Inventory Management
What is an Inventory?
• Inventory
– Is a stock of items kept to meet demand.
– Forms of Inventory
•
•
•
•
•
•
•
•
Finished goods
Raw materials
Labor
Purchased parts & supplies
Work-in-process
Component parts
Working capital
Tools, machinery & equipment
Different Types Of Inventory
Inventory must be managed at various locations in the production
process.
. Raw materials or purchased parts
. Partially completed goods, called “work-in-progress (WIP)”
. Finished goods inventories (manufacturing organizations)
. Merchandise (retail organizations)
. Replacement parts, tools and supplies
. Goods-in-transit between locations (either plants,
warehouses, or customers)
Inventory Management Locations
Production Process
Work center
Work
center
Work center
Work
center
WIP
WIP
Receiving
Raw Materials
Finished
Goods
Inventory Management
• Determining the amount of inventory to keep in
stock
• How much to order and when to replenish, or
order.
Customer Demand
• The starting point for the management of
inventory.
• Inventory exists to meet customer demand.
• Two types of customers:
– Internal customers
– External customers
Demand & Inventory
• Determinant of effective inventory management is
an accurate forecast of demand.
• Forecasting & inventory are interrelated.
Independent Demand
Inventory: a stock or store of goods
Dependent Demand
A
C(2)
B(4)
D(2)
E(1)
D(3)
F(2)
Independent demand is uncertain.
Dependent demand is certain.
Inventory Costs
• 4 types of inventory costs:
– Purchasing Costs
– Carrying Costs (Holding costs)
– Ordering Costs
– Shortage Costs (Backorder costs)
Inventory Costs
• Carrying Costs (Holding costs)
– Are the costs of holding an item in inventory.
– The greater the level of inventory over a period of time, the higher the
carrying costs.
– Includes:
• Cost of losing the use of funds tied up with inventory
• Direct storage costs such as rent, heating, cooling, lighting, security, refrigeration,
record keeping & transportation
• Interest on loans used to purchase inventory
• Depreciation
• Obsolescence
• Product deterioration & spoilage
• Breakage
• Taxes
• Pilferage
Inventory Costs
• Ordering Costs
– Are the costs of replenishing inventory.
– As the number of orders increases, the ordering cost increases.
– Includes:
•
•
•
•
•
•
Requisition & purchase orders
Transportation & shipping
Receiving
Inspection
Handling & storage
Accounting & auditing costs
Inventory Costs
• Shortage Costs (Backorder Costs)
–
–
–
–
–
Also referred to stockout costs
Are temporary or permanent loss of sales when demand cannot be met.
Demand cannot be met because of insufficient inventory.
Difficult to determine.
Includes:
•
•
•
•
•
•
Loss of profits
Loss of goodwill
Price discounts or rebates
Work stoppages which can cause delays
Downtime costs
Cost of lost of production
– As the amount of inventory on hand increases, the carrying cost increases,
whereas shortage costs decrease.
Functions of Inventory
• To meet anticipated demand
• To smooth production requirements
• To decouple operations
• To protect against stock-outs
Functions of Inventory (Cont’d)
• To take advantage of order cycles
• To help hedge against price increases
• To permit operations
• To take advantage of quantity discounts
Objective of Inventory Control
• To achieve satisfactory levels of customer service
while keeping inventory costs within reasonable
bounds
– Level of customer service
– Costs of ordering and carrying inventory
Effective Inventory Management
•
•
•
•
A system to keep track of inventory
A reliable forecast of demand
Knowledge of lead times
Reasonable estimates of
– Holding costs
– Ordering costs
– Shortage costs
• A classification system
Economic Order Quantity Models
• Economic order quantity model
• Economic production model
• Quantity discount model
The Cost Model
To employ an inventory control system that will indicate how much should be ordered and
when orders should take place so that the sum of the costs will be minimized.
The EOQ Cost Model
• Is the optimal order quantity that will minimize
total inventory costs.
• Assuming that:
– Demand is known with certainty and is constant over
time
– No shortages allowed
– Lead time for the receipt of orders is constant
– The order quantity is received all at once.
