Chapter 6
Inventory Control Models
To accompany
Quantitative Analysis for Management, Tenth Edition,
by Render, Stair, and Hanna
Power Point slides created by Jeff Heyl
© 2008 Prentice-Hall, Inc.
© 2009 Prentice-Hall, Inc.
Introduction
 Inventory is an expensive and important





asset to many companies
Lower inventory levels can reduce costs
Low inventory levels may result in stockouts
and dissatisfied customers
Most companies try to balance high and low
inventory levels with cost minimization as a
goal
Inventory is any stored resource used to
satisfy a current or future need
Common examples are raw materials, workin-process, and finished goods
© 2009 Prentice-Hall, Inc.
6–2
Introduction
 Basic components of inventory
planning
 Planning what inventory is to be
stocked and how it is to be acquired
(purchased or manufactured)
 This information is used in
forecasting demand for the inventory
and in controlling inventory levels
 Feedback provides a means to revise
the plan and forecast based on
experiences and observations
© 2009 Prentice-Hall, Inc.
6–3
Introduction
 Inventory may account for 50% of the total
invested capital of an organization and 70% of the
cost of goods sold
Energy
Costs
Capital
Costs
Labor Costs
Inventory
Costs
© 2009 Prentice-Hall, Inc.
6–4
Introduction
 All organizations have some type of inventory
control system
 Inventory planning helps determine what
goods and/or services need to be produced
 Inventory planning helps determine whether
the organization produces the goods or
services or whether they are purchased from
another organization
 Inventory planning also involves demand
forecasting
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6–5
The Inventory Process
Suppliers
Customers
Inventory Storage
Raw
Materials
Finished
Goods
Fabrication/
Work in
Assembly
Process
Inventory Processing
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6–6
Introduction
 Inventory planning and control
Planning on What
Inventory to Stock
and How to Acquire
It
Forecasting
Parts/Product
Demand
Controlling
Inventory
Levels
Feedback Measurements
to Revise Plans and
Forecasts
Figure 6.1
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6–7
Importance of Inventory Control
 Five uses of inventory
 The decoupling function
 Storing resources
 Irregular supply and demand
 Quantity discounts
 Avoiding stockouts and shortages
 The decoupling function
 Used as a buffer between stages in a
manufacturing process
 Reduces delays and improves efficiency
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6–8
Importance of Inventory Control
 Storing resources
 Seasonal products may be stored to satisfy
off-season demand
 Materials can be stored as raw materials,
work-in-process, or finished goods
 Labor can be stored as a component of
partially completed subassemblies
 Irregular supply and demand
 Demand and supply may not be constant over
time
 Inventory can be used to buffer the variability
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6–9
Importance of Inventory Control
 Quantity discounts
 Lower prices may be available for larger orders
 Cost of item is reduced but storage and insurance
costs increase, as well as the chances for more
spoilage, damage and theft.
 Investing in inventory reduces the available funds
for other projects
 Avoiding stockouts and shortages
 Stockouts may result in lost sales
 Dissatisfied customers may choose to buy from
another supplier
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6 – 10
Inventory Decisions
 There are only two fundamental decisions
in controlling inventory
 How much to order
 When to order
 The major objective is to minimize total
inventory costs
 Common inventory costs are
 Cost of the items (purchase or material cost)
 Cost of ordering
 Cost of carrying, or holding, inventory
 Cost of stockouts
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6 – 11
Inventory Cost Factors
ORDERING COST FACTORS
CARRYING COST FACTORS
Developing and sending purchase orders
Cost of capital
Processing and inspecting incoming
inventory
Taxes
Bill paying
Insurance
Inventory inquiries
Spoilage
Utilities, phone bills, and so on, for the
purchasing department
Theft
Salaries and wages for the purchasing
department employees
Obsolescence
Supplies such as forms and paper for the
purchasing department
Salaries and wages for warehouse
employees
Utilities and building costs for the
warehouse
Supplies such as forms and paper for the
warehouse
Table 6.1
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6 – 12
Inventory Cost Factors
 Ordering costs are generally independent
of order quantity
 Many involve personnel time
 The amount of work is the same no matter the
size of the order
 Carrying costs generally varies with the
amount of inventory, or the order size
 The labor, space, and other costs increase as
the order size increases
 Of course, the actual cost of items
purchased varies with the quantity
purchased
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6 – 13
Economic Order Quantity
 The economic order quantity (EOQ)
model is one of the oldest and most
commonly known inventory control
techniques
 It dates from a 1915 publication by
Ford W. Harris
 It is still used by a large number of
organizations today
 It is easy to use but has a number of
important assumptions
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6 – 14
Economic Order Quantity

Assumptions
1. Demand is known and constant
2. Lead time (the time between the placement and
receipt of an order) is known and constant
3. Receipt of inventory is instantaneous

Inventory from an order arrives in one batch, at
one point in time
4. Purchase cost per unit is constant throughout
the year; no quantity discounts
5. The only variable costs are the placing an order,
ordering cost, and holding or storing inventory
over time, holding or carrying cost, and these are
constant throughout the year
6. Orders are placed so that stockouts or shortages
are avoided completely
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6 – 15
Inventory Usage Over Time
 Inventory usage has a sawtooth shape
 Inventory jumps from 0 to the maximum when the shipment arrives
 Because demand is constant over time, inventory drops at a uniform
rate over time
Inventory
Level
Order Quantity = Q =
Maximum Inventory Level
Minimum
Inventory
0
Figure 6.2
Time
© 2009 Prentice-Hall, Inc.
6 – 16
EOQ Inventory Costs

The objective is to minimize total costs



The relevant costs are the ordering and
carrying/holding costs, all other costs are constant.
Thus, by minimizing the sum of the ordering and
carrying costs, we are also minimizing the total costs
The annual ordering cost is the number of orders per
year times the cost of placing each order
As the inventory level changes daily, use the average
inventory level to determine annual holding or carrying
cost


The annual carrying cost equals the average inventory
times the inventory carrying cost per unit per year
The maximum inventory is Q and the average inventory
is Q/2.
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6 – 17
Inventory Costs in the EOQ Situation
 Objective is generally to minimize total cost
 Relevant costs are ordering costs and carrying
costs
Average inventory
level 
Q
2
INVENTORY LEVEL
DAY
BEGINNING
ENDING
AVERAGE
April 1 (order received)
10
8
9
April 2
8
6
7
April 3
6
4
5
April 4
4
2
3
April 5
2
0
1
Maximum level April 1 = 10 units
Total of daily averages = 9 + 7 + 5 + 3 + 1 = 25
Number of days = 5
Average inventory level = 25/5 = 5 units
Table 6.2
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6 – 18
Inventory Costs in the EOQ Situation
 Develop an expression for the ordering cost.
 Develop and expression for the carrying cost.
 Set the ordering cost equal to the carrying
cost.
 Solve this equation for the optimal order
quantity, Q*.
© 2009 Prentice-Hall, Inc.
6 – 19
Inventory Costs in the EOQ Situation
 Mathematical equations can be developed using
Q
EOQ
D
Co
Ch
= number of pieces to order
= Q* = optimal number of pieces to order
= annual demand in units for the inventory item
= ordering cost of each order
= holding or carrying cost per unit per year
Annual ordering cost 

Number of
Ordering
orders placed  cost per
per year
order
Annual Demand
Cost per order
Number of units per order

D
Q
Co
© 2009 Prentice-Hall, Inc.
6 – 20
Inventory Costs in the EOQ Situation
 Mathematical equations can be developed using
Q
EOQ
D
Co
Ch
= number of pieces to order
= Q* = optimal number of pieces to order
= annual demand in units for the inventory item
= ordering cost of each order
= holding or carrying cost per unit per year
Average
Annual holding cost  inventory

Q
2
Carrying
 cost per unit
per year
Ch
 Total Inventory Cost =
D
Q
Co 
Q
Ch
2
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6 – 21
Inventory Costs in the EOQ Situation
Optimal Order Quantity is when the Total Cost curve is at its
lowest . This occurs when the Ordering Cost = Carrying Cost
Cost
Curve of Total Cost
of Carrying
and Ordering
Minimum
Total
Cost
Carrying Cost Curve
Ordering Cost Curve
Figure 6.3
Optimal
Order
Quantity
Order Quantity
© 2009 Prentice-Hall, Inc.
6 – 22
Finding the EOQ
 When the EOQ assumptions are met, total cost is
minimized when Annual ordering cost = Annual
holding cost
D
Q
Co 
Q
2
Ch
 Solving for Q
 Q Ch
2
2 DC
o
2 DC
o
Q
o
 Q  EOQ  Q
2
Ch
2 DC
Ch
*
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6 – 23
Economic Order Quantity (EOQ) Model
 Summary of equations
Annual ordering cost 
D
Q
Annual holding cost 
EOQ  Q 
*
Q
2
Co
Ch
2 DC
o
Ch
© 2009 Prentice-Hall, Inc.
6 – 24
Sumco Pump Company Example
 Company sells pump housings to other
companies
 Would like to reduce inventory costs by finding
optimal order quantity
 Annual demand = 1,000 units
 Ordering cost = $10 per order
 Average carrying cost per unit per year = $0.50
Q 
*
2 DC
Ch
o

2 ( 1,000 )( 10 )
0 . 50

40 ,000  200 units
© 2009 Prentice-Hall, Inc.
6 – 25
Sumco Pump Company Example
Total annual cost = Order cost + Holding cost
TC 
D
Q

Co 
1, 000
200
Q
2
Ch
( 10 ) 
200
2
( 0 .5 )
 $ 50  $ 50  $ 100
© 2009 Prentice-Hall, Inc.
6 – 26
Sumco Pump Company Example
Program 6.1A
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6 – 27
Sumco Pump Company Example
Program 6.1B
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6 – 28
Purchase Cost of Inventory Items
 Total inventory cost can be written to include the
cost of purchased items
 Given the EOQ assumptions, the annual purchase
cost is constant at D  C no matter the order
policy
 C is the purchase cost per unit
 D is the annual demand in units
 It may be useful to know the average dollar level
of inventory
Average
dollar level 
(CQ )
2
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6 – 29
Purchase Cost of Inventory Items
 Inventory carrying cost is often expressed as an
annual percentage of the unit cost or price of the
inventory
 This requires a new variable
I
Annual inventory holding charge as
a percentage of unit price or cost
 The cost of storing inventory for one year is then
C h  IC
thus,
Q 
*
2 DC
o
IC
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6 – 30
Sensitivity Analysis with the
EOQ Model
 The EOQ model assumes all values are know and
fixed over time
 Generally, however, the values are estimated or
may change
 Determining the effects of these changes is
called sensitivity analysis
 Because of the square root in the formula,
changes in the inputs result in relatively small
changes in the order quantity
EOQ 
2 DC
o
Ch
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6 – 31
Sensitivity Analysis with the
EOQ Model
 In the Sumco example
EOQ 
2 ( 1,000 )( 10 )
0 . 50
 200 units
 If the ordering cost were increased four times from
$10 to $40, the order quantity would only double
EOQ 
2 ( 1,000 )( 40 )
0 . 50
 400 units
 In general, the EOQ changes by the square root
of a change to any of the inputs
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6 – 32
Reorder Point:
Determining When To Order
 Once the order quantity is determined, the next
decision is when to order
 The time between placing an order and its
receipt is called the lead time (L) or delivery
time
 Inventory must be available during this period to
met the demand
 When to order is generally expressed as a
reorder point (ROP) – the inventory level at
which an order should be placed
ROP 
Demand
per day 
Lead time for a
new order in days
dL
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6 – 33
Determining the Reorder Point
 The slope of the graph is the daily inventory
usage
 Expressed in units demanded per day, d
 If an order is placed when the inventory level
reaches the ROP, the new inventory arrives at
the same instant the inventory is reaching 0
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6 – 34
Procomp’s Computer Chip Example
 Demand for the computer chip is 8,000 per year
 Daily demand is 40 units
 Delivery takes three working days
ROP  d  L  40 units per day  3 days
 120 units
 An order is placed when the inventory reaches
120 units
 The order arrives 3 days later just as the
inventory is depleted
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6 – 35
Inventory Level (Units)
The Reorder Point (ROP) Curve
Q*
Slope = Units/Day = d
ROP
(Units)
Lead Time (Days)
L
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6 – 36
EOQ Without The
Instantaneous Receipt Assumption
 When inventory accumulates over time, the
instantaneous receipt assumption does not apply
 Daily demand rate must be taken into account
 The revised model is often called the production
run model
Inventory
Level
Part of Inventory Cycle
During Which Production is
Taking Place
There is No Production
During This Part of the
Inventory Cycle
Maximum
Inventory
t
Time
Figure 6.5
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6 – 37
EOQ Without The
Instantaneous Receipt Assumption
 Instead of an ordering cost, there will be a
setup cost – the cost of setting up the
production facility to manufacture the desired
product
 Includes the salaries and wages of employees
who are responsible for setting up the
equipment, engineering and design costs of
making the setup, paperwork, supplies,
utilities, etc.
 The optimal production quantity is derived by
setting setup costs equal to holding or
carrying costs and solving for the order
quantity
© 2009 Prentice-Hall, Inc.
6 – 38
Annual Carrying Cost for
Production Run Model
 In production runs, setup cost replaces ordering
cost
 The model uses the following variables
Q  number of pieces per order, or
production run
Cs  setup cost
Ch  holding or carrying cost per unit per
year
p  daily production rate
d  daily demand rate
t  length of production run in days
© 2009 Prentice-Hall, Inc.
6 – 39
Annual Carrying Cost for
Production Run Model
Maximum inventory level
 (Total produced during the production run)
– (Total used during the production run)
 (Daily production rate)(Number of days production)
– (Daily demand)(Number of days production)
 (pt) – (dt)
Total produced  Q  pt
since
we know
t 
Q
p
Maximum
Q
Q
d 

