This document discusses inventory management concepts such as what inventory is, why companies hold inventory, and how to determine optimal inventory levels. It defines key inventory management terms like economic order quantity (EOQ), reorder point (ROP), safety stock, and service level. The document provides examples to illustrate how to calculate EOQ, determine ROP given lead time and demand variability, and optimize total inventory costs by balancing ordering, holding, and stockout costs. The overall aim of inventory management is to match supply and demand in the most cost-effective way.

1. Inventory Management

2. What Is Inventory?
• Stock of items kept to meet future demand
• Purpose of inventory management
– how many units to order
– when to order

3. Inventory Management System

4. Supply Chain Management
• demand information is distorted as it moves away
from the end-use customer
• higher safety stock inventories to are stored to
compensate
• Seasonal or cyclical demand
• Inventory provides independence from vendors
• Take advantage of price discounts
• Inventory provides independence between stages and
avoids work stoppages
1
3
-
4

5. – Sales growth: right inventory at the right place
at the right time
– Cost reduction: less money tied up in inventory,
inventory management, obsolescence
Higher profit
Why do we care?
At the firm level:

6. How do you manage your inventory?
How much do you buy? When?

7. What Do you Consider?
• Cost of not having it.
• Cost of going to the ordering or going to shop.
• Cost of holding and storing, lost interest.
• Price discounts.
• How much you consume.
• Some safety against uncertainty.

8. Inventory Costs
• Carrying cost
– cost of holding an item in inventory
• Ordering cost
– cost of replenishing inventory
• Shortage cost
– temporary or permanent loss of sales when
demand cannot be met
1
3
-
8

9. • Hedge against uncertain demand
• Hedge against uncertain supply
• Economize on ordering costs
• Smoothing
Benefits of Inventory
To summarize, we build and keep inventory in
order to match supply and demand in the most
cost effective way.

10. Modeling Inventory in a Supply Chain…
Warehouse
Industry
Supplier

11. Inventory Control Systems
.
1
3
-
1
1 • Continuous system (fixed-order-quantity)
– constant amount ordered when inventory declines
to predetermined level
• Periodic system (fixed-time-period)
– order placed for variable amount after fixed
passage of time

12. Multiperiod model
• Key questions:
– How often to review?
– When to place an order?
– How much to order?
– How much stock to keep?
orders
Supply
On-hand
inventory
• Ordering costs
• Holding costs

13. Multiperiod model – The Economic Order Quantity
• Demand is known and deterministic: D units/year
• We have a known ordering cost, S, and immediate replenishment
• Annual holding cost of average inventory is H per unit
• Purchasing cost C per unit
Supplier Demand
Industry

14. What is the optimal quantity to order?
Total Cost = Purchasing Cost + Ordering Cost + Inventory Cost
Purchasing Cost = (total units) x (cost per unit)
Ordering Cost = (number of orders) x (cost per order)
Inventory Cost = (average inventory) x (holding cost)

15. Finding the optimal quantity to order…
Let’s say we decide to order in batches of Q…
Number of
periods will be
D
Q
Time
Total Time
Period over which demand for Q has occurred
Q
Inventory position
The average
inventory for
each period is…
Q
2

16. Finding the optimal quantity to order…
Purchasing cost = D x C
Inventory cost =
Ordering cost =
D
Q
x S
Q
2
x H

17. So what is the total cost?
TC = D C + +
In order now to find the optimal quantity we need to
optimize the total cost with respect to the decision
variable (the variable we control)
D
Q
S
Q
2
H

18. What is the main insight from EOQ?
There is a tradeoff between holding costs and ordering costs
Order Quantity (Q*)
Cost
Total cost
Holding costs
Ordering costs

19. Economic Order Quantity - EOQ
Q* =
2SD
H
Example:
Assume a mining company that faces demand for 5,000 te
explosive per year, and that it costs Rs. 15,000 to have the
explosive shipped to the mines. Holding cost is estimated at Rs.
500 per te per year. How many times should the company order,
and what should be the order size?
te
Q 548
500
)
000
,
5
)(
000
,
15
(
2
*



20. Receive
order
Time
Inventory
Order
Quantity
Q
Place
order
Lead Time
If delivery is not instantaneous, but there is a lead
time L:
When to order? How much to order?

21. ROP = LxD
Receive
order
Time
Inventory
Order
Quantity
Q
Place
order
Lead Time
Reorder
Point
(ROP)
If demand is known exactly, place an order when
inventory equals demand during lead time.
D: demand per period
L: Lead time in periods
Q: When shall we order?
A: When inventory = ROP
Q: How much shall we order?
A: Q = EOQ

22. Example (continued)…
What if the lead time to receive explosive is 10
days? (when should you place your order?)
10
365
D =
R =
10
365
5000 = 137
So, when the explosive on the stock reaches 137 te,
order 548 te more.
Since D is given in years, first convert: 10 days = 10/365yrs

23. Time
Time
Inventory
Inventory
Level
Level
Order
Order
Quantity
Quantity
But demand is rarely predictable!
Demand???
Receive
order
Place
order Lead Time
ROP = ???