How Much? When? To Order
How much and when to order depends on many factors including:
ordering costs, carrying costs, lead times, variability in lead times,
variability in demand, variability in production, etc..
The Economic Order Quantity (EOQ) is the order size (how much?)
that minimizes the total cost of inventory.
How Much?
The Reorder Point (ROP) is the inventory
When! point
(when?) which triggers a reorder.
The Inventory Cycle
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place
order
Receive
order
Lead time
Place
order
Receive
order
Time
In the real world
In practice, demand is rarely known with certainty. Demand
is a random variable.
Inventory
level
(I)
Demand
Time (T)
Basic EOQ – Carrying Cost
Carrying Cost
Q
Carrying Cost =
H where
2
Q = Order quantity
H = Holding (carrying) cost per unit
Annual Carrying Cost is linearly
related to the Order Quantity
Order Quantity (Q)
Ordering Cost
Basic EOQ – Ordering Cost
D
Ordering Cost =
S where
Q
Q = Order quantity
D = Demand, usually in unit per year
S = Ordering Cost
Ordering Cost decreases as
Order Quantity increases;
however not linearly
Order Quantity (Q)
17 - 24
Basic EOQ – Total Cost
Total Cost
Total Cost (TC )  Carrying Cost  Ordering Cost
Q
D
TC  H  S
2
Q
Order Quantity (Q)
Cost Minimization Goal
Annual Cost
The Total-Cost Curve is U-Shaped
Q
D
TC  H  S
2
Q
Ordering Costs
QO (optimal order quantity)
Order Quantity (Q)
Basic EOQ – Instantaneous Delivery
The Basic Economic Order – instantaneous delivery model EOQ
is the quantity which minimizes Total Cost = Carrying Cost +
Ordering Cost. It is where Carrying Cost = Order Cost and is
calculated by:
Total Cost
2DS
Basic EOQ = Q0 =
H
Q0
Length of Order Cycle =
D
Basic EOQ
The EOQ Formula
Q = 2DS
H
where
Q = economic order quantity/size
D = annual demand
S = ordering/transaction cost
H = annual holding cost
Minimum Total Cost
The total cost curve reaches its minimum where
the carrying and ordering costs are equal.
Q OPT =
2DS
=
H
2(Annual Demand )(Order or Setup Cost )
Annual Holding Cost
Basic EOQ – Example
Example a: A local distributor for a national tire company
expects to sell 9,600 steel belted radial tires of a certain size and
tread design next year. Annual Carrying Cost is $16 per tire, and
Ordering Cost is $75. The distributor operates 288 days per year.
What is the EOQ?
2DS
Econmic Order Quantity = Q 0 =
H
2(9,600)(75)
=
= 300
16
Basic EOQ – Example
Example b: How many times per year does the tire distributor reorder
tires?
D 9,600
Number of Orders Per Year =
=
= 32
Q0
300
Example c: What is the length of the order cycle (Cycle Time)?
Q0
300
Length of Order Cycle =
=
=.03125
D 9,600
therefore
since there are 288 days in the year the
Order Cycle = .03125*288 = 9 days
Basic EOQ – Example
Example d: What is the Total Annual Cost if the EOQ is ordered?
Q
D
300
9,600
TC 
H S 
(16) 
(75)
2
Q
2
300
= $2,400 + $2,400 = $4,800
Basic EOQ – Example
Example a: Piddling Manufacturing assembles security monitors. It
purchases 3,600 black and white cathode ray tubes (CRT’s) at $65
each. Ordering costs are $31, and annual carrying costs are 20% of the
purchase price. Compute the optimal order quantity.
S = $31
H = .20($65) = $13
2DS
Econmic Order Quantity = Q0 =
H
2(3,600)(31)
=
= 131
13
Basic EOQ – Example
Example b: Compute the total annual ordering cost for the optimal
order quantity.
Q
D
131
3,600
TC 
H S 
(13) 
(31)
2
Q
2
131
= $852 + $852 = $1,704
EOQ – Non-instantaneous Delivery Model
The basic EOQ model assumes instantaneous delivery;
however, many times an organization produces items
to be used in the assembly of products. In this case
the organization is both a producer and user. Orders
for items may be replenished (non-instantaneously)
over time rather than instantaneously.