inventory  pt  dt  p  d  Q  1  
p
p
p

level
© 2009 Prentice-Hall, Inc.
6 – 40
Annual Carrying Cost for
Production Run Model
 Since the average inventory is one-half the
maximum
Average inventory
Q
d 

1 
2 
p
and
Q
d 
Annual holding cost 
 1  C h
2 
p
© 2009 Prentice-Hall, Inc.
6 – 41
Annual Setup Cost for
Production Run Model
 Setup cost replaces ordering cost when a
product is produced over time
(independent of the size of the order and
the size of the production run)
Annual setup cost 
D
Q
Cs
and
Annual ordering cost 
D
Q
Co
© 2009 Prentice-Hall, Inc.
6 – 42
Determining the Optimal
Production Quantity
 By setting setup costs equal to holding costs, we
can solve for the optimal order quantity
Annual holding cost  Annual setup cost
Q
d 
D
1

C

Cs

 h
2 
p
Q
 Solving for Q, we get
Q 
*
2 DC
s
d 

Ch1 
p

© 2009 Prentice-Hall, Inc.
6 – 43
Production Run Model
 Summary of equations
Q
d 
 1  C h
2 
p
D
Annual setup cost 
Cs
Q
Annual holding cost 
Optimal production
quantity Q 
*
2 DC
s
d 

Ch1 
p

If the situation does not involve production but receipt
of inventory over a period of time, use the same model
but replace Cs with Co
© 2009 Prentice-Hall, Inc.
6 – 44
Brown Manufacturing Example
 Brown Manufacturing produces commercial
refrigeration units in batches
Annual demand  D  10,000 units
Setup cost  Cs  $100
Carrying cost  Ch  $0.50 per unit per year
Daily production rate  p  80 units daily
Daily demand rate  d  60 units daily
How many refrigeration units should Brown produce
in each batch?
How long should the production cycle last?
© 2009 Prentice-Hall, Inc.
6 – 45
Brown Manufacturing Example
1.
2.
Q 
*
Q 
*

