The newsboy problem is as follows : we have the opportunity to purchase
now an amount 
of some good, at unitary cost 
(regardless
of the quantity purchased). It is assumed that the future demand distribution
is exactly known by its cumulative density function 
,
that the future unit selling price 
is known and independent
of the number sold and that non sold units are discarded. In any case,
considering a salvation value 
is only transforming 
into 
.
The satisfied demand will be 
, leading
to the actual gain : 
.
The usual criterion used to fix the optimal quantity 
to buy
is maximizing the expected value of this gain. Denoting this expectation
by 
and taking the derivatives, one obtains the condition 
,
i.e. the well known:
Let us introduce the following notations. 
is the maximal gain when the demand is known, 
(resp.
) is the expected value of the demand knowing that the
demand is over (resp. under) the inventory, i.e. 
(resp.
), and 
is the probability that the demand
exceeds the inventory (). When the value of is
clear from the context, the dependence on will not be emphasized.
In other words:
A straightforward computation leads to:
what is
the cost generated by the order : you have a risk 
that
the demand 
overflows your inventory and then you miss
the opportunity to sell 
units and, on the
other hand, you have a risk 
that your inventory overflows
the demand and then you have to deal with 
leftover units.
This formula is often written as 
where 
. Introducing Eq. 1
into Eq. 2 we obtain the remaining cost
associated with the best decision. This cost, that can be interpreted
as the cost of uncertainties, is given by:
