Let us consider what is known as "the newsboy problem"
and use the notations of [Scarf 1958]. 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 
.
In this Section, we will only examine what can be inferred from these
hypotheses and postpone the discussion of their validity to later
Sections. 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 
, we have:
,
i.e. the well known:
is known, the best quantity you can buy
is not the expectation of the demand.
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. Let us denote by 
(resp. 
) the expected value of the demand knowing that the demand is over
(resp. under) the inventory, i.e. 
(resp. 
),
and denote by 
the probability that the demand exceeds
the inventory (). When the value of is clear from
the context, the dependence from will not be emphasized. In
other words:
A straightforward computation leads to:
is the Holy
Grail of the problem. Without uncertainties you can choose
so that will be the expectation of your gain (the solution
being 
. But in presence of uncertainties, there is no choice
of that allows your expected gain to reach this value.
Moreover, this quantity 
has a
clear meaning in terms of risks evaluation. You have a risk 
that the demand 
overflows your inventory . And in this
case, your score is burdened by the fact that you miss the opportunity
to sell 
units, leading to an average miss
to gain of 
. On the other hand, you have
a risk 
that your inventory overflows the demand. And in that
case, your 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
measures the cost of uncertainty. A better choice for can decrease
this cost, but it will never vanish. Reporting eq:Q-optimal
into eq:cost-of-uncertainty leads to the following expression
for the cost of uncertainties (where 
are relative
to ):
The usefulness of this expression will appear clearly in sec:How-to-summarize,
suggesting both how to define a new parameter 
and an upper
bound for the cost of uncertainties.
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Before introducing another family of distributions (the triangular
ones), let us recall briefly what happens when using the most current
distributions. The corresponding results are summarized in fig:gain-depending-onctp-and-shape
where is fixed and the cost to price ratio is either 
(lefts) or 
(rights). Each subfigure draws the graphs of 
relative to 
(the upper broken line) and to 
(the three curves, the lowest being relative to the greatest 
).
The widely used normal law (see fig:normal-g
and fig:normal-d) is a sound model when assuming that
consumers are acting as additive processes whose decisions are taken
independently... but this is quite never the case. Many decision criteria
are the same for everybody, the welfare of the general economy among
them. On the other hand, when using a Gaussian distribution only for
convenience (as it is often the case in the literature), it must be
clear that this choice implies that 
remains small in order
to ensure that 
.
A better candidate to model a positive quantity is the lognormal
law (see fig:log-g and fig:log-d) since
it doesn't introduce artificial negative values. Nevertheless, it
should be kept in mind that using this model is roughly equivalent
to assume that the solvable demand is the product of many independent
random positive factors. Clearly, this is not ever the case. Moreover,
when using a lognormal distribution only for convenience, one introduces
an extra dependence between variance and skewness since, for this
distribution, 
where 
is
defined as 
.
When using one of the former distributions, the parameters
and are the only degrees of freedom available. In order
to test how robust are the conclusions drawn from partial knowledge,
it is of interest to use models that depend at least on three parameters.
Such is the "two Dirac's" distribution
that has been introduced in [Scarf 1958] to obtain the general max-min
formula. In this model, the parameters 
are defined by:
and 
,
we obtain 
and 
.
Since 
should remain positive, the range 
of the allowed values for shortens when increases.
One has: 
and
the other parameters as before.
In each of the six subfigures of fig:gain-depending-onctp-and-shape,
we have drawn the locus of the extremal points. All of them start
from 
. In the normal case, the loci are
two small rectilinear segments. In the lognormal case, can
grow to infinity, inducing the existence of a ``fat tail'' for
the distribution. Thus, quite all the mass should concentrate towards
in order to equilibrate the ``fat tail'' since the mean
has to remain constant. This is the reason why 
when 
, as it can be seen on both graphs. But
while, in fig:log-g, the locus shifts ever to the left,
we can see, in fig:log-d, that the locus starts shifting
to the right (due to the value of 
) and afterwards turns
to the left (and tends to the origin). In the "two Dirac's"
case, the loci are a part of the upper broken line.
We will now introduce another law, namely the triangular distribution,
whose name has been coined from the shape of their pdf. The rationale
to introduce this new law is to provide an easy to use 3-parameters
distribution that seems more realistic than the "two Dirac's".
Denoting respectively, as in fig:trian-pdf,
the min, 
the mode and 
the 
of the distribution,
the function is given by:
as great 
.
In comparison, it has been seen that using a normal model is unrealistic
unless 
.
The triangular model is a simple way to deal with the fact that often
the demand is not symmetrical around it's mean. This lack of symmetry
is usually measured by the 
, where 
is the third centered moment. This moment has a nice expression over
:
(the mean),
(the width) and 
(the barycentric position of in 
,
verifying 
) to characterize the distribution. We
obtain: 
. Conversely, the
skewness gives the shape (i.e. ), then gives the
size (i.e. 
) and finally fixes the position of the triangle
along the axis. By this method, one can deal with skewness up to 
.
fig: trian has been drawn using 
. In fig:trian-g,
the skewness of the distribution and the cost to price ratio are acting
in conjunction, and the locus of the extremal points shifts clearly
to the left. In fig:trian-d, these two factors are acting
in opposition, and the shift to the right of the corresponding locus
is not so strong.
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