Importance of Inventory
A typical hospital spends about 20% of its budget on medical, surgical, and
pharmaceutical supplies. For all hospitals it adds up to $150 billion annually.
The average inventory in US economy about 1.13 trillion on 9.66 trillion of
sales. About $430 billion in manufacturing, $230 billion in wholesaler, $411
billion in retail.
Inventory management has a trade-off decision between level of Customer
Service and Inventory Cost.
How do we measure Customer Satisfaction? Number and quantities of sales
lost, back orders, customer complains.
What happens when a company with a large WIP and FG inventory finds a
market demand shift to a new product? Two choices:
Fire-sell all WIP & FG inventories and then quickly introduce the new
product  Significant losses
Finish all WIP inventory and sell all output before introducing the new
product  Delay and reduced market response time
Inventory Classified
Inputs Inventory
– Raw materials and Parts
In-process inventory
– Flow units that are being processed
– Flow units to decouple operations (line balancing inventory).
– Flow units produced to take advantage of Economics of Scale (batch
inventory).
Outputs Inventory
– To meet anticipated customer demand (average inventory and safety
stock).
– To smooth production while meeting seasonal demand (seasonal
inventory).
– In transit to a final destination to fill the gap between production and
demand lead times (pipeline inventory).
Input Inventory, In Process Inventory, Output Inventory
Engine Plant
Assembly Plant
Dealership
Automobile Firm
Process
Input Inventory
In-Prcess Inventory
Output Inventory
Engine plant
Castings
Unfinished engines
Finished engines
Assembly plant
Finished engines,
chassis, etc
Unfinished Automobile
Automobile
Dealership (sales)
Automobile
Automobile
-
Inventory
Operations, finance, and marketing have interest in
inventories.
Poor inventory management hampers operations, diminishes
customer satisfaction, and increases operating costs.
A typical firm probably has tied in inventories about
30 percent of its current assets
90 percent of its working capital (Current Assets – Current Liabilities)
Both Understocking and Overstocking are undesirable;
Understocking; lost sales, dissatisfied customers, production
lost.
Overstocking; tied up funds, physical holding cost,
obsolescence.
Objectives of Inventory Control
Inventory management has a trade-off decision between level of Customer
Service and Inventory Cost.
How do we measure Customer Satisfaction? Number and quantities of sales
lost, back orders, customer complains.
Inventory Turns Per Year
Industry
Upper Quartile
Median
Lower Quartile
Dairy
34.4
19.3
9.2
Electronic Components
9.8
5.7
3.7
Computers
9.4
5.3
3.5
Publishing
9.8
2.4
1.3
Consumer Electronics
6.2
3.4
2.3
Appliances
8.0
5.0
3.8
Industrial Chemical
10.3
6.6
4.4
Periodic Inventory (Counting) Systems
Physical count of items made at periodic intervals.
Disadvantage: no information on inventory between two
counts.
Advantage: order for several items are made at the
same time.
At each count, the inventory level is identified and the
required volume to satisfy the demand during the
period (until the next count) is ordered.
The quantity of order is variable but the timing of order is
fixed.
Re-Order Point (ROP) is defined in terms of time.
Perpetual Inventory Systems
Keeps track of removals from inventory continuously,
thus monitoring current levels of each item.
A point-of-sales (POS) system may record items at
the time of sale.
When inventory reaches ROP an order of EOQ
(Economic Order Quantity) units is places.
The quantity of order is fixed but the timing of order is
variable.
ROP is defined in terms of quantity.
Bin Systems
One-Bin System (Periodic)
Order Enough to
Refill Bin
Two-Bin System (Perpetual)
Order One Bin of
Inventory
Full
Empty
Economics of Scale
Economies of Scale (EoS): when average unit cost of output
decreases with volume. Such as large quantity discounts
(Economies from price discounts), or a total fixed cost
which is independent from volume (Economies from fixed
cost of procurement)
– Fixed order cost of purchasing or fixed setup cost of production
does not depend on the volume, the larger the volume the smaller
the cost per unit.