2 DC
s
Production
d 

Ch1 
p

cycle 
Q
p

2  10 , 000  100
4 , 000
80
 50 days
60 

0 .5  1 

80 

2 ,000 ,000
 4
0 .5 1

16 ,000 ,000
 4,000 units
© 2009 Prentice-Hall, Inc.
6 – 46
Brown Manufacturing Example
Program 6.2A
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6 – 47
Brown Manufacturing Example
Program 6.2B
© 2009 Prentice-Hall, Inc.
6 – 48
Quantity Discount Models
 Quantity discounts are commonly available
 The basic EOQ model is adjusted by adding in the
purchase or materials cost
Total cost  Material cost + Ordering cost + Holding cost
Total cost  DC 
D
Q
Co 
Q
2
Ch
where
D  annual demand in units
Cs  ordering cost of each order
C  cost per unit
Ch  holding or carrying cost per unit per year
© 2009 Prentice-Hall, Inc.
6 – 49
Quantity Discount Models
Because
unitare
cost
is now variable
 Quantity
discounts
commonly
available
Holding
 Ch  IC
 The basic EOQ
modelcost
is adjusted
by adding in the
purchase
materials
cost
I  holdingorcost
as a percentage
of the unit cost (C)
Total cost  Material cost + Ordering cost + Holding cost
Total cost  DC 
D
Q
Co 
Q
2
Ch
where
D  annual demand in units
Cs  ordering cost of each order
C  cost per unit
Ch  holding or carrying cost per unit per year
© 2009 Prentice-Hall, Inc.
6 – 50
Quantity Discount Models
 A typical quantity discount schedule
DISCOUNT
NUMBER
DISCOUNT
QUANTITY
DISCOUNT (%)
DISCOUNT
COST ($)
1
0 to 999
0
5.00
2
1,000 to 1,999
4
4.80
3
2,000 and over
5
4.75
Table 6.3
 Buying at the lowest unit cost is not always the
best choice
© 2009 Prentice-Hall, Inc.
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Quantity Discount Models
 Total cost curve for the quantity discount model
Total
Cost
$
TC Curve for Discount 3
TC Curve for
Discount 1
TC Curve for Discount 2
EOQ for Discount 2
0
Figure 6.6
1,000
2,000
Order Quantity
© 2009 Prentice-Hall, Inc.
6 – 52
Brass Department Store Example
 Brass Department Store stocks toy race cars
 Their supplier has given them the quantity
discount schedule shown in Table 6.3
 Annual demand is 5,000 cars, ordering cost is $49, and
holding cost is 20% of the cost of the car
 The first step is to compute EOQ values for each
discount
EOQ
EOQ
EOQ
1

( 2 )( 5 , 000 )( 49 )
( 2 )( 5 , 000 )( 49 )
2


( 2 )( 5 , 000 )( 49 )
3
( 0 . 2 )( 5 . 00 )
( 0 . 2 )( 4 . 80 )
( 0 . 2 )( 4 . 75 )
 700 cars per order
 714 cars per order
 718 cars per order
© 2009 Prentice-Hall, Inc.
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Brass Department Store Example
 The second step is adjust quantities below the
allowable discount range
 The EOQ for discount 1 is allowable
 The EOQs for discounts 2 and 3 are outside the
allowable range and have to be adjusted to the
smallest quantity possible to purchase and
receive the discount
Q1  700
Q2  1,000
Q3  2,000
© 2009 Prentice-Hall, Inc.
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Brass Department Store Example
 The third step is to compute the total cost for
each quantity
DISCOUNT
NUMBER
UNIT
PRICE
(C)
ORDER
QUANTITY
(Q)
ANNUAL
MATERIAL
COST ($)
= DC
ANNUAL
ORDERING
COST ($)
= (D/Q)Co
ANNUAL
CARRYING
COST ($)
= (Q/2)Ch
TOTAL ($)
1
$5.00
700
25,000
350.00
350.00
25,700.00
2
4.80
1,000
24,000
245.00
480.00
24,725.00
3
4.75
2,000
23,750
122.50
950.00
24,822.50
Table 6.4
 The fourth step is to choose the alternative
with the lowest total cost
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6 – 55
Brass Department Store Example
Program 6.3A
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6 – 56
Brass Department Store Example
Program 6.3B
© 2009 Prentice-Hall, Inc.
6 – 57
Use of Safety Stock
 If demand or the lead time are uncertain,
the exact ROP will not be known with
certainty
 To prevent stockouts, it is necessary to
carry extra inventory called safety stock
 Safety stock can prevent stockouts when
demand is unusually high
 Safety stock can be implemented by
adjusting the ROP
© 2009 Prentice-Hall, Inc.
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Use of Safety Stock
 The basic ROP equation is
ROP  d  L
d  daily demand (or average daily demand)
L  order lead time or the number of
working days it takes to deliver an order
(or average lead time)
 A safety stock variable is added to the equation
to accommodate uncertain demand during lead
time
ROP  d  L + SS
where
SS  safety stock
© 2009 Prentice-Hall, Inc.
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Use of Safety Stock
Inventory
on Hand
Time
Figure 6.7(a)
Stockout
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Use of Safety Stock
Inventory
on Hand
Safety
Stock, SS
Stockout is Avoided
Time
Figure 6.7(b)
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