24. X
X
Inventory at time of receipt
Receive
Receive
order
order
Time
Time
Inventory
Inventory
Level
Level
Order
Order
Quantity
Quantity
Place
Place
order
order
Lead Time
Lead Time
Actual Demand < Expected Demand
ROP
Lead Time Demand

25. Stockout
Point
Unfilled demand
Receive
Receive
order
order
Time
Inventory
Order
Order
Quantity
Quantity
Place
Place
order
order
Lead Time
Lead Time
If Actual Demand > Expected, we Stock Out

26. ROP = Expected Demand
Average
Time
Inventory
Level
Order
Quantity
If ROP = expected demand, service level is
50%. Inventory left 50% of the time, stock
outs 50% of the time.
Uncertain Demand

27. To reduce stockouts we add safety stock
Receive
Receive
order
order
Time
Time
Place
Place
order
order
Lead Time
Lead Time
Inventory
Level
ROP =
Safety
Stock +
Expected
LT
Demand
Order Quantity
Q = EOQ
Expected
LT Demand
Safety Stock

28. Service level
Safety
Stock
Probability
of stock-out
Decide what Service Level you want to provide
(Service level = probability of NOT stocking out)

29. Service level
Safety
Stock
Probability
of stock-out
Safety stock =
(safety factor z)(std deviation in LT demand)
Read z from Normal table for a given service level

30. Caution: Std deviation in LT demand
Variance over multiple periods = the sum of
the variances of each period (assuming
independence)
Standard deviation over multiple periods is
the square root of the sum of the variances,
not the sum of the standard deviations!!!

31. Average Inventory =
(Order Qty)/2 + Safety Stock
Receive
Receive
order
order
Time
Time
Place
Place
order
order
Lead Time
Lead Time
Inventory
Level
Order
Quantity
Safety Stock (SS)
EOQ/2
Average
Inventory

32. How to find ROP & Q
1. Order quantity Q =
2. To find ROP, determine the service level (i.e., the
probability of NOT stocking out.)
 Find the safety factor from a z-table or from the graph.
 Find std deviation in LT demand: square root law.
 Safety stock is given by:
SS = (safety factor)(std dev in LT demand)
 Reorder point is: ROP = Expected LT demand + SS
3. Average Inventory is: SS + EOQ/2
2SD
EOQ
H


33. Example (continued)…
Back to the explosive requirement… recall that the lead
time is 10 days and the expected yearly demand is 5000 te.
You estimate the standard deviation of daily demand to be
d = 6. When should you re-order if you want to be 95%
sure you don’t run out of explosive?
168
)
36
(
10
65
.
1
137 


ROP
Since the expected yearly demand is 5000 te, the expected
demand over the lead time is 5000(10/365) = 137 te. The z-
value corresponding to a service level of 0.95 is 1.65. So
Order 548 te explosives when the inventory level drops to 168 te.

34. Problem 1: Assume you have a product with the following parameters:
Demand  360
Holding cost per year  $1.00 per unit
Order cos : $100
t  per order
What is the EOQ?
P2: Given the data from Problem 1, and assuming a 300-day work year; how many orders should be processed
per year? What is the expected time between orders?
P3: What is the total cost for the inventory policy used in Problem 1?
P4: Assume that the demand was actually higher than estimated (i.e., 500 units instead of 360 units). What will
be the actual annual total cost?
P5: If demand for an item is 3 units per day, and delivery lead-time is 15 days, what should we use for a re-order
point?

35. Problem 1:
2*Demand*Order cost 2*360*100
EOQ 72000 268 items
Holding cost 1
   
Problem 2:
Demand 360
1.34 orders per year
268
N
Q
  
Working days
300/1.34 224 days between orders
Expected number of orders
T   

36. Problem 3:
Demand*Order Cost (Quantity of Items)*(Holding Cost)
TC
Q 2
360*100 268*1
134 134 $268
268 2
 
    
Problem 4:
Demand*Order Cost (Quantity of Items)*(Holding Cost)
TC
Q 2
500*100 268*1
186.57 134 $320.57
268 2
 
    
Note that while demand was underestimated by nearly 50%, annual cost increases
by only 20% ( / . )
320 268 120
 an illustration of the degree to which the EOQ model
is relatively insensitive to small errors in estimation of demand.
Problem 5:
ROP Demand during lead-time units
  
3 15 45
*

37. Problem :
We need 1,000 electric drills per year. The ordering cost for these is $100 per order and the carrying cost is
assumed to be 40% of the per unit cost. In orders of less than 120, drills cost $78; for orders of 120 or more,
the cost drops to $50 per unit.
Should we take advantage of the quantity discount?
* (2)(1000)(100)
($78) 80 units
(0.4)(78)
p
Q  
* (2)(1000)(100)
($50) 100 units 120 to take advantage of quantity discount.
(0.4)(50)
p
Q   
Ordering 100 units at $50 per unit is not possible; the discount does not apply until 120 the
order equals 120 units. Therefore, we need to compare the total costs for the two alternatives.
Demand*Order Cost (Quantity of Items)*(Holding cost)
Total cost Demand*Cost
2
Q
  
(1000)(100) (80)(0.4)(78)
Total cost($78) (1000)(78) $80,498
80 2
   
(1000)(100) (120)(0.4)(50)
Total cost($50) (1000)(50) $52,033
120 2
   
Therefore, we should order 120 each time at a unit cost of $50 and a total cost of $52,033.