– EoS of Procurement, EoS of Production, EoS of Transportation.
– We often refer to the order or production in response to the
economies of scale as batch; production batch, procurement batch,
transfer batch.
Inventory Costs
Opportunity cost of capital tied up in inventory; The foregone return on the
funds invested in inventory which could have been invested in alternative
projects.
Physical holding costs; warehouse rent, insurance, security, lighting,
heating, cooling, spoilage, obsolescence
Obsolescence costs; cost of a market demand shift to a new product (we
may also include it in opportunity cost).
Opportunity cost of inventory is rC, where r is firm’s rate of return
Physical holding costs per unit of time (typically a year) is expressed as a
fraction h of the variable cost of C of acquiring (or producing) one flow
unit of inventory.
Physical holding cost = hC
Cumulative cost of holding one flow unit of inventory is therefore
H= Physical Holding Cost + Opportunity Cost = (h+r)C
11
Basic Inventory Model
Only one product
Annual demand is known
Demand is constant throughout the year
Each order is received in a single delivery
Lead time does not vary
No quantity discount
-Two costs
 Holding or Carrying Costs: Cost to carry an item in inventory for
one year
 Ordering Costs: Costs of ordering and receiving inventory
Unit cost of product is not incorporated because we assume it is fixed. If
there is quantity discount, then we need unit cost of product.
If inventory carrying cost is stated in terms of a percentage of the unit cost of
the product, then we need unit cost of product.
Ordering Policy
The optimal order quantity reflects a trade-off between
carrying cost and order cost.
As order size increases, the order cost decreases, while
carrying cost increases.
When the quantity on hand is just sufficient to satisfy
demand in lead time (ROP), an order for EOQ is placed.
Since there is no variation neither in usage rate nor in
lead time, the order will be received at the instant that
the inventory on hand falls to zero.
The Basic Inventory Model
Annual demand for a product is 9600
R = 9600
Annual carrying cost per unit of product is 16$
H = 16
Ordering cost per order is 75
S = 75
a) How much should we order each time to minimize our
total cost
b) How many times should we order
c) What is the length of an order cycle (working days
288/year)
d) What is the total cost
Do NOT worry if you do not get integer numbers
Discussion
Discuss with the students
The Inventory Cycle
Usage
rate
Usage
rate
Q
Profile of Inventory Level Over Time
Quantity
on hand
Reorder
point
Receive
order
Place Receive
order order
Lead time
Place
order
Receive
order
TimeTime
The Inventory Cycle
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place Receive
order order
Lead time
Place
order
Receive
order
Time
The Inventory Cycle
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place Receive
order order
Lead time
Place
order
Receive
order
Time
The Inventory Cycle
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place Receive
order order
Lead time
Place
order
Receive
order
Time
The Inventory Cycle
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place Receive
order order
Lead time
Place
order
Receive
order
Time
Ordering Cost
R = Demand in units / year
Q = Order quantity in units / order
Number of orders / year =
R
Q
S = Order cost / order
Annual order cost =
R
S
Q
Annual
Cost
Annual Ordering Cost
R
S
Q
Ordering Costs
Order Quantity
(Q)
Carrying cost
Q = Order quantity in units / order
At the beginning of the period we get Q units.
At the end of the period we have 0 units.
Q
Q/2
0
Average Inventory / Period
Q = Order quantity in units / order
At the beginning of the period we get Q units.
At the end of the period we have 0 units.
Q0 Q

2
Average inventory is = 2
This is average inventory / period.
