Managing InventoryManaging Inventory

Why do we have inventory?

Inventory Decisions (When & How Much)

Economic Order Quantity

Safety Stocks & Service Levels

The Newsboy Problem

Problems with Inventory

The Big PictureThe 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

Inventory CostsInventory 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

THREE BASIC DECISIONS

THREE BASIC DECISIONS

When toreview?

When toorder?

How much to order?

Continuous

Reorderpoint

FixedEOQ

(Economic orderquantity)

Continuousreview(Q,R)

Periodic

At reviewtime

VariableOrder-up-to

Periodicreview(T, TI)

4

Policy

Simple Continuous ReviewSimple Continuous Review

Continuous Review (Q,R) Policy in a deterministic system with no delivery lead-time:

time

inve

ntor

y

Q

0

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?

Order CT

More Complicated (Q, R)More Complicated (Q, R)

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).

time

inve

ntor

y

Q

0

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.

SS

1stOrderPlaced

RQ

2ndOrderPlaced

3rdOrderPlaced

Q

Q+SS

LTD LTD LTD

The EOQ FormulaThe EOQ Formula

Under certain conditions, we can calculate Q directly from a simple formula...

let D be the avg. demand per unit timelet C be the per-unit cost (to you) of inventorylet 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:

Optimal Reorder Point – ROptimal 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)?

SS and Service LevelSS 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,

n Service Level0 50.0%1 84.0%2 97.7%3 99.9%

Service Level n75% 0.6790% 1.2895% 1.6599% 2.33

2~ ( , )UU Norm U

Aside on Standard DeviationsAside 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:

(Q,R) Example(Q,R) Example

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

Diagram of (T, TI) PolicyDiagram of (T, TI) Policy

time

inve

ntor

y

TI-I1

0

SS

1stMonth’sOrderPlaced

TI

2ndMonth’sOrderPlaced

3rdMonth’sOrderPlaced

Q+SS

T

TI-I2 TI-I3

T T

LTDLTD LTD

I3

I2I1

Periodic ReviewPeriodic 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?

SSUQTI

(T,TI) Example(T,TI) Example

Policy ComparisonPolicy 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

Single-Period InventorySingle-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

Newsboy ProblemNewsboy 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:

Newsboy Example INewsboy 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?

Demand Probability0 0.101 0.152 0.303 0.204 0.155 0.10

Newsboy Example INewsboy Example I

Newsboy Example IINewsboy 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?

Newsboy Example IINewsboy Example II

Not everybody thinks we need inventory:

• The Goal

• Toyota Production System

• Make-to-Order Movement (Dell)

Other ConceptsOther Concepts