EOQ – Non-instantaneous Delivery Model
Consider the situation where a toy manufacturer makes dump trucks.
. The manufacturer also produces (production rate) the rubber
wheels that are used in the assembly of the dump trucks.
Let’s consider 500 per day for example.
. In this case the ordering costs associated with an order for
rubber wheels would be the cost associated with the setup
and delivery of the rubber wheels to the dump truck
assembly area.
. The manufacturer makes the dump trucks at constant rate
per day (production rate). Let’s consider 200 per day for
example.
EOQ – Non-instantaneous Delivery Model
Non-Instantaneous Inventory (p=500, u=200)
6000
5000
Units
4000
3000
2000
1000
0
0
5
10
15
20
Time
Cumulative Production
Cumulative Usage
Inventory On Hand
25
EOQ – Non-instantaneous Delivery Model
As you see in this example, the inventory (shown in yellow) depends
on the production rate (shown in blue) and the usage rate (shown in
pink).
How much to order depends on setup costs and carrying costs. The
Economic Order Quantity (EOQ) is the order size that minimizes the
total cost of inventory. Sometimes this is referred to as the Economic
Run Quantity because it is dependent on the cumulative
manufacturing production quantity.
Total cost = Carrying costs + Setup costs.
A schematic of the non-instantaneous considerations are shown on the
next slide.
EOQ – Non-instantaneous Delivery Model
Production/Usage
(EOQ - Run Size)
Maximum Inventory
Cumulative
Production
Amount
on hand
Usage Only
Economic Production Quantity (EPQ)
• Production done in batches or lots
• Capacity to produce a part exceeds the part’s
usage or demand rate
• Assumptions of EPQ are similar to EOQ except
orders are received incrementally during
production
Economic Production Quantity
Assumptions
•
•
•
•
•
•
•
Only one item is involved
Annual demand is known
Usage rate is constant
Usage occurs continually
Production rate is constant
Lead time does not vary
No quantity discounts
EOQ – Non-instantaneous Delivery Model
Econmic Run Quantity = Q 0 =
p = production rate
S = ordering cost
D = annual demand
2DS
H
p
p-u
u = usage rate
H = carrying cost
Q0
Maximum Inventory = I max =
(p - u)
p
I max
Average Inventory =
2
where
EOQ – Non-instantaneous Delivery Model
Minimum Total Cost = TC min = Carring Cost + Setup Cost
I max
D
=
( H) +
(S)
2
Q0
Q0
Cycle Time =
u
Q0
Run Time =
p
Non-instantaneous - Example
Example a: A toy manufacturer uses 48,000 rubber wheels per
year for its popular dump truck series. The firm makes its own
wheels, which it can produce at a rate of 800 per day. The toy
trucks are assembled uniformly over the entire year. Carrying
cost is $1 per wheel per year. Setup cost for a production run of
wheels is $45. The firm operates 240 days per year. Determine
the optimal run size.
Non-instantaneous - Example
p = production or delivery rate = 800 wheels per day
D = annual demand = 48,000 wheels
48,000
u = usage rate =
= 200 wheels per day
240
S = ordering cost = $45
H = carrying cost = $1 per wheel per year
2(48,000)45)
800
Econmic Run Quantity = Q0 =
1
800 - 200
= 2,400 wheels
Non-instantaneous - Example
Example b: Compute the minimum total cost for carrying and setup.
2,400
Maximum Inventory = I max =
(800 - 200)
800
= 1,800 wheels
TC min = Carring Cost + Setup Cost
I max
D
=
( H) +
(S)
2
Q0
1,800
48,000
=
(1) +
(45)
2
2,400
= 900 + 900 = $1,800
Non-instantaneous - Example
Example c: Compute the cycle time for the optimal run size.
2,400
Cycle Time =
 12 days
200
thus a run of wheels will be made every 12 days
Example d: Compute the run time for the optimal run size.
2,400
Run Time =
= 3 days
800
thus each production of wheels will take 3 days
When to Reorder with EOQ Ordering
• Reorder Point - When the quantity on hand of an
item drops to this amount, the item is reordered
• Safety Stock - Stock that is held in excess of
expected demand due to variable demand rate
and/or lead time.
