory System
A periodic, or fixed-time period inventory system is one in
which time between orders is constant and the order size
varies.
Vendors make periodic visits, and stock of inventory is
counted.
An order is placed, if necessary, to bring inventory level
back up to some desired level.
Inventory not monitored between visits.
At times, inventory can be exhausted prior to the visit,
resulting in a stockout.
Larger safety stocks are generally required for the periodic
inventory system.
Chapter 16 - Inventory Management
62
Order Quantity for Variable Demand
For normally distributed variable daily demand:
Q  d (tb  L)  Z d tb  L  I
where:
d  average demand rate
tb  the fixed time between orders
L  lead time
 d  standard deviation of demand
Z d tb  L  safety stock
I  inventory in stock
Chapter 16 - Inventory Management
63
Order Quantity for Variable Demand Example
Corner Drug Store with periodic inventory system.
Order size to maintain 95% service level:
d  6 bottles per day
 d  1.2 bottles
tb  60 days
L  5 days
I  8 bottles
Z  1.65 for 95% service level
Q  d (tb  L)  Z d tb  L  I
 (6)(60  5)  (1.65)(1.2) 60  5 8
 398 bottles
Chapter 16 - Inventory Management
64
Order Quantity for the Fixed-Period Model
Solution with Excel
Exhibit 16.6
Chapter 16 - Inventory Management
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Example Problem Solution
Electronic Village Store (1 of 3)
For data below determine:
Optimal order quantity and total minimum inventory cost.
Assume shortage cost of $600 per unit per year,
compute optimal order quantity and minimum inventory
cost.
Step 1 (part a): Determine the Optimal Order Quantity.
D 1,200 personal computers
Cc  $170
Co  $450
Q  2CoD  2(450)(1,200)  79.7 personal computers
Cc
170
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Example Problem Solution
Electronic Village Store (2 of 3)




Q
1,200 
 79.7 
D
Total cost  Cc  Co 170

  450


 79.7 
Q
2
2




 $13,549.91
Step 2 (part b): Compute the EOQ with Shortages.
Cs  $600




2
(
450
)(
1200
)
 600 170 
2
CoD
 Cs  Cc 
Q



 



Cc  Cs 
170
600


 90.3 personal computers
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67
Example Problem Solution
Electronic Village Store (3 of 3)





Cc
170

S Q
 19.9 personal computers
  90.3
170  600 
Cc  Cs 







2 CoD
2
(
Q

S
)
CsS
Total cost 
 Cc

Q
2Q
2Q


2
2
(
600
)(
19
.
9
)
(
90
.
3

19
.
9
)
1,200 

170
 450



2(90.3)
2(90.3)
 90.3 
 $11,960.98
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Example Problem Solution
Computer Products Store (1 of 2)
Sells monitors with daily demand normally distributed with
a mean of 1.6 monitors and standard deviation of 0.4
monitors. Lead time for delivery from supplier is 15 days.
Determine the reorder point to achieve a 98% service level.
Step 1: Identify parameters.
d  1.6 monitors per day
L 15 days
 d  0.4 monitors per day
Z  2.05 (for a 98% service level)
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Example Problem Solution
Computer Products Store (2 of 2)
Step 2: Solve for R.
R  d L  Z d L  (1.6)(15)  (2.05)(.04) 15
 24  3.18  27.18 monitors
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Chapter 16 - Inventory Management
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