What is average inventory / year
Cycle Inventory: The average inventory
Average Inventory / year
Time
Time
Inventory Carrying Cost
Q = Order quantity in units / order
Q
2
Average inventory / year =
H = Inventory carrying cost / unit / year
Annual carrying cost =
Q
H
2
Annual
Cost
Annual Carrying Cost & Annual Ordering Cost
Q
H
2
Carrying Costs
Order Quantity
(Q)
Total Cost
Annual
Annual
Total cost = carrying + ordering
cost
cost
TC =
Q
H
2
+
RS
Q
Example
Annual demand for a product is 9600
D = 9600
Annual carrying cost per unit of product is 16$
H = 16
Ordering cost per order is 75
S = 75
a) How much should we order each time
to minimize our total cost
b) How many times should we order
c) what is the length of an order cycle (working days 288/year)
d) What is the total cost
EOQ
Q OPT =
2RS
=
H
2(Annual Demand )(Order or Setup Cost )
Annual Holding Cost
TC  (Q / 2) H  ( R / Q) S
Using calculus, we take the derivative of the total
cost function with respect to Q and set the
derivative (slope) equal to zero and solve for Q.
What is the Optimal Order Quantity
2RS
EOQ 
H
R = 9600, H = 16, S = 75
2(9600)(75)
EOQ 
 300
16
How many times should we order
Annual demand for a product is 9600
R = 9600
Economic Order Quantity is 300
EOQ = 300
Each time we order EOQ
What is Cycle inventory?
Cycle inventory is Average inventory. EOQ/2 = 150
How many times should we order ?
R/EOQ
9600/300 = 32
what is the length of an order cycle
working days = 288/year
9600 is required for 288 days
300 is enough for how many days?
(300/9600)(288) = 9 days
Compute Flow Time
9600T = 150
T = 4.5 days
What is the Optimal Total Cost
The total cost of any policy is computed as
TC  (Q / 2) H  ( R / Q) S
The economic order quantity is 300
TC  (300 / 2)16  (9600 / 300)75
TC  2400  2400
TC  4800
This is the total cost of the optimal policy
Other EOQ Examples
Joe Smith needs to drive 2 miles to the closest ATM. He
withdraws money weekly.
Ordering cost? Driving time and cost. This periodic
withdrawal leaves a cycle inventory of money for Joe.
Carrying cost? The interest on the average cash he has.
Big Blue runs a Bus service. Running the bus on a specific
route has fixed costs. Batch Size = The people who
arrive between 2 consecutive trips. Cycle Inventory =
Average number of people waiting to board the bus.
The City of Pittsburg collects trash from its residents every
week on Monday. The average inventory of trash in the
household constitutes that house’s cycle inventory.
Centura Health Hospital
A Centura health hospital processes a demand of 600 units of IV starter kit
each week and places an order of 6000 units at a time.
What is the ordering cycle?
6000/600 =10 week
How many orders per year? (year = 52 weeks)
52/10 = 5.2
Or
R = 600×52 = 31200 units per year
31200/6000 = 5.2
What is cycle inventory
6000/2 = 3000
How long a typical IV unit stays in inventory
RT = I
600T = 3000
T = 5 weeks
31200T=3000
T = .096 year or 5 weeks
Centura Health Hospital: Traditional Order Size
A Centura health hospital incurs a cost of $130 regardless the
quantity purchased each time it places an order. S = $130. Cost
of each unit is C = $3, and R = 600/week, or 31200/year
assuming 52 weeks per year. Inventory carrying cost is $0.9 per
unit per year H = $0.90, Q = 6000
Total annual fixed order cost =
S(R/Q) = 130(31200/6000) = $676
Total annual holding cost =
H(Q/2) = 0.90(6000/2) = $2700
Total annual purchasing cost =
CR = 3(31200) = $93600