Reorder Point
ROP = d x LT
where:
d = demand rate (units per day or week)
LT = lead time in days or weeks
Determinants of the Reorder Point
•
•
•
•
The rate of demand
The lead time
Demand and/or lead time variability
Stockout risk (safety stock)
Quantity
Safety Stock
Maximum probable demand
during lead time
Expected demand
during lead time
ROP
Safety stock reduces risk of
stockout during lead time
Safety stock
LT
Time
When To Reorder
If variability in demand or lead time is present the ROP is calculated
using the following general formula:
ROP  Expected demand during lead time + Safety Stock
Safety Stock - stock that is held in excess of expected demand due to
the variability in demand rate and/or lead time
For example: If the expected demand during lead time is 100 units
and the desired amount of safety stock is 10 units then
ROP = 100 + 10
ROP - Demand & Lead Time Variability
It is rarely the case in business where demand & lead time are
constant.
Variability can exist because of many reasons
(customers, transportation, etc.); therefore, we
Q
must
consider these impacts on inventory.
Quantity
on hand
Usage/demand
rate
Reorde
r
point
Receive
order
Place
order
Receive Place
order order
Lead time 1
Receive
order
Lead time 2
Place
order
Receive
order
Lead time 3
Time
Safety Stock
The calculation of safety stock depends on the variability of demand,
lead time and the service level the organization desires.
Service Level – is the proportion of customer orders that are serviced
on-time. Customers usually understand that 100% of their orders will
not be serviced on-time and will establish standards for service.
By developing a probability distribution of demand during lead time,
a company can use statistical calculations which determine how
much safety stock is necessary to meet customer service
requirements. In this case, the supply of inventory on hand a company
must have to meet customer requirements is calculated by supply
(inventory on hand) = expected demand + safety stock. This is
depicted on the next slide.
Safety Stock
Probability
distribution of
quantity of demand
during lead time
“Service Level”
probability of no stock out
Expected
demand
Safety Stock
Risk of
a stock out
EOQ With Quantity Discount
EOQ with Quantity Discount is very important because price
reductions are frequently offered to induce customers to order in
larger quantities.
Why do you think this is done?
In this model the purchasing cost will vary depending on the quantity
purchased. Purchasing cost was omitted in the previous EOQ models
because the price per unit was the same for all units; thus, the
inclusion of the purchase cost would only increase the total cost
function by the purchase cost amount. Thus, it would have had no
effect on the EOQ calculation.
EOQ Without Quantity Discount
Adding Purchasing cost
w/o quantity discount
doesn’t change EOQ
TC with Purchasing
Cost
Cost
TC without Purchasing
Cost
Purchasing Cost – same for all
units
EOQ
Quantity
EOQ With Quantity Discount
In this model, how much to order depends on purchase costs, setup
costs and carrying costs. The Economic Order Quantity (EOQ) is the
order size that minimizes the total cost of inventory. Sometimes this is
referred to as the Economic Run Quantity because it is dependent on
the cumulative manufacturing production quantity.
Total cost = Carrying costs + Ordering costs + Purchasing Cost
Total Cost (TC )
 Carrying Cost  Ordering Cost  Purchasing Cost
Q
D
TC  H  S  PD
2
Q
where P  unit price
EOQ With Quantity Discount
The Method of Computing EOQ with Quantity Discount is a step wise
process.
. First, compute the common EOQ using the earlier formula
. Second,
.. Identify the price range where the common EOQ lies
.. If the common EOQ is in the lowest quantity
range then the EOQ with quantity discount is the
common EOQ
.. Otherwise, the EOQ with quantity discount is the
quantity where the total cost is minimum when
considering the cost for the common EOQ and the
cost for all minimum quantities of price breaks
greater than the common EOQ.
EOQ With Quantity Discount - Example
Example a: The maintenance department of a large hospital uses 816
cases of liquid cleanser annually. Ordering costs are $12, carrying costs
are $4 per case per year, and the price schedule for ordering is listed
below. Determine the optimal order quantity and the total cost.
There are 240 days in a year.
