Let us suppose that we have the opportunity of ordering now a quantity
of some good, at unitary cost 
. Let us additionnaly
suppose that we know what is the probability distribution of the future
demand 
, as well as the exact unitary price 
at which
the selling will occur (non sold products being discarded). This situation
is known as ``the news' boy problem''.
As usual, we note 
the probability density, and 
the
cumulative distribution. The satisfied demand 
will be
if it happens that 
and otherwise, leading to
the gain 
. The
usual criterion to fix the optimal value 
of the ordered quantity
is maximizing the expected value of this gain. Denoting this
quantity by 
, we have:
Derivating, one obtains the condition 
,
i.e.
Let us now compare the eventual gain 
with the naive value 
, i.e.
with the gain that will occur if we buy right now the expectation
of the future demand and if, by chance, it
happens that we effectively sell these 
units. Defining 
as the probability of a too small order quantity and 
(resp.
) as the expected value of the demand knowing that 
(resp. knowing that 
), we have :
Since the right hand side is obviously positive, 
is the
Holy Graal of the problem. Without uncertainties, the value of
can be chosen in order to obtain for the expected gain (the
solution being 
. But in presence of uncertainties, the
value becomes unreachable.
Moreover, the difference
has
a clear meaning in terms of risks evaluation. A merchant has a risk
that the demand overflows his inventory .
And in this case, his score is burdened by the fact that he misses
the opportunity to sell 
units, leading to
an average miss to gain of 
. On the
other hand, the merchant has a risk 
that his inventory exceeds
the actual demand. And in that case, his score is burdened by the
resulting 
leftover units, leading to an
average extra cost of 
.
Therefore, the right hand side of eq:cost-of-uncertainty
is nothing but the cost of uncertainty. The best choice for
can decrease this cost, but it never vanishes.