Total annual cost = TC = S(R/Q) + H(Q/2) + CR
= 676 + 2700 + 93600 = $96976
Centura Health Hospital: EOQ
2RS
EOQ 
H
2(130)( 31200)
EOQ 
 3002
0.9
Compute Cycle Inventory
I = EOQ/2 = 1501 unit
Compute Total Cost
TC = 130(31200/3002) + 0.9(3002/2) + 3(31200)
TC = 96302
Compute Average Flow Time
T = I /R = 1501/600 = 2.5 weeks
cycle
i
cycle
Managerial Insight
How Managers Could Reduce EOQ
2RS
EOQ 
H
Insight: How Managers Could Reduce EOQ
Fixed Order Cost Reduction: In order to decrease the
optimal order size we only have two ways:
– Reduce S
– Centralize
Current Situation: Batches of 3,002 Starter Kits
– Cycle inventory of 1,501
– Adds 2.5 weeks to flow time of IV Starter Kits
New Situation: reduce cycle inventory by half
– Reduce order size by 1,501
– Changes flow time to 1.25 weeks
– Must reduce S to $32.50 from $130
Fixed Order Cost Reduction
Is not only applied to order cost in Procurement,
but also fixed costs in Transportation, and
Production
– Reduce Procurement Fixed Cost
eCommerce; electronic purchase orders
– Reduce Transportation Fixed Cost
Changing the transportation mode, ex. Ship to large truck,
truck to air
– Reduce Production Fixed Cost
Setup cost reduction: a major factor in lean operations, JIT
systems
Inventory vs. Sales Growth
Optimal batch is proportionate to the square root of
outflow rate. Doubling company’s annual sales
does not require a doubling of cycle inventories, i.e.,
inventory growth should not track sales growth
Quadruples outflow rate  doubles EOQ
Doubles Cycle inventory and Flow time
With increase in company’s annual sales , EOQ
increases, however, we need to order more
frequently
Centralization
• Decentralized
– Nine hospitals order supplies independently
• Centralization
– Centralized purchasing of all supplies
• Must order for total output flow rate 9 times the output
flow rate of each hospital.
• Store supplies in central warehouse
• Average inventory only three times (equal to sq. rt. of 9)
that of decentralized warehouse
• Consolidated order can be split and delivered to meet
requirements of respective hospitals.
Centura Health Hospital: Decentralized
Nine hospitals, each orders independently, S =
$130/order, H = $0.90/unit/year, Flow Rate = 600
units/week (31,200 per year)
2 RS
EOQ 
 3002
H
Holding cost = .9(3,002/2) = 1,351
Ordering cost = 130(600×52)/3,002 =1,351
Total cost per hospital = 2,702
Total cost of all hospitals = 9(2,702) = 24,318
Cycle Inventory per hospital = EOQ/2 = 1,501 units
Cycle Inventory for Centura Health = 9×1,501 = 13,509
Average flow time I /R = (9×1501)/(9×600) = 2.5 weeks
cycle
Centura Health Hospital: Centralized
Centura switches purchasing via central warehouse,
total flow rate to be met from new order process is
9×600×52= 280,800, S = 130, H = $0.90/unit/year.
EOQ 
2 RS
2(280800)(130)

 9006
H
.9
Holding cost = .9(9,006/2) = 4,053
Ordering cost = 130(280,800)/9,006 =4,053
Total cost of all hospitals = 8,106
Total cost of all hospitals under decentralized ordering: 24,318
Cycle Inventory for Centura Health = 9,006/2 = 4503
Cycle Inventory per hospital = (4,503/9) = 500 unit
Average flow time I /R = 500/600 = less than one week
cycle
Total Cost of EOQ
TC 
TCEOQ
SR
Q
H
Q
2
SR
EOQ

H
EOQ
2
TCEOQ 
S 2 R2 H
2 H 2 RS

2 RS
4H
TCEOQ  2
RSH
2
2 RS
EOQ 
H
TC EOQ
TCEOQ 
SR
H
2 RS
H
2 RS
H
2
RSH
RSH

2
2
TCEOQ  2 RSH
Assignment
Problem 6.3. Note that h is .3 or 30%. (Do NOT use 20%. The 20% is inside the
30%).
Problem 6.10.