1. Managing Inventory Why do we have inventory?
Inventory Decisions (When & How Much)
Economic Order Quantity
Safety Stocks & Service Levels
The Newsboy Problem
Problems with Inventory

2. The Big Picture Inventory can improve ROI by various means
Improve utilization (i.e. lower investment)
Protects against blockage and starvation
Decouples downtimes of different operations
Increase sales
Decreases order-to-delivery cycle time
Decreases Stockouts
However, holding costs are very expensive (on the order of 20% of VC per year) which decreases ROI
Hence, the wrong amount of inventory in either direction could be disastrous

3. Inventory Costs
When deciding how much inventory to carry, we must balance three basic costs:
ordering cost - how much an order costs to process, independent of order size
holding cost - cost of capital tied up in
inventory, warehousing costs,
shrinkage, deterioration, obsolescence,
etc.
backorder cost - cost of alienating customers

4. THREE BASIC DECISIONS When to review? When to order? How much
Continuous
Reorder
point
Fixed
EOQ
(Economic order
quantity)
review
(Q,R)
Periodic
At review
time
Variable
Order-up-to
review
(T, TI)
When to
order?
How much
to order?
Policy 4

5. Simple Continuous Review
Continuous Review (Q,R) Policy in a deterministic system with no delivery lead-time: Q
inventory
time
Order CT
If D is the demand per unit time, and Q is the order quantity, what is the order cycle time?
What is the average inventory?

6. More Complicated (Q, R) Q+SS Q Q inventory R Q SS time
Let’s add in a little bit of randomness and a delivery lead-time (LT). Now we’ll need a reorder point (R), and a safety stock (SS).
Q+SS Q Q R
inventory Q SS
LTD
LTD
LTD
time
1st
Order
Placed
2nd
Order
Placed
3rd
Order
Placed
What is a stockout?
What is the average inventory (assuming no stockouts)?
Now we need to figure out a reasonably optimal order quantity and safety stock.

7. The EOQ Formula
Under certain conditions, we can calculate Q directly from a simple formula...
let D be the avg. demand per unit time
let C be the per-unit cost (to you) of inventory
let h be the inventory holding rate (watch time units)
let S be the ordering cost
We write total cost as follows
then take the derivative and set it equal to zero to find the optimum order quantity:

8. Optimal Reorder Point – R
What are the costs associated with R?
The general form for R is given by
Why do we need safety stock (SS)?

9. SS and Service Level The general form for SS is given by
where n depends on the target service level.
service level – Likelihood of not running out of stock over a reorder cycle
e.g. a 95% service level means, at most, a 5%
chance of a stock-out
If lead time demand U is normally distributed,

10. Aside on Standard Deviations
SS is a function of the standard deviation of lead time demand. . .
and you cannot simply add them together.
For example, if we have a std. deviation in daily demand of 20 units, and a delivery lead time of 12 days, then the std. deviation of lead-time demand is:

11. (Q,R) Example

12. (Q,R) Example (cont.)

13. Diagram of (T, TI) Policy
Q+SS I3
inventory I2 I1
TI-I2
TI-I3
TI-I1 SS
LTD
LTD
LTD
time
1st
Month’s
Order
Placed
2nd
Month’s
Order
Placed
3rd
Month’s
Order
Placed

14. Steps to determine T and TI
Periodic Review
Inventory is checked every T time units. Then a quantity is ordered to bring I back up to a target inventory level (TI).
Steps to determine T and TI
First we figure out Q like in the continuous model.
Then we calculate a reasonably optimal review period (T). T = Q/D.
To calculate our order point and safety stock, all we need to do is let
LTD+T = delivery lead time + review period
Then we calculate the SS as before.
Finally,
Question: Will the safety stock be larger or smaller under periodic review?

15. (T,TI) Example

16. Policy Comparison Continuous Periodic Review Review Order quantity
Order interval
Inv. monitoring
Mgmt. effort
Service Level for
same SS
Average Inventory
Applicability
(Hi or Lo Mix)
Supplier Mgmt. 5

17. Single-Period Inventory
Sometimes you must make a one-time decision as to how many lawnmowers to order for this season or how many airline seats to reserve on this flight...
 the Newsboy Problem
The basic tradeoffs are simple:
too high and you risk not selling them all
too few and you lose an opportunity to sell
(and maybe lose the customer for good!)
We use marginal analysis to find the best solution:
 stock Q if the opportunity cost of understocking exceeds the risk cost of overstocking

18. Newsboy Problem D = stochastic demand Q = purchase quantity
P = selling price (to customer) per unit
C = material cost (to you) per unit
G = unit cost of loss of goodwill due to shortage
V = salvage value
Co = cost of overstocking a unit = C - V
Cu = cost of understocking a unit = P - C + G
Expected marginal opportunity cost of understocking:
Expected marginal risk cost from overstocking:
Selecting the purchase quantity:

19. Newsboy Example I Example 1:
The McCormick Hardware Store places one order for riding lawn mowers each February. The lawn mowers being purchased this year cost $300 and sell for $425. In the past, McCormick has always been able to sell all of its surplus mowers during the September “end-of-summer” sale. The sales price for any surplus mowers this year is $250. If the following probability distribution for demand is assumed, how many mowers should be ordered?

20. Newsboy Example I

21. Newsboy Example II Example 2:
A retail outlet store sells bathing suits for $10 each. The cost to the store is $8 a suit. All suits not sold during the summer season are sold for half-price in an end-of-season clearance sale. Demand for the suits has historically been normally distributed with a mean of 500 and a standard deviation of 100.
a) What is the recommended order quantity?
b) What is the probability that someone will attempt to purchase a suit after the outlet is sold out?
c) Suppose that the owner’s policy is that, in order to keep customers happy and returning to the store, stockouts should be avoided if at all possible. What is your recommended order quantity if the owner wants no more than a 15% probability of stockout?
d) Using your answer to part c), what is the implicit goodwill cost you are assigning to a stockout? That is, how many dollars per suit is the owner implying that she would pay in order to avoid a stockout?

22. Newsboy Example II

23. Toyota Production System
Other Concepts
Not everybody thinks we need inventory:
The Goal
Toyota Production System
Make-to-Order Movement (Dell)