Order Price Per
Quantity
Box
1 to 49
20.00
50 to 79
18.00
80 to 99
17.00
100 or more 16.00
D = annual demand = 816 cases / year
S = ordering cost = $12
H = carrying cost = $4 per case per year
EOQ With Quantity Discount - Example
2DS
Common EOQ = Q0 =
H
Order Price Per
Quantity
Box
1 to 49
20.00
50 to 79
18.00
80 to 99
17.00
100 or more 16.00
2(816)(12)
=
= 70 cases
4
Common EOQ = 70
Is in the second price break
EOQ With Quantity Discount - Example
Order Price Per
Quantity
Box
1 to 49
20.00
50 to 79
18.00
80 to 99
17.00
100 or more 16.00
Common EOQ = 70
Is in the second price break
Therefore; the
EOQ with quantity discount = minimum (TC(70), TC (80), TC (100))
EOQ With Quantity Discount - Example
TC70
70
816

(4) +
12 + 18(816) = $14,968
2
70
TC80
80
816

(4) +
12 + 17(816) = $14,154
2
80
TC100
100
816

(4) +
12 + 16(816) = $13,354
2
100
Therefore; the total cost is minimum for an order quantity of 100
and the
EOQ with quantity discount = 100
Operations Strategy
• Too much inventory
– Tends to hide problems
– Easier to live with problems than to eliminate them
– Costly to maintain
• Wise strategy
– Reduce lot sizes
– Reduce safety stock
The Friendly Sausage Factory (FSF) can produce hot dogs at a rate of
5,000 per day. FSF supplies hotdogs to local restaurants at a steady
rate of 250 per day. The cost to prepare the equipment for producing
the hot dogs is $66. Annual holding costs are $.45 per hot dog. The
factory operates 300 days per year. Answer the following questions.
a. What is the annual demand for hot dogs?
b. What is the production run which will minimize total costs?
c. Based on the optimal production run in b., what is the maximum
inventory?
d. What is the average inventory?
e. How many times will FSF have to make hot dogs every year?
f. When FSF starts a manufacturing run of hot dogs, how long will
they run the machine (i.e. what is the run time)?
g. How long will it be from the end of a manufacturing run, until
another run needs to be made (i.e. what is the pure consumption
time – the difference between the order cycle time and the run
time)?
A chemical company produces sodium bisulfate in 100
pound bags. Demand for the product is 20 tons per day.
The capacity for production is 50 tons per day. Setup
cost is $100, and storage and handling costs are $5 per
ton per year. The firm operates 200 days per year.
Answer the following questions.
a. What is the annual demand in tons?
b. How many bags per manufacturing run are optimal?
c. What is the average inventory in bags for the optimal
run size?
d. What is the manufacturing run time?
e. What is the pure consumption time?
A mail-order house uses 18,000 boxes a year. Carrying costs are
$.60 per box per year, and ordering costs are $96. The following
price schedule applies.
Number of Boxes
Price per Box
1,000 to 1,999
$1.25
2,000 to 4,999
$1.20
5,000 to 9,999
$1.15
10,000 or more
$1.10
Answer the following questions.
a. What is the optimal order quantity?
b. What is the common EOQ?
c. How many orders will the mail-order house have to make during
the year?
d. What is the carrying cost, ordering cost, purchase cost and total
cost for the optimal quantity?
A jewelry company buys semiprecious stones to make bracelets and rings. The
supplier quotes a price of $5 per stone for quantities of 600 stones or more, $9 for
orders of 400 to 599, and $10 per stone for lesser quantities. The jewelry firm
operates 200 days per year. Usage is 25 stones per day, and ordering costs are $48.
a. If carrying costs are $2 per year for each stone, answer the following questions.
i. What is the optimal ordering quantity?
ii. What is the common EOQ?
iii. How many orders per year will the company make?
iv. What is the total ordering cost for the optimal solution?
b. If carrying costs are 30 percent of the purchase price per year for each stone,
answer the following questions.
i. What is the carrying cost per unit when 400 to 599 stones are ordered?
ii. What is the optimal ordering quantity?
iii. What is the common EOQ?
iv. How many orders per year will the company make?
v. What is the total purchase cost for the optimal solution?
c. If the lead time is 6 working days, at what point should the company reorder?