Statistical Inventory
control models II
Newsboys Model
Learning objective

After this class the students should be
able to:
• Apply optimization techniques to inventory model
• calculate the appropriate order quantity in the
•
face of uncertain demand.
analyze the implication of the “Newboys Model”
Time management

The expected time to deliver this module
is 50 minutes. 30 minutes are reserved
for team practices and exercises and 20
minutes for lecture.
Classic model



Wilson (1934), in this classic paper, he breaks
the inventory control problem into two distinct
parts:
1. Determining the order quantity, which is
the amount of inventory that will be purchased
or produced with each replenishment.
2. Determining the reorder point, or the
inventory level at which a replenishment
(purchase or production) will be triggered.
First part: replenishment

In this section, we address this two-part problem in
three stages (two classes): First, we consider the
situation where we are only interested in a single
replenishment, so that the only issue is to determine
the appropriate order quantity in the face of
uncertain demand. This has traditionally been called
the newsboy model because it could apply to a
person who purchases newspapers at the beginning
of the day, sells a random amount, and then must
discard any leftovers.
Second part: reorder point

Second, we consider the situation where
inventory is replenished one unit at a
time as random demands occur, so that
the only issue is to determine the reorder
point. The target inventory level we set
for the system is known as a base stock
level, and hence the resulting model is
termed the base stock model.
Third part: (Q, r) model

Third, we consider the situation where
inventory is monitored continuously and
demands occur randomly, possibly in batches.
When the inventory level reaches (or goes
below) r, an order of size Q is placed. After a
lead-time of l, during which a stockout might
occur, the order is received. The problem is to
determine appropriate values of Q and r. The
model we use to address this problem is
known as the (Q, r) model
The Newboys model

Consider the situation a manufacturer of Christmas lights
faces each year. Demand is somewhat unpredictable and
occurs in such a short burst just prior to Christmas that if
the inventory is not on the shelves, the demand will be
lost. Therefore, the decision of how many sets of lights to
produce must be made prior to the holiday season.
Additionally, the cost of collecting unsold inventory and
holding it until next year is too high to make year-to-year
storage an attractive option. Instead, any unsold sets of
lights are sold after Christmas at a steep discount.
Appropriate quantity production

To choose an appropriate production
quantity, the important pieces of
information to consider are:
(1) anticipated demand, and
(2) the costs of producing too much or too
little.
Notation
X  Demand (in units), a random variable
dG ( x )
G x  
 density function of demand
dx
co  Cost ( in dollar ) per unit left over after demand is realized
c s  Cost ( in dollar ) per unit of shortage ; and
Q  Production/order quantity (in units); is the decision variable
Christmas Light Example

Suppose that a set of lights costs $1 to make and distribute and is
selling for $2.

Any sets not sold by Christmas will be discounted to $0.5. In terms of
the above modeling notation, this means that the unit overage cost is
the amount lost per excess set or co = $(1- 0.5) = $0.5.


The unit shortage cost is the lost profit from a sale or cs = $(2 - 1) = $1.
Suppose further that demand has been forecast to be 10,000 units
with a standard deviation of 1,000 units and that the normal distribution
is a reasonable representation of demand.
Christmas Light Example

The firm could choose to produce 10,000 sets of lights. But, the
symmetry (i.e., bell shape) of the normal distribution implies
that it is equally likely for demand to be above or below 10,000
units. If demand is below 10,000 units, the firm will lose co =
$0.5 per unit of overproduction. If demand is above 10,000
units, the firm will lose cs = $1 per unit of underproduction.
Clearly, shortages are worse than overages.

This suggests that perhaps the firm should produce more than
10,000 units. But, how much more?
Overage

If we produce Q units and demand is X units, then the number of
units of overage is given by
Units Over  max Q-X, 0
Q  X
Units over  
0
if Q  X   0 (overage)
if Q  X
( shortage)
Expected overage
E (Units over)   max Q  x, 0g ( x)dx
Q
0
  max Q  x g ( x)dx
Q
0
Shortage
Units short  max X-Q, 0
X  Q
Units over  
0
if X  0 ( shortage)
if Q  X
(overage)
Expected Shortage

E (Units short )   max x  Q, 0g ( x)dx
Q

  max x  Q g ( x)dx
Q
The expected cost
Y (Q)  co  max Q  x g ( x)dx
Q
0

 cs  max x  Q g ( x)dx
Q
Optimization

“Q” that minimizes the expected cost can be
find applying the Leibnitz's rule:
a2 ( Q ) 
d a2 ( Q )

f  x, Q dx  
f  x, Q dx

a1 ( Q ) Q
dQ a1 (Q )
da2 (Q)
 f a 2 (Q), Q 
dQ
da1 (Q)
 f a1 (Q ), Q 
dQ
Optimization

Applying the Leibnitz’s rule
Q

dY (Q)
 co  (1) g ( x)dx  cs  (1) g ( x)dx
0
Q
dQ
 coG (Q)  cs 1  G (Q)   0
cs
 G (Q ) 
co  cs
*
To minimize expected overage plus shortage cost, we
have to choose a production or order quantity Q* that
satisfies:
cs
G (Q ) 
co  cs
*
When G(x) increases in x, so that anything that makes
the right-hand side of the equation larger will result in a
larger Q*. This implies that increasing cs will increase Q*,
while increasing co will decrease Q*, as we would
intuitively expect.
Christmas Light Example

As its demand is normally distributed,
 Q*  10,000 

G (Q )  
 1,000 
1

 0.67
1  0.5
*
where Ф represents the cumulative distribution function of the standard
normal distribution. From a standard normal table, we find that Ф(0.44) =
0.67. Hence, we have that
G (0.44)  0.67
Q * 10,000
 0.44
1,000
or Q*  10,440

Reflections
Each team is invited to analyze the
implications of the Newsboys Model,
based on the results found (20).
Reference

Factory Physics. Hopp & Spearmen,
Irwin, 1996. Chapter 2.